FORMULAS of NONCOMMUTATIVE ANALYSIS Them, and S O On

Recent Titles in This Series

119 M . V. Karasev and V. P. Maslov, Nonlinea r Poisso n brackets . Geometry and quantization, 199 3 118 Kenkich i Iwasawa , Algebrai c functions, 199 3 117 Bori s Zilber, Uncountabl y categorica l theories, 199 3 116 G . M. Ferdman, Arithmeti c of probability distributions, and characterization problem s on abelian groups, 199 3 115 Nikola i V. Ivanov, Subgroup s of Teichmuller modular groups, 199 2 114 Seiz o ltd, Diffusio n equations , 199 2 113 Michai l Zhitomirskii, Typica l singularities of differential 1-form s and Pfama n equations, 199 2 112 S . A. Lomov, Introductio n to the general theory o f singular perturbations, 199 2 111 Simo n Gindikin, Tub e domains and the Cauchy problem, 199 2 110 B . V. Shabat, Introductio n t o complex analysis Part II . Functions of severa l variables, 1992 109 Isa o Miyadera, Nonlinea r semigroups , 199 2 108 Take o Yokonuma, Tenso r spaces and exterior algebra, 199 2 107 B . M. Makarov, M. G . Goluzina, A. A. Lodkin, and A. N. Podkorytov , Selecte d problem s in real analysis, 199 2 106 G.-C . Wen, Conforma l mapping s and boundary value problems, 199 2 105 D . R. Yafaev, Mathematica l scatterin g theory: Genera l theory, 199 2 104 R . L. Dobrushin, R. Kotecky, and S. Shlosman, Wulf f construction: A global shape fro m local interaction, 199 2 103 A . K. Tsikh, Multidimensiona l residue s and their applications, 199 2 102 A . M. Il'in, Matchin g of asymptotic expansions o f solutions o f boundary valu e problems, 199 2 101 Zhan g Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Don g Zhen-xi, Qualitativ e theory o f differential equations , 199 2 100 V . L. Popov, Groups , generators, syzygies, and orbits i n invariant theory , 199 2 99 Nori o Shimakura, Partia l differential operator s o f elliptic type, 199 2 98 V . A. Vassiliev, Complement s o f discriminants o f smooth maps : Topolog y and applications, 199 2 97 Itir o Tamura, Topolog y o f foliations: A n introduction, 199 2 96 A . I. Markushevich, Introductio n t o the classical theory o f Abelian functions , 199 2 95 Guangchan g Dong, Nonlinea r partial differentia l equation s o f second order , 199 1 94 Yu . S . Il'yashenko, Finitenes s theorems fo r limi t cycles , 1 991 93 A . T. Fomenko and A. A. Tuzhilin, Element s of th e geometry and topolog y o f minima l surfaces i n three-dimensional space , 199 1 92 E . M. Nikishin an d V. N. Sorokin, Rationa l approximation s an d orthogonality , 199 1 91 Mamor u Mimura and Hirosi Toda , Topolog y o f Li e groups, I and II , 199 1 90 S . L. Sobolev, Som e applications o f functional analysi s i n mathematical physics , third edition, 199 1 89 Valeri i V. Kozlov and Dmitrii V. Treshchev, Billiards : A genetic introduction t o the dynamics o f systems with impacts , 199 1 88 A . G. Khovanskii, Fewnomials , 199 1 87 Aleksand r Robertovic h Kemer , Ideal s of identitie s o f associativ e algebras, 199 1 86 V . M. Kadets and M. I. Kadets, Rearrangement s o f serie s in Banac h spaces , 199 1 85 Miki o Ise and Masaru Takeuchi, Li e groups I , II, 199 1 {Continued in the back of this publication) This page intentionally left blank 10.1090/mmono/119

Translations o f MATHEMATICAL MONOGRAPHS

Volume 11 9

Nonlinear Poisson Bracket s Geometry and Quantizatio n

M. V. Karasev V. P. Maslov

o America n Mathematical Societ y 3 Providence , Rhode Islan d M. B. KAPACEB, B. II. MACJIO B HEJIHHEHHBIE CKOBK H nYACCOHA . TEOMETPHil H KBAHTOBAHM E

Translated fro m th e Russia n b y A . Sossinsk y an d M . Shishkov a Translation edite d b y Simeo n Ivano v

2010 Mathematics Subject Classification. Primar y 58-XX , 81S10 ; Secondary 81Q20 , 16S32 .

ABSTRACT. The authors consider mechanisms o f the arising o f nonlinear degenerate Poisson brack - ets in Hamiltonian mechanics, deformations o f brackets, and their cohomology. A geometric objec t that i s the analo g o f a Li e grou p fo r nonlinea r bracket s i s studie d i n detail . A constructio n o f asymptotic quantizatio n o n genera l symplecti c an d Poisso n manifolds , an d i n particular , a rul e for quantization o f two-dimensional surfaces , propose d b y the authors, i s presented with complet e proofs. In addition, the book contains an elementary introduction to the theory o f semiclassical approx - imation, considerabl e referenc e materia l o n the calculu s o f functions o f noncommuting operators , and a summary o f results o n algebra s with non-Li e commutin g relations . This work is intended fo r mathematicians, includin g graduate students, specializin g i n differen - tial geometry , algebra , mathematica l physics , an d asymptoti c methods .

Library o f Congres s Cataloging-in-Publicatio n Dat a Karasev, M . V. (Mikhai l Vladimirovich ) [Nelineinye skobk i Puassona . English ] Nonlinear Poisso n brackets . Geometr y an d quantization/M . V . Karasev , V . P . Maslov . p. cm . — (Translation s o f mathematical monographs ; v . 119. ) Includes bibliographica l references . ISBN 0-8218-4596- 9 1. Hamiltonia n systems . 2 . Poisso n brackets . 3 . Poisso n manifolds . I . Maslov , V . P . II. Title. III . Series .

QA614.83.K3713 199 3 92-4206 1 514/.74—dc20 CI P

AMS softcover ISB N 978-0-8218-8796- 7

© 199 3 by the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retains al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America .

Information o n copyin g an d reprintin g ca n b e foun d i n the bac k o f this volume . @ Th e pape r use d i n this boo k i s acid-free an d fall s within th e guideline s established to ensur e permanenc e an d durability . This publication wa s typeset usin g Aj^fS-T^, the America n Mathematica l Society' s Tg X macr o system . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 7 1 6 1 5 1 4 1 3 1 2 Contents

Preface i x Introduction 1 CHAPTER I . Poisson Manifold s 7 §1. Poisson brackets related to Li e groups 7 1.1. Symplectic leaves and the Darboux theorem (8 ) 1.2. Linear brackets. Phase space over a Lie group (14 ) 1.3. Brackets generated by 1-forms . Cocycle s of Lie bialgebras (18 ) 1.4. Example s of compatible brackets. The Yang-Baxter equation i n Lie algebras (23) §2. Reduction an d deformatio n o f brackets 2 7 2.1. Lagrangian and coisotropic submanifolds. Hamiltonia n flows (27) 2.2. Bifibrations an d brackets on their bases (30) 2.3. Lie-Cartan reduction. Action-angle variables (35 ) 2.4. Examples of reduced brackets (38 ) 2.5. Brackets generated by 2-forms. The Dirac bracket (44 ) §3. Perturbations an d cohomolog y o f Poisso n brackets 5 3 3.1. The infinitesimal deformatio n problem . Examples (53) 3.2. Structure o f the Poisson manifold nea r nondegenerate leave s (56) 3.3. Free brackets. Nonisotropic deformations (61 ) 3.4. Anomalies in the Jacobi identity (67 ) 3.5. Tower of obstructions. General outline for the calculation o f tensor cohomology, cocyles, and coboundaries (71 ) CHAPTER II . Analo g o f the Group Operatio n fo r Nonlinea r Poisso n Brackets 7 5 §1. Phase spac e over a Poisson manifol d 7 5 1.1. Symplectic groupoids (75 ) 1.2. Analogs of direct Li e theorems (78 ) 1.3. System o f Lie equations (81 ) 1.4. Gluin g of the phase space . An analog of the third invers e Lie theorem (83 ) 1.5. Multiplication i n phase space. Analogs of the 1s t and 2n d invers e Lie theorems (87 ) §2. Examples o f symplecti c goupoids 9 1 2.1. Actions of groupoids and bifibrations (91 ) 2.2. Polar groupoid (94 ) vi CONTENT S

2.3. Nilpotent an d solvabl e brackets (96 ) 2.4. The Cartan structur e (99 ) 2.5. The groupoid fo r the Cartan structure . Affin e bracket s (103 ) §3. Finite-dimensional pseudogioup s an d connection s o n Poisso n manifolds 10 6 3.1. Actions of finite-dimensional pseudogroup s (108 ) 3.2. Reconstruction o f a pseudDgroup fro m canonica l vector fields and structure functions (112 ) 3.3. Canonical actions on symplectic manifolds (115 ) 3.4. Linear connections and basis of the pseudoalgebra (117 ) 3.5. Poisson brackets on groups and pseudogroups compatibl e wit h them (121 ) 3.6. Adjoint almos t brackets and almos t Poisso n action s (126 ) 3.7. Local vanishing of torsion an d non-Hamiltonian action s (130 ) 3.8. The symplectic groupoid generate d b y a pseudogroup (133 )

CHAPTER III . Poisso n Bracket s i n R n an d Semiclassica l Approximation 139 § 1. Lagrangian submanifold s a s fronts o f wav e packet s 139 1.1. Quantum densit y o f a packet (140 ) 1.2. Gaussia n an d oscillating packets (143 ) 1.3. Theorem o n the Lagrangian propert y o f fronts (146 ) 1.4. Functoria l propertie s o f density (149 ) 1.5. Localization o f wave packets (153 ) 1.6. Holograph y (154 ) §2. The correspondenc e principl e i n the languag e o f Lagrangia n geometry 157 2.1. Intertwining of classical an d quantum variable s (157 ) 2.2. One-dimensional obstructions . Path inde x (162 ) 2.3. Formulas for the intertwining operator (168 ) 2.4. Quantization o f solutions t o Hamiltonian systems . The eigenvalue problem (173 ) 2.5. The Cauchy problem. The oscillator and 90 ° rotation s (177 )

CHAPTER IV . Asymptoti c Quantizatio n 185 § 1. Review o f genera l approache s t o quantizatio n 185 1.1. General idea s and notation (185 ) 1.2. Quantizatio n o f symplectic manifold s (188 ) 1.3. Quantization o f degenerate Poisso n bracket s (190 ) §2. Sheaf o f wav e packets ove r a symplectic manifol d 192 2.1. Actio n o f Poisso n mapping s on wav e packets (192 ) 2.2. Nonlocal cocycl e over th e groupoi d o f Poisso n mapping s (197 ) 2.3. Two-dimensional obstruct : ons to gluing a sheaf. Global •-product o f symbol s (203 ) 2.4. Relationship wit h the theory o f geometric quantization (211 ) 2.5. Torus, sphere, and spher e wit h horns (214 ) CONTENTS vn

§3. Quantization of two-dimensional surface s 22 6 3.1. Index of two-dimensional surface s (226 ) 3.2. Rule of quantization (231 ) 3.3. Intertwining operators in quantized symplectic manifolds (233 ) 3.4. Example. Asymmetric SO(3)-top (234 ) 3.5. Quantization of Poisson mappings. Lifting o f asymptotics fro m reduced spaces (237 ) §4. Nonlinear commutatio n relation s in semiclassica l approximation 24 5 4.1. Quadrati c relations with a small parameter (246) 4.2. Quantu m corrections to Poisson brackets (247 ) 4.3. Generators of the *-produc t on oscillating symbols (249) 4.4. Representation of commutation relations by ft-pseudodifferential operators (255 ) 4.5. Convolution correspondin g to nonlinear Poisson brackets (258) Appendix I . Formulas of Noncommutative Analysi s 26 5 1.1. Ordered functions of operators and Weyl functions (265 ) 1.2. Formulas of differentiation an d disentangling (271 ) 1.3. Permutation of operators. Commutation with the exponent (277 ) 1.4. Functions of functions of operators (284) 1.5. Reduction to normal form (287 ) 1.6. Paradoxe s of forma l calculation s with functions of operator s (292) Appendix II. Calculus of Symbol s and Commutation Relation s 29 7 2.1. Generalized Jacobi conditions and Poincare-Birkhoff-Wit t property (297 ) 2.2. Chang e of order and *-produc t over the Heisenberg algebra (306) 2.3. Semilinear commutation relations (309 ) 2.4. Strongly nonlinear and solvable relations (313 ) 2.5. Quantum Yang-Baxter equation (321 ) 2.6. Reduction to triangular form (324 ) 2.7. Spectru m and cospectrum of quadratic-linear relations (328 ) 2.8. Transformation of scal e and structure constants (335) 2.9. Algebras equivalent to Lie algebras (346) References 353 This page intentionally left blank Preface

Our book deals with two old mathematical problems. Th e first is the problem o f constructin g a n analo g o f a Li e group fo r genera l nonlinear Poisso n brackets. Th e secon d i s the quantizatio n proble m fo r suc h brackets i n th e semiclassical approximation (whic h is the problem of exact quantization fo r the simplest classe s of brackets). Progressively, these problems are coming to the fore in the modern theory of differentia l equation s an d i n quantu m theory , sinc e the approac h base d on constructions o f algebra s and Lie groups seems, in a certain sense , to be exhausted. Our mai n goa l i s to describ e in detai l the ne w objects tha t appea r i n th e solution of these problems. Man y ideas of algebra, modern differential geom - etry, algebraic topology, operator theory are combined and synthesized here. And of course, it is not difficul t t o predict that this field will rapidly develo p further. At present, th e questio n o f whethe r analog s o f th e theory o f Li e groups, the theor y o f representation s an d harmoni c analysi s fo r nonlinea r Poisso n brackets ca n be constructed ha s an affirmativ e answer . Recently , structure s that ca n be taken a s the basis of this analogy have been obtained . Naturally, on e must distinguis h the pure quantum, the semiclassical , an d the classica l level s o f thi s problem . A t th e quantu m level , fe w fact s ar e known abou t genera l nonlinea r brackets , an d w e have mad e som e progres s only for the simplest quadratic-linear brackets. Bu t at the semiclassical level, we already hav e a self-consistan t theor y fo r a n arbitrar y nonlinea r bracket . In particular , arbitrar y symplecti c manifold s (i.e. , nondegenerat e brackets ) are quantized i n the semiclassical approximation . And finally, at the classical level (i.e., at the level of differential geometry) , the pictur e i s clea r i n al l details , althoug h man y importan t relation s hav e not ye t bee n investigated . Thi s geometry i s presented i n Chapter s I and II , where w e conside r i n detai l th e relationshi p between Poisso n bracket s an d symplectic groupoids. W e describe an analog of Lie algebras-pseudoalgebra s of Poisson manifolds , an d a n analog of Li e groups-finite-dimensional pseu - dogroups (specia l families o f nonassociative loops).

ix X PF:EFACE

It is shown how these structures mutually interact with different geometri c and algebrai c anomalies: i n the Jacobi identities, in the la w of associativity , in th e relation s betwee n Poisso n bracket s an d connections , i n relation s between bracket s an d grou p multiplication , i n th e fac t tha t action s o f group s of symmetrie s ar e Hamiltonian. Alon g the way , new properties o f symplec - tic leave s o f Poisso n manifold s an d thei r cohomolog y appear , obstruction s preventing th e deformation o f these brackets ar e calculated, generalization s of Dira c brackets an d thei r interaction s wit h th e Yang-Baxter equatio n ar e investigated. Th e classical scheme o f Lie-Cartan reductio n i s also given. Chapters II I an d I V dea l wit h semiclassica l approximation . The y con - cern asymptotic s wit h respec t t o "D e Brogli e wav e length " a s h — • 0 or , more precisely , th e propertie s o f noncommutativ e algebra s i n approxima - tions when the commutators between the generators are assumed to be small. In contras t t o th e theor y o f perturbation s (deformation) , semiclassica l ap - proximation preserve s th e basi c topologica l characteristic s o f th e spectru m of th e inita l algebr a i n th e limi t a s h -+ 0. An d sometime s thi s passag e to the limit yield s unexpected geometri c structures. W e demonstrate thi s i n Chapter I V for genera l phas e manifold s wher e there i s no globa l separatio n of variables into "coordinates" and 'momenta" . Fo r example, instead o f the Bohr-Zommerfeld conditions , the rules of quantization fo r two-dimensiona l nonclosed surface s appea r here; thus a notion o f integer-valued inde x gener - alizing the notion o f path inde x arises for suc h surfaces . Moreover, in Chapters III and IV, specialists will find new results concerning, it would seem , completely settle d subjects , for example , theorems abou t the geometry of fronts o f oscillations, about the half-integer path index (gen - eralizing th e so-calle d "Maslo v index" ) o n Lagrangia n submanifold s no t i n general position, about the nonlocal cocycl e of groups o f Poisson mappings , etc. The technique o f asymptotic quantization o f symplectic manifolds devel - oped i n §§1- 3 of Chapte r I V generalizes an d combine s the well-known con - structions of geometric and deformation quantization . I t does not rely on the results o f Chapter s I and II . At the sam e time, the asymptotic quantizatio n of degenerat e Poisso n bracket s (§ 4 Chapter IV ) i s essentiall y base d o n th e geometric investigations carried out in Chapters I and II . All the results on the pure quantum level (i.e., not related directly to geometry and semiclassics) are collected in Appendix II. There we give a number of technical methods for the exact calculation o f the regular representation an d cospectrum (a n analo g o f a group) fo r differen t algebra s with non-Li e com - mutation relations . W e are mainly concerne d wit h quadratic an d quadratic - linear relations. Actually, the topic of this book i s now rapidly developin g and i s far fro m being completed . Her e w e ar e a t th e ver y beginnin g o f a lon g road . Th e authors' onl y goa l wa s to presen t som e o f thei r result s obtaine d durin g th e PREFACE XI last fifteen years. Al l this material i s published i n monograp h for m fo r th e first tim e except subsections 1.1 , 1.2 , 2.1 of Chapter I; 1.1, 2.3-2.5 of Chapter III and 1.1-1.3 , 2.1 , 2.5 o f the Appendices. The book is accessible to graduate students. Al l statements ar e proved i n detail. Appendi x I i s presente d specificall y t o serv e a s a referenc e o n for - mulas of noncommutative analysis . Th e calculations give n in som e section s can be regarded a s exercises in the application o f these formulas. Moreover , Chapter II I is , basically, a n elementar y introductio n t o the theory o f semi - classical approximation . There , fo r example , the mathemaica l background of the process of plane holography is presented a s an illustration . We express ou r gratitud e t o D . V . Anosov , V . I . Arnold , V . S . Buslaev , I. V. Cherednik, B. A. Dubrovin, L. D. Faddeev, A. T. Fomenko, P. I. Golod, D. I. Gurevich, A. A. Kirillov, V. V. Kozlov, G. L. Litvinov, M. A. Semenov- Tyan-Shanskii, V . V. Trofimov, V . G. Turaev , A . M. Vershik, A . M. Vino- gradov, and V. S. Vladimirov with whom we have discussed the topics of this book at different times . We also expres s ou r gratitud e t o Professor s Miche l Audin , Janus h Czyz , Pere Dazord, Andr e Lichnerovich , an d Ala n Weinstei n fo r usefu l contacts , comments, and advice . The discussion s wit h V . V . Belov , A . M . Chebotarev , V . G . Danilov , S. Yu. Dobrokhotov, an d Yu. M. Vorob'ev hav e also been ver y useful. Th e authors express their sincere gratitude to them and also to all their colleagues in the Department o f Applied Mathematics of the Moscow Institute of Elec- tronic Engineering who helped a great deal in preparing this book. This page intentionally left blank This page intentionally left blank APPENDIX I

Formulas of Noncommutative Analysi s

1.1. Ordere d functions o f operators and Weyl functions. GENERAL DEFINITIONS. Conside r a function f of n scala r variables (symbol) as well as a set of n element s A, B,..., C o f some noncommutativ e algebra, for example operators in a linear space. W e denote by / th e forma l Fourier transform o f the symbol /

fix) = (2^//« )*"'*# > xt = x/+.-- + x n?.

12 n We define a function of the ordered elements A, B , .. . , C b y putting

/(i,l,...,C) = j f{x)e ix»c-.-eix>Beix>Adx. (1.1 )

Thus the number s ove r noncommuting element s ge t their orde r fro m righ t to lef t (th e order o f their actio n a s linear operators) . Thes e notations wer e proposed i n [96] . Apparently, Feynma n [137 ] was the first to systematicall y use functions o f ordere d operators . I n this appendi x w e have collecte d th e main formulas o f the calculus developed in [96] and in subsequent works , as well as those of the calculus of Weyl functions [57 , 58, 149]. The Weyl functions i n th e sam e element s A , B , .. . , C ar e define d a s follows f(A, B, .. . , C) = ff(x)e i{XlA+X2B+'"+XnC)dx. (1.2 ) Here the se t o f al l elements i s symmetrized, i.e. , the y pla y a n equa l rol e i n the sens e o f orde r o r in the sens e of order o f their action a s operators. On e can regard them as "acting" simultaneously and this is shown in (1.2 ) by the same number co over all the elements. In the general case one must consider severa l sets of elements at the same time: A = (A 1, .. . , Ak), B = (B l, .. . , Bm), ... , C = (C1, ... , C l) an d define mixed function s o f them; for example ,

12 n 1 , 1.2 , 2 „, n , n 7 f{A;B;... ; C) = f(A l, .. . , Ak ; Bl, .. . , Bm ; ... ;C 1,...,C/). Here all the elements (operators) A J ac t simultaneously first, all the B J afte r 265 266 FORMULAS OF NONCOMMUTATIVE ANALYSIS them, and s o on. Th e formal definitio n i s the followin g f(A\B\... ; C) =/.../ /(x;y : .. . ; z)elz'C • • -eiy'Belx'A dxdy • • -dz, (1.3) where x-A = x^1 H 1-** ^ • The initial problem of noncommutative analysis is that of the exact meaning of integrals (1.1)—(1.3 ) and the description o f possible classes of symbol s and classe s o f algebra s to whic h the element s A J\ B J\ .. . , CJ ca n belong . Here a lot of versions appear. W e shall restrict ourselves to only two of them. The first one involves selfadjoint operator s in a Hilbert space, the second one involves operators in Banach scale s and with general poly-Banach algebras . J J J SELFADJOINT OPERATORS . W e assum e tha t al l A \ B \ .. . , C ar e lin - ear operator s define d o n a commo n dens e invarian t linea r subspac e D i n a Hilber t spac e %?. Al l th e operator s ar e essentiall y selfadjoint ; more - k over, al l linea r combination s x-A,y-B, .. . , z-C (wher e x e R , y e m I R , ... , z e R ) are als o essentiall y selfadjoint . W e conside r th e produc t

Q(x 9 y, ... , z) = e --e e an d it s iterations

{N) kN mN lN where J C = (JC (1), ... , x ) € R , y e R , .. . , z e R . W e shal l as - sume that the following conditio n holds . CONDITION (a). Fo r an y N ^ 1 an d an y vecto r u e D th e functio n QN(x, y , .. . , z)u i s infinitel y differentiabl e i n jc , y9 .. . , z i n th e stron g topology an d th e norms o f al l its derivatives increas e a t mos t polynomiall y as |jc | H h \z\ —> oo. We also introduce the space S°°(R d), d = (k + m-i h l)N, consistin g of all complex smooth function s o n R satisfyin g th e estimat e

r 3rV5|m|r^sup|(l + |^|) |^y/(0|}

We endow this space with th e natural convergenc e wit h respec t to the poly - norm | | • ||r s . (Th e spac e obtaine d wil l no t b e a topologica l one ; se e (1.8 ) below).

LEMMA 1.1 . [68] . Under the condition (a ) there exists a linear manifold D^ such that DcD^c/ and for any f e S°°(R d) formula (1.3 )

f(A;B;... ; C) = (/, Q) defines a linear operator D^ - > D co which admits a closure in %f. Ifue 1 n D^, then the mapping f H + f(A, .. . , C) from S°° into %? is continuous. The manifol d D^ i n thi s lemm a i s compose d o f al l th e vector s o f th e d form (g , Q Nu), wher e u e D, g e S°°(R ), d = (k + m + • • • + /)J Y , and TV is any number . FORMULAS O F NONCOMMUTATIVE ANALYSI S 267

Naturally, i f al l th e operator s fro m th e set s A , B , .. . , C ar e bounde d (and selfadjoint) , the n th e conditio n ( a) hold s automaticall y an d Lemm a 1.1 i s trivial.

LEMMA 1.2 . Suppose the sum of squares (A1)2 + • • • + (A k)2 + (B 1)2 + • • • + {B mf + • • • + (C 1)2 + • • • + (C7)2 is essentially selfadjoint, the operators A1, B\ .. . , C s generate a Lie algebra on D , and all the matrices of the adjoint representation of this algebra possess only real eigenvalues. Then condition (a) holds. In particular, it holds if the Lie algebra mentioned above is nilpotent. The proof follow s fro m th e Nelson theorem [108] . Note that i f the symbol / split s into a product

/(x,y,...,z)-/Q(x)//y).../7(z), then the corresponding operator (1.3 ) also splits into a product of Weyl functions with respect to the sets A , B , .. . , C:

f{A\B\... \C) = f7{C).--ff{Byfa{A). (1.4 ) Moreover, i f the symbol i s a polynomial, fo r exampl e

a> to then th e Wey l function f{A) - f{A l, .. . , A ) coincide s o n D^ wit h th e 1 k completely symmetrize d polynomia l i n the generators A , ... , A (1) ( Il, /(^)=ao./+x:^^|7^E^ •••^ ' • d-5 ) |s|>0 ' '* o s ta en over Here the sum Yl a * ^ ^e mapping s a : {1, ... , \s\) —> { 1, .. . , k} which take the value j a t exactl y s. points . Fo r example,

W CO . CO CO . A A 2 1 , A A 2 , .2 A \s A , .2, 2 1 . A A 2 A 2 , A 2 A A 2 , A 2 A 2 A . A A = -(A A +A A ) , A (A ) = -(A A A A-A A A +A A A ). Details concerning the definition o f Weyl functions wit h repect to selfadjoin t operators and example s can be found i n [58 , 149 , 183]. OPERATORS I N BANAC H SCALES . NO W w e shal l presen t anothe r approac h to the operator calculus based on the theory o f Banach scales . Suppose | | • || , / / = 0 , ±1, ±2, ... , i s a famil y o f norm s define d o n a linear spac e 2 an d | | • || j ^ | | • || fo r an y \i ; le t the followin g conditio n hold: i f {u n} c 3 i s fundamental * i n the norm | | • || an d tends to zero in the nor m | | -1| j , then i t als o tends to zer o i n the nor m | | • || . Denot e b y

* or Cauchy (Editor' s note) . 268 FORMULAS OF NONCOMMUTATIVE ANALYSI S

Bfl th e completion o f 2 i n the norm | | • || . Then there is a chain o f dens e imbeddings which is called a Banach scale. The standard example is given by the Sobolev scale generate d b y th e operato r o f differentiation . Th e theor y o f scale s i s developed i n detail, for example , in [83]. The operator A : 2J — • 21 i s called a generator in a scale if it is continuous in the scale, i.e.,

V ii3v\\A\\B_^B

eM{A) = e t{A)et(A)9 N V/i3c3N\\e {A)\\ ^

V u 3 v li m -(et(A)-I)-A 0. B. -^B.. Generators A = A(e) dependin g o n a paramete r e ar e sai d t o b e generators uniform i n e i f th e constant s c an d N i n th e estimat e (1.6 ) ar e independent o f e . A se t o f operator s A 1, .. . , A define d o n 2J i s calle d a Weyl set in a scale if the operator OJ*A — CO XA + ^^^.A i s a generator over the scale uniform wit h respect to co € R * , \co\ = 1 . Fo r x € R w e denote

-M:*/* =*W(R*4 The properties of generators in Banach scale s are described i n [96] . Her e we present onl y one example. EXAMPLE 1.1 . Suppos e the real functions a^ , b satisf y th e estimate s

! sup(l + |*|) |a;(x)|

A = iJ2aj(x)— + b(x)

l n is a generator over the scale {H k(R )}, I, Ic = 0, ±1, ±2, ... generate d by the set of norm s

2 //2 /2 \\u\\Hi =||(1 + M ) (1-A)* M||L2, U € 3 = C 0°°(R"). k FORMULAS O F NONCOMMUTATIVE ANALYSI S 26 9

We now consider set s of real functions a™ , b m o n R n satisfyin g th e estimates (1.7 ) for each m = 1,...,«. W e introduce the matrices Q m(x) = l n {{ida™ldxs)). The n if the set Q (x), ... , Q {x) i s the Weyl set in the algebra of matrices uniform i n x e Rn , then the set of operators

A =l Z^aj^^V + b M ' m=l,..., «

l n is a Weyl set in the scale {H k(R )} . For genera l Wey l set s A, B, ... , C an d symbo l / e S°°(R d), formul a 1 2 n (1.3) defines the operator f(A ; B; ... ; C) whic h is continuous in the scale. One can introduce a natural structure of convergence (but not a topology) in the algebra 3?{B\ o f all operators continuous in the scale, so that the map- 1 2 n . ping / i- > f(A; B; ... ; C) act s continuousl y fro m S ( R ) int o -^{ 5 } (see [58]). The relations (1.4 ) and (1.5) for polynomial symbol s / als o hold for operators in a scale. W e note that polynomial functions ca n be constructed, of course, not only fro m generator s o r Weyl set s o f operators, bu t even fro m arbitrary operator s fro m 2"{B\ . Thus , i t is convenient t o generalize for - mula (1.3) to a certain extent. Suppos e T {, .. . , Tn e £?{B } . We define an operator continuous in a scale 2 2 n 1 3 2n- l Tx.~THf(A;B;... ; C ) lz c lx A = f(x\y; ... ; z)Tne~ ' •- T {e~ ' dx---dz.

The Wey l calculu s (1.2 ) als o admit s suc h a generalization . Le t A = 1 k (A , .. . , A ) b e a Wey l se t of operators L {, .. . , Lk e ^{B^ an d / e S°°{Rk). B y definition [57] , we set

w , \ de f _ 2 2 k k 1 2k+ l i,-V(>« ^ ) = ELa(l)-V)^ A )> a where the sum is taken over all permutations o o f the set of indices 1 , ... , k, and the operation y k i s defined b y the formula

= I dr Q f °dT {... Jo Jo

./0 1 If the complete set of operators L {, .. . , Lk , A , .. . , A i s a Weyl set, then the definition give n above turns into an identity. 270 FORMULAS OF NONCOMMUTATIVE ANALYSIS

POLY-BANACH ALGEBRAS . Beside s the initial algebras of symbols S°° an d of operator s &{B } , we consider belo w the algebras £?S°° , 2C{2"{B^) of all continuous operators on the initial algebras. I n order to describe them in a unifor m wa y and construct th e operator calculus , i t i s convenien t t o introduce the general notion o f a poly-Banach algebra . Let 7 b e a partially ordere d nonempt y se t directed t o both sides , i.e., if /, j e I, the n 3k, I e I: / < k, j < k an d / < /, / < j. We add the point at infinity to the numerical half-axis: R + = [0, oo)U{oo} . Suppose X i s a linear space . I t wil l be called polynormed i f the mappin g (polynorm) X x I - * E+, denote d (x , /) — • \\x\lj , is defined an d possesses the followin g propertie s (a) II^H,. = kHMI / VceC , (b) ll x + yll^H^. + lMi.,

(e) VJC 3 / : 11J» C 11/ < oo . The set I(x) o f indices / fo r which ||x||. < o o will be called the finiteness set. Al l possible intersections o f finiteness sets form th e basis of the filter of sections in the sense of [16]. Thi s filter will be denoted b y A 0 . Suppos e A is a filter of sections majorizing it , i.e., we have A D AQ . W e introduce the convergence in X generate d b y the filter A. W e shall sa y that th e directed set {JC Q} tend s t o zer o i f ther e exist s a a e A suc h tha t th e polynorm s ||JC || f- (beginnin g wit h a certain a 0) ar e finite for eac h / e o an d tend to zero. Thi s convergenc e i s compatible wit h linea r operations . Th e space X endowed wit h thi s convergenc e wil l be called polynormalized over the filter A. The properties o f polynormed space s are described i n detail i n [35, 58] . An algebra M wil l be called a poly-Banach algebra if it is supplied with a poly-norm ove r a certain filter A, and this polynorm i s separable, complet e and the operation o f multiplication i s continuous i n it. Examples of poly-Banach algebra s are • th e completion o f the algebra &{B } o f al l operators continuou s i n the Banach scale; • th e algebr a J?(X) o f al l continuou s linea r operator s o n a complet e separable polynormed spac e X ; • th e algebra S^ R ) (wit h repec t t o the usual multiplicatio n o f func - tions). We note that the latter poly-Banach algebr a i s certainly no t a topological space, and the first two algebras are topological spaces only in the trivial cas e when th e scale {B } reduce s to one Banach spac e o r when X i s a Banac h space. W e must inevitably leave the category of topological spaces if we want to construct a logical calculus of unbounded operators . FORMULAS OF NONCOMMUTATIVE ANALYSI S 27 1

If M i s a poly-Banach algebra, then the generating elements in this algebra and th e Wey l set s o f element s ar e define d jus t a s i n th e particula r cas e o f algebra 5C{B } . Function s o f thes e element s ar e define d accordin g t o th e formulas (l.l)-( 1.3) . 1.2. Formula s o f differentiatio n an d disentangling . I n noncommutativ e analysis, th e rol e o f ordinar y derivative s i s playe d b y differenc e relations . They will be called differenc e derivatives . Suppose / i s a smoot h functio n i n on e variable. B y 8f w e denote th e difference derivative

and by 8 /w e denot e its JV-fol d iteration: 8 f = 8(8 f) o r

7=1 ,= 1 * C

dz2 • • • Jo Jo

Jo If / € S°°(R), the n 8 Nf e S°°(R N+l). We note that th e differenc e derivativ e doe s not var y whe n it s argument s are permuted, an d it coincides on the diagonal with the usual one

Moreover, w e have the Leibniz identity N 8(fg)(£,...,£ ) = 2sS /(£>•••> £ )&g{S >•••> £ ) > j=o the Taylor identity

r U) N N f(^rj) = Y^ jf ^) + rj 8 f(^rl9^...^); ^rjeR 7=0 J ' and the Newton identity

j=0 0 l N N N + (i -z)-"{?-z )s Az°9z\...,z ). All these identities for m th e basis for al l other formulas i n which the scala r variables will be replaced b y noncommuting operators . 272 FORMULAS OF NONCOMMUTATIVE ANALYSIS

Without specia l mention , b y calculus o f functions i n noncommuting operators we shall mean on e of the two constructions describe d in 1.1 . All the symbols are taken from the class S co .

THEOREM 1.1 . Let A and B be two operators. Then

f(A + B) - f(A) = Bdf(A + B,A) = B6f(A + B,A). (1.9 ) Moreover, if A(t) is a smooth family of operators, then

Ttf(A(t)) = -^df(A(t),A(t)). (1.10 ) The proof is very simple [96]

3 i 3 i 3 i f(A + B)- f(A) = f(A + B) - f{A) = (A + B- A)Sf(A + B, A)

= [A + B-A]df(A + B,A).

2 1 2 The factor i n square brackets i s equal to A + B — A = B; thi s yields (1.9) . We apply (1.9 ) to the differenc e

±[f(A{t + At))-f(A(t))] and, passing to the limit as At —• 0, obtain (1.10) . The histor y o f the differentiation formul a (1.10 ) i s a long one ; see, for example, [39, 172] . O f course, th e main effort s ar e applied t o obtai n th e maximal extention of the class of admissible symbols / an d operators A(t) ; see [96]. We presen t her e a n analo g o f the differentiation formul a i n the case of Weyl calculus, directly for functions i n several variables [33, 57] :

l k 1 k %-tg(A (t),..., A (t)) = ^fdjgiA ,...,A ). We alway s assum e summatio n ove r repeate d indices . Th e Weyl calculu s is convenient precisely because both differentiation an d commutation formula s (see (1.21) ) contai n th e ordinary derivative s o f symbols , no t the differenc e ones as in (1.10) or (1.14). EXAMPLE 1.2 . Derivative of the trace. Suppos e A(t) i s Hermitian matri x depending smoothly on the parameter / . The n (1.10 ) yield s

2 2 dA 1 3 dA l 1 ^tr[/M(0)] = tr ffif(A,A) tr %SAA,A) tr £/<,) Here we used the invariance o f the trace t r wit h respec t to cyclic permutations. FORMULAS OF NONCOMMUTATIVE ANALYSIS 273

EXAMPLE 1.3 . Derivative of the natural logarithm In . Suppos e U(t) i s a smooth family o f invertible operators. The n by (1.10),

d f TT , N dU\nU-\nU dt dt 3 I 17-£7 2 rfJ7 1 _i 3 1 i _ i 3 1, 2 3 1 , = ~U {UU -1 ) ln(UU l ) = VQ(UU ) , (1.11 ) where Q({ ) = ln{/({ - 1) , K = (dU/dt)U~ l. Naturally , ou r calculation i s formal sinc e the functio n I n doe s not belong to the clas s S° ° . Bu t w e can give a rigorou s meanin g t o it . Th e condition s unde r whic h formul a (1.11 ) holds are , fo r example , th e followin g [58] : U{t) i s a differentiat e famil y of element s o f a Banac h algebr a an d fo r al l t th e poin t 0 belong s t o th e unbounded connecte d componen t o f a subset o f the complex plan e c\(l(t)l(trlnz(t)),

1 1 where 1( 0 i s the spectrum Sp(£/(/)) , 22T = {^A" ^,, A 2 € X}. In th e cas e whe n U(t) i s a matrix , afte r takin g th e trace , w e obtain b y (1.11) ^ trl n * 7 = ^-In det£/ = tr K dt dt since £2 ( 1) = 1 . Thu s we come to the well-known relatio n

-r-det£/ = det(7t r dt £"- We now retur n t o formul a (1.9) . I t ca n b e continue d an d a n expansio n of f(A + B) i n power s o f th e incremen t B ca n b e obtaine d b y differen t methods. Fo r example, as follows [54]:

N-\ . 2 1 3 3 f(A + 5) = £ fik)(A)D + D dNf(A + B,A,...,A), T] k\ k N k=0 where \k —±— 3 2 Dk = {A + B - Ay = [A + £ - ^)^_ 1 3 1 2 *-l - D k_xB -{A- A)D k_{ = (R B - ad A)Dk_{ = (*, - a d J 5 .

Here by i? 5 an d ad ^ w e denote the operators of multiplication b y B fro m the right and commutation wit h A fro m th e left respectively . Thi s noncommutative Taylor series is effective, fo r example , in the case when B i s small and an y commutato r o f orde r r o f th e operator s A an d B ha s the valu e r+l k 0(£ ); i n this case D k = 0(B ). The following theore m hold s also [53 , 204]. 274 FORMULA S OF NONCOMMUTATIVE ANALYSIS

THEOREM 1.2 . Suppose the commutator of order r + 1 is equal to zero [•••[B, A],...,A] = 0 .

r+\ Then f(A + eB)

N 1 k+w = /(^) +- Ek * ^ E / \A)

x[---[B,A],...,A]-A],...,A]+0(eN). * v ' ' v ' fik fit Naturally, the remainder 0{e ) can be calculated explicitly. Another way to generalize (1.9 ) is to apply it to itself severa l times [96 ]

*lz} 2 2 k t 1 2k+ l f(A + B) = f(A)+J2B..-BSkf(A,..., A) k=\ 2 2 N x r 1 2N- 1 2N+ 1 + B-..BS f(A,... 9 A 9 A + B). (1.9a ) If the operator B i s a small perturbation wit h respect to A , the n this serie s gives a procedure fo r approximat e calculatio n o f the function f(A + B). EXAMPLE 1.4 . Expansion for the exponent. A classica l an d perhap s th e oldest version o f formula (1.9a ) i s the expansio n N MA+B) it itA A ){A+B) = e + E**(/) + i f e^ BRN(r) dx, k=\ in which

i{t x )A i(x )A **(0 = / * f'dz k Pdx k_y-- rdr xe - « Be *-^ B.-. Jo Jo Jo

i(Xj—T.)A n ix.A x e 2 ' Be l . EXAMPLE 1.5 . Summing the Campbell-Hausdorff series. I t is well know n that the product of exponential functions e lAelB ca n be formally represente d in the for m e l , where the exponent C i s a linear combinatio n o f th e operators A , B an d al l their possible commutator s

C = A + B + ±[A 9B] + ±([A 9[A,B]] + [[A 9B],B]) + .--. This formal Campbell-Hausdorf f serie s is usually derived by means of a very cumbersome procedure; see , for example , [122] . Th e applicatio n o f the operator calculu s simplifies this procedure to a large extent [73] . We note that

-, 1 f d itA iB. , D c = ln{e e ) + B lJ 0 Tt - FORMULAS OF NONCOMMUTATIVE ANALYSIS 275

For the family o f operators 17(f ) = eltAel b y formula (1.11) , we obtain

~ln U = AQ(UU~ l) = Q(e itiidAeUdB)(A).

Here the relation s

l l 1 a ad AQ(UU~ ) = Q(LuR- )(A)i L^" = e" V' * are use d an d ad^ , L u, i? ^ denot e th e operation s o f commutatio n an d multiplication

ad^(r) = [A, 71 , L^r ) = U.T, R V{T) = 7* . £7, r i s a n elemen t o f th e initia l algebr a o f operators . Thu s w e finally ge t eiAeiB = e iC, wher e [73 ]

C= f Q(e itad'eiadB)(A)dt + B, Q({ ) = ^-. (1.12 ) Jo C — 1 A formula closel y related to the one above was obtained i n [176] . If w e expand the exponents in (1.12 ) into a formal serie s and expand the function Q int o a power serie s i n a neighborhood o f th e point f = 1 , w e arrive at an analog of a well known Campbell-Hausdori f serie s

oo (-l)" v - (ad^ad/«...(ad B)^ad/>(i?)

'1^1*

where / ! = / }! • • • ln\, |/ | = / } H h ln . The Campbell-Hausdorif serie s itself in the form give n in [122] can be obtained by a certain modification o f (1.12)

C = C n(e itad*eitad»)(A + e iiad*(B)) dt. Jo Thus, by using an expansio n simila r to (1.13) , w e can obtain on e more for - mula for the operator C i n the form o f a series in powers of the commutators of A an d B . Th e number o f suc h formulas i s infinite, an d al l of them ca n be enumerated i n a certain sens e (se e [58]) . Ther e ar e als o generalization s of suc h formula s t o th e cas e o f th e produc t o f mor e tha n tw o exponentia l functions. Belo w w e shal l giv e the final resul t fo r th e cas e o f a continua l family o f exponents. EXAMPLE 1.6 . Multiplicative integrals and the disentangling formula. Sup - pose A(t) i s a family o f elements of a Banach algebra (for example, a family of bounde d operators) , dependin g continuousl y o n th e paramete r t £ R . Define a famil y o f operator s U(t) a s the solutio n o f th e followin g Cauch y problem

*£ = A(t)U, U\ l=0 = I. 276 FORMULA S OF NONCOMMUTATIVE ANALYSI S

This solution is called a multiplicative integral or a /-exponen t and is denoted by U(t) = Expf'A = f\e Air)d\

Here w e giv e som e o f th e simples t formula s relate d t o suc h multiplicativ e integrals

(I) ||Exp/'4| (t)Mt; Jo Xl (II) Exp / A = I+ f A(T)dt+ [ dx x [ dT2A(Tl)A(T2) + --' ; Jo Jo Jo Jo A{r)dr (III) i f [A(T) , A{T")} = 0 VZ',T" , the n Ex p f A = e^ ; Jo

(IV) (EXP ^ AYB-UXVJ A\ = (EXPJ 2Ld A\B).

The so-calle d disentangling formula i s a n importan t relatio n whic h i s use d very often (thi s is the terminology proposed by Feynman [137], who obtained certain deep generalizations o f this formula) :

Exp f {A + B)= ( Exp f AY(EXV f BY where -l B(t)= (Expf A \ -B{t). (Ex p / " A\

An important particula r cas e of this formula i s

et{a+b)=eta-Exp[lB, B(t) = e' tBd'(b). Jo We now consider the proble m o f representin g a multiplicative integra l i n the exponential for m

Exp / (iA) = e iC{t) o r C(t) = \ I n Exp / iA. Jo l Jo In particular, if the family A(t) i s piecewise constant, then the multiplicative integral Ex p /J A turn s int o th e produc t o f a finite number o f exponentia l functions; the n the operator C i s the logarithm o f this product . We obtain a formula fo r C(t) b y the same method a s formula (1.12 ) (se e [73]):

C(t)= f Q(Ex p / i2L& A)A{T)dr. Jo Jo A condition fo r this formula t o hold is , for example , the followin g

r(t) = ef«llad^lldx <2. FORMULAS OF NONCOMMUTATIVE ANALYSIS 27 7

In that cas e [73]

»c«»«75prT(ln23W)i'IM(t)l|rft- 1.3. Permutatio n o f operators. Commutatio n wit h the exponent. Togethe r with the formulas o f the perturbations theory , combinatoric s (changin g the order of action, permuting operators) plays an important role in noncommutative analysis. Suppose A , B , C are operators . B y definition, [A , B] = AB - BA i s the commutator o f A and B ; [A , B\C] — AB - CA i s the quasicommutator of A with the pair (B , C).

THEOREM 1.3 . There are the following commutation formulas [96]

[A,f(B)) = [A 2,B]Sf(B,B), 3 [g(A)9f(B)] = [^, 5]^(j(, Vfl£, I); (1.14 ) 2 1 12 3 /S2^2415

g{A,B) - g{A,B) = [A,B]—f-(A,A,B, B) (1.15 ) and the quasicommutation formula [58 , 100 ]

[A,f{B)\f(Q] = M,2£|C]

2 ;V + [..-[5, ^],...,^]<5 /(l.-. ,A,A),

n C [^, f(A)\f(C)] = £ ~/ \ )V ~[B,A\C],... 9 A\C] n=\

+ [---[B*A\C],... , A\C]dNf(A, C, ... , C), 278 FORMULA S OF NONCOMMUTATIVE ANALYSI S

EXAMPLE 1.7 . Similarity mapping. Suppos e U i s an invertible operator . Then Uf(A)U~l =f(UAU~ l). (1.17 ) Indeed, denot e C = UAU~ l. The n [U, A\C] = 0 and , by formula (1.16) , we get [U,f(A)\f(C)] = 0. EXAMPLE 1.8 . Commutation with the exponential function

iA iA [A,[A9B]] = 0* f(B)e = e f(B + i[B, A]). (1.18 ) Indeed, formul a (1.14 ) yield s

iA iA iA [B,e ] = ie [B,A] o r [e 9 {B + i[B, A])\B] = 0. Hence, by the quasicommutation formula (1.18) , we get the required relation

iA [e ,f(B + i[B9A])\f(B)] = 0. The following statemen t [57 ] is a direct generalizatio n o f (1.18).

THEOREM 1.4 . Suppose S e C 0°°(R), ImS = 0 . Then [A, [A,B]] = 0 implies the following relation

f(B)eiS{A) = eiS{A)f(B + 1{B] A]SS(A,A)). (1.19 ) For example , suppos e A = x, B = -ihd/dx ar e the operators o f multiplication b y the independent variabl e an d o f differentiatio n i n the scal e {Hk(R)} (se e Example 1.1) , h — • 0 i s a small parameter. The n (1.19 ) yields

t w = E #(i)'-/ <^J))+H',)X

N~l k 1 r / x \*- / i i / i \ /

A:=0 /= 0

+ (-ift)"*iV, where the remainder R N ha s the form *»=i'^(l)>(-4-<<>h It i s precisely this expansio n i n powers o f the parameter h whic h i s the basis of the theory of semiclassical approximation. A s we can see, the operator technique yields this expansion simply by using purely algebraic methods without applyin g the stationary phas e method [97 , 102] . FORMULAS OF NONCOMMUTATIVE ANALYSIS 279 Another approach: b y (1.18) , we have 9-im •/(-O^-/(-»£+*<.>)• and further w e can apply formula (1.9a ) (se e [96]) or Theorem 1. 2 (se e [53]) in order to obtain an expansion in powers of % . Naturally, the expansions obtained also hold in the multidimensional case. Consider the sets of operators q = (q {, .. . , qm) an d

d d d -ih ( • * • * \ dQ„ l m 2m in the scale {H k(R )} . Suppos e g e S°°(R ). W e construct the followin g operators continuous in this scale *(J'-,'*£)' *( J'-/R^)' 8 ^-ih^)- The last operator i s a Weyl operator. Thes e operators wil l be called pseudodifferential (mor e exactly, h-pseudodifferential) operators [97, 147 , 184 , 192, 261]. We have the following important identity:

2 *{«--ih£d=*(*¥'-**£)=g(i: -H ih-t 0- We give the first terms of the expansion o f the commutation formula wit h the exponential functions : • fo r the ordered case I "**•*('• ~/*£)"*SsB*(*'if)

8 C + 0(f t ) . +(-/») jfsH* '*"*PP" (Here the arguments q an d p — dS/dq i n the derivatives of the function g are omitted.) • fo r the Weyl case

s ,~i (*> - fc 9 \ is (*> OS d \ g q l e g q lh { >- %)- - \ ^q- d-q) d dq

CO CO CO 1 2 2 (m dS. A d + 2(-m d BBg[ + • ( ds\ , .. Adg d l dfdgf ds\\ + 0(ft) . 280 FORMULAS O F NONCOMMUTATIVE ANALYSI S

EXAMPLE 1.9 . An abstract differentiation formula. Le t M b e a certai n algebra. An y linear operator D : M— • M satisfyin g th e Leibniz identit y D(AB) = AD(B) + D(A)B VA.BeM. will be called a differentiation o f M.

THEOREM 1.5 . Let D be a continuous differentiation of a poly-Banach algebra M (for example, of the algebra M = £?{B } of continuous operators in a Banach scale). Let A be a generating element in M and f e S°°(R) . Then I)(f(A)) = D(A)df(A,A). (1.20 )

PROOF [54]. Denote by L A th e operator of multiplication by A fro m th e left. I t is a generator in the poly-Banach algebra Jt?(M) an d f(L A) = Lr, A). By the commutation formul a (1.1) , we have

Bf(LA) - f(L A)B = [T)?LA]df(LA , L A). Applying the left-hand sid e to the unit 1 o f the algebr a M an d taking int o account tha t D(l ) = 0 b y th e Leibni z identity , w e obtai n th e followin g element D(f(L A)l) = D(f(A)). Applyin g th e right-han d sid e t o 1 takin g into account that [D , L A] = LD,A), w e obtain the elemen t

8 h{A) fCLA , LA)(\) = L7 . 3 (1 ) = D(^) 5f(A, A).

The formulas (1.10 ) and (1.14 ) obtained above are actually other versions of formula (1.20) . There are also analogs of the formulas o f Theorem 1. 3 fo r the Weyl func - tions, an d the y ca n b e writte n i n sufficientl y compac t for m i n th e cas e o f functions o f severa l variables. Below we shall assume that A = (A 1, .. . , An) i s a Weyl set of operators . w 1 ** n We us e the notatio n f(A) = f(A , ... , A ) a s i n 1.1 . Th e partia l deriva - tives o f th e symbol s f{£ { ,...,£" ) ar e denoted a s follows djf = df/di j . Repeated indice s indicate summing . THEOREM 1.6 . We have • the commutation formula

J [B9f(A)] = [B9A )djf(Ay9 (1.21 ) • the formula of consistence with ordered calculus

Bf{A)= [ Bf(AT + A(l-T))dr; (1.22 ) ./o * The ter m derivatio n seem s to be more commo n (Editor' s note) . FORMULAS OF NONCOMMUTATIVE ANALYSIS 28 1

• the composition formula f(A).g(A) = (fg)(A)

1 j + f fidulA , A ]dif(nA + ( 1 - fi)A)d jg(fiA + ( 1 - fi)A); Jo (1.23 ) • the substitution formula

g(A)f(A) = (fg)(A)

+ f ndn ( vdv\A l, A^dJ^l-^A + fiuA + fiil-^AJ

x (djg(uA + ( 1 - v)A) - d jg{vA + ( 1 - u)A)). (1.24) The proof i s given i n [57] . Th e relation (1.8 ) whic h w e used before i s a generalization o f formula (1.22) . EXAMPLE 1.10 . Weyl calculus and Lie algebras. Suppose that the operators A , ... , A satisf y th e relations [AJ,Ak] = a k/As, (1.25 ) where kj ar e structur e constant s o f a Li e algebra , i.e. , rea l numbers suc h that X kj = -k[ k an d

By applying the commutation formula (1.21 ) to the Weyl function i n generators A , ... , A , we obtain

[Ak, f(A)] = D {k)f(A), D {k) = i^Af^ .

This yields, in particular, that i f the symbo l / i s annihilated b y all the operators Z ) , then the operator f(A) commute s with all the generators A . It is called a Casimir operator of the Lie algebra (1.25) . Th e Jacobi identit y (1.26) implies that the syste m o f equations D {1) f = • • • = D^ n) f = 0 ca n be solved locally . (I t ma y b e that there ar e no globa l solutions , differen t fro m constants, as in "wild" Lie algebras [78]). We now conside r th e compositio n formul a (1.23) . Fo r an y operato r B , we have 2 Bf(A) = Bf(A) + [B, A j]Vjf(A, A), (1.27 ) where VjAe,V) = f djATtl + {\-T)i)TdT. 282 FORMULAS OF NONCOMMUTATIVE ANALYSI S

By passing to the Weyl ordering of operators according to formula (1.23 ) i n the right-hand sid e of (1.27) , we get [57 ]

m=0

2 1 3 J } + [A >,..., [A « ,£]••• }F N{dh • • • dJNf)(A, A), where b m ar e the Bernoulli numbers, and F N i s a certain linear continuou s mapping S°°{R n)— • S°°(R2n). I n particular , thi s expansio n yield s a well- known formul a o f compositio n i n a n associativ e envelop e o f a Li e algebr a [78] for operator s satisfying th e relations (1.25 ) and fo r polynomial symbol s s

Akf(A) = (L kf){A), L k = lk(-id)di,h, k=l,...,n, (1.28 ) where

00 h fp K(x) - T\~ l

Mx) = ((A*)U=>,.... > A * s xAk > xeR "- 1 n LEMMA 1.3 . The operators L , ... , L satisfy the relations (1.25). From the terminology o f the theory o f Lie algebras, we see that the A(x) are matrices of the adjoint representation and the 31 (x) ar e matrices of right shifts i n the coordinates o f the firs t kind ; se e [17 , 78] . Th e matrices 31 [x) are define d i n a sufficientl y smal l neighborhoo d o f th e poin t x = 0 (unti l the spectru m o f th e matri x A(x ) lie s insid e a circl e o f radiu s 2% ). Thu s the formula (1.28 ) ca n be violated fo r thos e (nonpolynomial ) function s th e Fourier spectru m o f which goe s out o f this neighborhood o f the zero point . For example , i t ma y happen tha t th e compositio n A e lxA fo r x eR n an d |JC| sufficientl y larg e cannot be represented in the form o f a Weyl function o f the generators A 1,... , An . We can avoi d this situatio n i n certai n case s by refusing t o conside r Wey l i j n n functions an d passing to ordered function s f{A , ... , A ) . By the identitie s

n 1 n j J l 1 . d . ix mA /'-M ! ix„A ix,A J ix: ,A ~ ix.A -i-—[e n -e l ] = e " ••• £ J Ae J ~x -e l

n ix^nd.n ix iadj/ A j, r ix„A ix.A , = e n A -e J Aj (A)-[e n --e l ] and ix k J XkAik) J {k) j j e ^ (A ) = (e ) sA\ (A ) s=^ 9 FORMULAS OF NONCOMMUTATIVE ANALYSIS 283 we obtain the formul a n n •^T7 \S 9 r ixA ix.A\ ,S r ix„A ix,A\ -I32(X).-—[e n -e { ] = A -[e n -e l ] , dXj where the matrix inverse to the matrix 3l(x) i s defined a s follows

Denoting as above I s (x, £) = 3l(x)s.^j, multiplyin g the formula obtaine d by f(x) , an d integrating wit h respec t t o x e R n, w e arrive a t a formul a similar to (1.28):

l l H s A'.f{A ,...,> ) = (L'f){A ,-A )9 L E E f(-id)dt J) . (1.28a ) Here the operators l) , ... , Ln als o satisfy the commutation relations (1.25) and the matrices 31 (x) ar e right shifts i n coordinates o f the secon d kin d [115]. The y can be well defined in a considerably wider neighborhood of the point x = 0 compare d wit h the case (1.28 ) of coordinates of the first kind.

(1) LEMMA 1.4 . If the Lie algebra (1.25) is nilpotent (all the matrices A , ..., A(w) are nilpotent), then the matrix of right shift & is defined globally (on all of R" ) both in coordinates of the first and of the second kind. If the Lie algebra (1.25) is solvable (i.e., all the matrices A 1, ..., A" are lower triangular in the chosen basis), then the matrix 3i is defined globally (over all of Rn) in coordinates of the second kind, but & can possess singularities in coordinates of the first kind. It is not so simple to get rid of singularities for other classes of Lie algebras, for exampl e compac t Li e algebras. Th e space R n, ove r whic h the Fourier images of the symbols / ar e defined, mus t be curved by substituting it by a Lie group G with structure constants X kJ . The space R n itsel f i s identifie d with the Lie algebra Q of this group. G and g are mutually connected by the exponential mapping ex p : g— • G. An d if T A(a), a e G, is a representation A lx A of th e grou p wit h generator s iA {, .. . , iAn , then T (exp(x)) = e ' . I f R(a) i s a differential o f the right shift b y the element a e G at the point e , then U(exp(jc) ) = 31 [x). I f dp(a) i s a left Haa r measure on G, then Aix) n-e- \ dp(zxp(x)) = det V A(x ) J dx. By formula (1.28) , we calculate

k ix ^V(exp(x)) = [l (-id/dt, Z)e '%=A = /*(*, -id/dx)T(exp(x)) or AkTA(a) = -iSf {k)TA(a), a £ G. 284 FORMULA S OF NONCOMMUTATIVE ANALYSI S

Here the differential operator s & ) o n the group G ac t as follow s

{k) (3f (p)(txp(x)) = l\i,d)dx)(p{txp(x))=^(x)^(p(tx V(x)).

Thus, (3fq>)(a) = R(a)*d(p(a), i.e. , «S ? ar e right-invarian t vecto r fields o n G. The y are antisymmetric with respect to the lef t measur e dp . 1, n Instead of the functions f(A) o r f(A , ... , A ) w e consider the integrals TlA((p) = f G

nA{y/)-TlA( i s the convolution of two functions (distributions ) o n the group:

(V* ?)(<*) = i V{P)

THEOREM 1.7 . [53 ] The following relations hold: • the formula for a compound function

fiQ^AgiA + BV + lA^B^giAtAiB^giA^B,^ xd2f(C,C,g(A + B)), (1.29 )

1 2 w/zere C = g(A 9 B), and S tg is the difference derivative with respect to the jth variable', • the formula for a function of a sum

f(A + B) = f(A + B) + [A *B]S2f{ATB, TVS\ A + B)

= /(A + B) + M * #]<52f(A + B, J ( + 5, J l +1); (1.30 ) • the formula for the exponent of a sum ei{A+B) = eiB(I-Q)eiA, (1.31 ) FORMULAS OF NONCOMMUTATIVE ANALYSIS 28 5 where Q= Cdx [ Tdnei{x~'l){A+B)e-,TB[A,B]ei',Be-'{x-'t)A. Jo Jo In particular, if the operators A and B commute , then

f{A + B) = f(A + B). If the operators A and B do not commute, then the formulas of Theorem 1.7 provide the possibility to expand the compound function s o f these operators in powers of their commutators. Fo r example, the following expansio n holds

{k N f(C) = f(g(A,B)) + JT,f \g(A , B))Zk + 0 [A,B]. (1.32 )

N 2 2 N 1 2N+ 1 Here O [A , B] denote s the summands o f the form S l--SN(p(Dl- • DN), where D. ar e certain operators , cp e S°°(RN), an d S- ar e commutators o f the generators A and B (i.e. , the elements of a Lie algebra generated by A and B ), and the order of each commutator S. i s not less than one, and the sum of orders of all S {, ... , SN i s not less than N. The operators 0 N[A, B] wil l be called commutator elements of order N with respect to A , B . One ca n sho w [54 ] that th e coefficient s Z k i n (1.32) ar e define d b y th e formula k Zk = (C-g(A,B)) , C = g(A,B) and satisf y th e recurrent chai n o f relation s

Z0=l, Z 1==0, .. k o o oo oo . Z k+\ = fc +i 2 ^ Z^ 2^ Z-, s \n\{m-n)\ n=lm=n /= l 5=0 v '

2 {m n) l +s L33 xKlm[A,[A,.. ,[A,Zk_n].--]](g ' d l g)(A, B) ( >

11O O 2 13 [ z ] ] jB) + FM^s=l d^^' * -" ^' ' where A ; > 1, d, = d/d^ ,

^m,n)n l>:\ t l _ m n K 2 2 r> \e,?) = (^) ~ (l[2(t >t >z '))' e'=e and the operators K f m are defined a s follow s

Ki^m^-^'^-^- (L34 ) 286 FORMULA S OF NONCOMMUTATIVE ANALYSIS

Subsection 1. 5 wil l be devoted to calculatin g the operator K t m . I t turns m3x{l m) out tha t K lm = O ' [A,B]. Thus , i n th e infinit e serie s (1.33 ) th e summing ove r m , /, s ca n b e bounde d fro m above , fo r example , b y th e N l{M)/2] number N i f we cut the terms O [A , B]. The n Z k = O [A, B] . W e give here explicit formulas fo r N = 3 :

2 2 2 Z2 = ±[A ,B](dlgd2g)(A,B) + ±[[A ,B],B](dlg(d2) g)(A, B)

1 + \[A Bf({dxfg-{d1f)g)CA,B) + 0\A,B],

2 2 l Z3 = \[[A ,B], B]{d lg(d2g) )(A, B) + -[A, [A, B]](d 2g(d1 gf)(A,B)

+ ^Bfiid.gfid^g + id^gid^f + d.gd^d^g^A, B)

+ 0 3[A,B],

2 2 Z, = \[A * B] ((5, gf{d2g) ) (A, B) + 0\A, B],

3 Zk = 0 [A,B] fo r k > 5. We no w writ e formula s simila r t o (1.29) , (1.32 ) i n th e cas e o f th e Wey l calculus [57]. 1 THEOREM 1.8 . Let A = (A ,... , A") be a Weyl set of operators in a CO . CO Banach scale, g € S°°(Rn), and let the operator C = g{A) = g(A , ... , A n) be a generator {see 1.1). Then for any f e S°°(R) f(C) = f(g(A))

i J + f vdv[A ,A ]Af(g(vA + (\-v)A),C)d ig(vA + (l-v)A) Jo

xdjg(vA + (l-v)A), _„„.,„,„. „..,„„ .„.„:;."i powers of commutators has the form f(C) = f(g(A))

1 j k 2 2 2 + [A , A ][A', A ] (±f"(g)d g • d kg + ^f"(g)d kgdjgd gj (A)

CO 1 j k 2 l + ^{[A , A ], A )(f"(g)digd kg)(A) + 0\A , .. . , A"]. We note that i f the function g i s linear, then the second summand i n the right-hand sid e o f (1.35 ) vanishes . Thus , w e obtain a n importan t propert y of the Weyl calculus, namely, its affine covarianc e [149] . FORMULAS OF NONCOMMUTATIVE ANALYSIS 28 7

COROLLARY 1.1 . Suppose M is the afflne mapping M : R — • R given by , {MZ) = m'JZ' + mQ. J Consider the set of operators {MA)' — m'jA + m0. The relation

f((MA)1, .. . , (MA)k) = (M*f){A l, A k) holds for any symbol feS°°(R k).

EXAMPLE 1.11 . Weyl pseudodifferential operators. Let us apply the functions obtained to the fc-pseudodifferential operators from Exampl e 1.8 . De- note g = g(q, -ih-j^) . B y Theorem 1. 6

= fe2 l fi4 £i*2 8\8i - y( S 2) + <*(ft > ft) ~ ~Y^Sx, ft) + °( )' ( ! 36) [ft, ft] = -/fi(ft> Si) - ih3 P(g\> ft) + °(^4)> where {f t , ft} = (Jdg {, dg 2) i s the Poisson bracket of the functions ft , ft G S^R ") , / = , and the operations a an d / ? are defined as follows

, v 1 T kk' Jl'l ~2 «(ft,ft) = "g^ / d kl8\-dk'l'g2> fitei> ft) = ^ k'^JJf»lj8r4n'Sr The structur e o f these formula s i s discussed i n more detai l fro m a general point of view in 2.2, Appendix II. We present one more consequence of Theorem 1.6 . Suppose sf l, .. . , sf n is a set of symbols on 5°°(R 2m) suc h that the set of operators sf x , ... , sf n l m n is a Weyl set in the scale H k(R ). The n for any / e S°°(R ) w e have

1 0 4 137 /(/ 5...,/VV^i + ^ )' ( ) where

l n F0 = f(s/ ,...,j/ ),

and the remainder 0(h 4) i n (1.37) has the form h 4R an d R i s continuous l to the right in the scale {H k} uniforml y i n h . 1.5. Reductio n t o norma l form . Suppos e A an d B ar e two operators; 2 1 consider the functions i n these operators g(A, B) an d f(A + B). T o reduce these functions t o normal for m mean s to set the operator A everywher e at the firs t place , for example, as it was done in (1.32) fo r the function f(C) , 288 FORMULA S OF NONCOMMUTATIVE ANALYSIS

1 2 where C = g(A, B) . Suc h permutations ar e ofte n use d in quantum theor y in the situation whe n A i s the annihilation operato r (Wick' s formulas (se e [126])). In this section we shall follow [54]. So A an d B belon g t o th e algebr a M (fo r example , t o th e algebr a o f continuous operator s i n a Banach scale ; se e 1.1) . Suppos e 1 i s the unit i n M an d / i s the identity operator on M. B y L^ , R Q, an d ad ^ w e denote as above the operators of multiplication b y Q fro m th e left, fro m th e right, and o f commutation wit h Q respectively .

THEOREM 1.9 . The following identity

g(A, B) = (g{L RA + LZAA , L LB - iLf)/) 1 , holds and the expansion of the right-hand side of this identity in powers of the operators Lad and ad L yields

N g(A,B)= J2 d[d™g{A,A)K lm + 0 [A,B]. (1.38 ) m,/=0

Here the operators K l m have the form (1.34 )

and can be calculated by the following recursive formulas

1 m

(1.39) , /

H=1

m x{l m) We not e tha t K lm = O * ' [A, B] . Thus , th e summin g ove r / , m in (1.38 ) ca n b e restricte d u p to N, an d th e remainde r O [A, B] ca n b e written explicitly (a s the remainder o f a Taylor series).

THEOREM 1.10 . The following identity

2 f(A + B) = f(RA + (LB + *d A))l9 holds and the expansion of the right-hand side of this identity into a series in powers of ad ^ by the perturbation theory (1.12) yields the formula

N f(A + B) = Y1 f^C* + B)Xk + 0 [A, B]. (1.40 ) FORMULAS OF NONCOMMUTATIVE ANALYSIS 28 9

Here the operators X k have the form

1 ,-^r \ l.k Xk = B (A + B-A-B)

1 "- i _Zi_

w/iere /_ , = 1 , 7 0 = ad^ , / = A' , 5, C ™ w f/ie binomial coefficient (^) , C_. = 1 . These operatorserators cancan bebe calculatcalculated by the following recurrent formulas

X0 = l, X,=0 , k-i

(1.41)

{k+i)/2] N We note that X k = d [A, B]. Thus , the summin g over 0 [A,B] in (1.40 ) ca n b e stoppe d afte r th e numbe r IN , an d th e remainde r ca n b e written explicitly. Th e calculation o f the first several elements X k give s

X2 = -[A,B],^,B], X, X^ = = \7 ([A,[A, B]] + [[A, B], B]),

X = l[A,B] 4 g + ±([A, [A,[A, B)]] + [A, [[A, B], B]] + [[[A, B], B], B]).

EXAMPLE 1.12 . Dirac's problem. Suppos e h^O. D o there exist operators CO 60 A, B fo r whic h th e relatio n / — • jrf(A, B) i s a homomorphism o f a Li e algebra o f function s o n R (supplie d wit h a Poisso n bracket ) int o a Li e algebra o f operators ? If suc h A an d B exist , then CO CO CO CO CO CO [f(A,B),g(A,B)) = -ih{f,g}(A,B), where v 8i ' a^dit d^di 2' We choose / = e -"'"1 , g = e~ail , t e R . The n the following relatio n mus t hold [e , e ] — -iht e . (1-42 ) 290 FORMULA S OF NONCOMMUTATIVE ANALYSIS

By (1.14), (1.30) , and b y comparing the coefficient s a t t 2 , we get [A, B] = ih 1. I n particular, al l the other commutators of A an d B ar e equal to zero. In this case the Theorems 1.9 , 1.1 0 yiel d v - M* Y - ( fft>* A/,m~ / , */,m > ^~2.4...(2fc ) * and relatio n (1.42 ) (afte r w e reduc e th e left - an d right-hand side s t o th e normal form (1.38) , (1.40)) is reduced to the followin g

.±2 -ih£ -iht 2 ! -iht e 2 = e - 1 . This is true only if h = 0. Thu s the answer to the Dirac problem i s negative: it is impossible to preserve the Poisson brackets in the quantum situation (a s a substitut e fo r th e commutator). Thi s ca n be don e onl y up to 0(h ) (se e (1.36)). We now show how calculations according to the recursive formulas (1.39) , (1.41) ca n be represented i n the form o f intuitive diagrams. Consider an integer-valued lattice on the plane and its first quadrant. Le t (m, n) b e a node (i.e. , a point o f the lattice). Th e integer m + n i f m ^ n, or 2( m + n) i f m = n (i.e. , if the nod e lie s on the bisector ) wil l be calle d the weight of the node (m , n) o f the lattice. We dra w al l possibl e interval s connectin g th e node s o f ou r lattic e an d satisfying th e following condition s (a)-(d ) (Figur e 29): (a) An interval does not intersect the bisector or the nodes of the lattice. (b) The intervals situated abov e the bisector may have only the followin g form (al l the intervals are directed upward and to the right):

FIGURE 2 9 Moreover, we admit horizontal intervals lying above the bisector and ending at it. (c) The intervals lying at the bisector itself are regarded as double intervals. (d) The whole picture is symmetric with respect to the bissector. A path o n th e lattic e obtaine d i n thi s wa y i s a connecte d polygona l lin e without self-intersection s wit h vertice s a t th e nodes , startin g a t th e origi n (0, 0 ) an d endin g a t a certai n node . A diagram i s th e unio n o f a finite number o f differen t paths . We associat e a certai n elemen t (operator ) wit h eac h pat h accordin g t o the followin g rule . Suppos e a n elemen t P is situate d a t a nod e a . W e associate the operation o f commutation ad ^ wit h the vertical interval going out o f a ; thu s the elemen t [A , P] appear s a t the end o f this interval. Th e operation ad 5 wil l b e associate d wit h th e horizonta l interval ; th e elemen t [P, B] appear s at the end of this interval. Th e operation of multiplication by FORMULAS OF NONCOMMUTATIVE ANALYSIS 291

_ 7 [A PJ f s(fi) ; QC

FIGURE 3 0

• • • 2~-** • . • /. • = = __L. _L. _L_. —[([CP BJ, 3J-K»~L BJ I s+5 s+4 s+7 s ' ' 2 >ly • >• >i • • QC FIGURE 3 1 the corresponding element K l m wil l be associated with the interval startin g from the diagonal (see Figure 30) (s{P) i s the weight of the node / ? throug h which our interval passes). Intervals connected successively will denote successive application of these operations (se e Figure 31). Here s i s the weight of the node following a . Suppose the unit 1 i s situated a t the node a — (0,0). W e consider a certain pat h L startin g fro m thi s node . Th e element associate d wit h the path L b y the rule mentioned abov e wil l be denoted b y L(l). Obviously , L(l) = O [A, B], wher e / i s the number of links in the path L . Let 2 - [Jj Lj b e a diagram. The n the element 3(1) = £V Lj(l) i s said to be the element generated from 1 by the diagram 31. Consider specia l diagram s X k compose d o f al l possible path s endin g at the nodes (m , n), wher e m + n = k. Fo r example, the diagram X 5 i s shown on Figure 3 2 on the following page .

THEOREM 1.11 . The elements X k in the expansion (1.40 ) are generated from the unit by the diagram X k, i.e., Xk = X k(l). We no w change th e weight s o f th e node s o f ou r lattice. Th e numbe r max(m, n) i f m ^ n o r 2m i f m = n wil l b e calle d th e weight o f the node (m , n). Conside r the diagram J^ m compose d o f all the paths start - ing from th e origin and ending at the node (/ , m).

THEOREM 1.12 . The elements K { m in expansion (1.38) are generated from the unit by the diagram X { m , i.e., Kf m = 3£{ m(l). 292 FORMULAS OF NONCOMMUTATIVE ANALYSI S

rT~

FIGURE 3 2

FIGURE 3 3

1.6. Paradoxe s of formal calculations with functions o f operators. W e now give some unexpected counterexample s whic h should make us very cautiou s when formally usin g the combinatorial relation s of the previous subsection . EXAMPLE 1.13 . (Nelson) . Nondifferentiability. W e tak e tw o coordinat e planes cut along the negative half-axis and glue them together into a two-leaf plane n wit h variables x , y , i.e. , th e Riemannia n surfac e o f the functio n \/x + iy . Th e operator s B = -id/dy, A = -id/dx ar e define d o n th e dense subspac e S = C£°(I I \ {0} ) o f th e Hilber t spac e L 2(ri). The y ar e essentially selfadjoin t an d commut e o n 2 : [A , B] = 0 . However , despit e Theorem 1.3 , function s o f th e operator s A an d B , generall y speaking , d o not necessaril y commute . Fo r example, i f the support o f a function u e 2J lies on the first leaf of n i n the square 0 < x < 1 , 0 < y < 1 and i f x > 1, t < -1, then th e suppor t o f th e functio n e ltAelxBu lie s o n the first lea f o f n, an d th e suppor t o f th e functio n e lxBeltAu lie s o n th e secon d lea f o f n (Figure 33). Thus [e itA , ehB] ± 0. Th e cause of this paradox i s the following: condi - tion (a ) o f 1. 1 i s violated sinc e the product o f groups Q(t , r ) = e lxBeltA i s not differentiat e a t th e poin t T = t = 0. Henc e i t i s impossibl e t o defin e increasing functions o f the operators A,B, an d i n particular, t o represen t these operator s themselve s b y formul a (1.1) . Therefore , i t doe s no t follo w that th e group s e lt , e lx commut e i f th e operator s A , B themselve s d o FORMULAS O F NONCOMMUTATIVE ANALYSI S 293

(same for arbitrar y functions f(A) an d g(B) , wher e f,g€ C(R) , i n general). EXAMPLE 1.1 4 [58] . Exponential growth. Conside r the two following op - erators: A = (ix) djdx + i/2, B — x whic h ar e i n essenc e selfadjoin t o n Qj = C^°(R ) i n the spac e L (R) . The y satisfy th e commutation relatio n [A,B]=iB (1.43 ) on 21 . Denot e by /(£ ) = f - / , g(£) = f(€)~l. Sinc e fg = 1, we have the relation 2 3 1 3 1 B = Bf(A-A)g(A-A). We transform th e right-hand side as in the proof o f Theorem 1. 1 2 3 14 0 2 4 0 B = Bf(A - A)g(A -A) = Qg(A - A) ,

2 3 1 2 3 1 where by (1.43 ) Q = Bf{A - A) = B{A - A - i) = 0. Thus , w e obtain th e false equality B = 0 ! The error here is in that w e used the symbol £ 2/(£3 -£l)g{€4 -£°) whic h is smoot h o n R , but not analytic o n C i n th e calculations . W e canno t use such symbol s in this cas e since the derivatives o f the product o f group s Q{t,T) = e nBeltA wit h respec t t o T increas e exponentiall y a s \t\ - > oo . Condition (a) fro m subsectio n 1. 1 i s violated. 2 3 1 Generally speaking, the operator B(p(A — A), wher e q> e C 0 (R) , does not exist. Actually , by (1.43 ) the commutation operatio n ad A : T — • [A, T] ha s the eigenvalu e / an d th e "eigenvector " B relate d t o it . Thus , th e relatio n 2 3 1

[A9...,[A,B]--] = XB9 InU^ O (1.44 ) n hold instead of (1.43). Jus t as above, by using the symbols /({ ) = C -X an d g = f~ , we obtain B = 0. Thu s i n an y Banac h scale , where the operato r A i s a generator, and B i s a continuous operator, the relation (1.44 ) implie s B = 0. I n particular, the operator (ix) djdx + i/2 canno t b e a generator in any Banach scale in which the operator of multiplication by x i s continuous l (for example , in the scal e H k(R)). Consider three more operators satisfyin g th e relation s

2 AB + BA = iCB, [C 9A] = 0, C = 1. (1.45 ) 294 FORMULA S OF NONCOMMUTATIVE ANALYSIS

Suppose p(±l ) = l,p = 0ina neighborhoo d o f zero and cp e C£°(R). We denote /({, rj) = £ - it] , g(£ , r\) =

B = ip(C)B = Bf(A - A , C)g(A - A , C) = Qg(A - A , C), 2 3 1 1 where Q = 5/(^ - A, C) = 0. Thus , (1.45 ) implie s 5 = 0. A t the same time it is easy to verify tha t the self adjoint operator s A = il{~\xd/dx + 1/2), B = x, C = 7(_), where 7 (_) i s inversion, satisf y th e relations (1.45) , i.e., I^u(x) = u(-x) . 2 EXAMPLE 1.1 5 [58]. Defect. Conside r the operators A = /(l +x )d/dx + ix, 5 = arctanj c o n the domain o f definition D = C£°(R ) c L 2(R). I t is easy to verify tha t [A,B] = il. (1.46 ) Thus by the shift formul a (1.18) , we have e'itAf{B) = f(B^t)e'itA. (1.47 ) Suppose t > 2n, an d the symbol / e C£°(R ) i s chosen s o that /({ ) = 1 for |{ | < 7 r and /({) = 0 fo r | f | > 37r/2. The n sinc e arctan x - f f > 37r/2, we have f(B + /) = 0 an d since | arctan*| > n/2, w e also have f(B) = 1 . However, these two relations contradict (1.47) . The erro r her e i s caused b y the fact tha t th e operator A , althoug h i t is symmetric, is not essentially selfadjoint. Actually , the equations (A* ± i)y/ — 0 posses s the solutions y/(x) = (e±arctan*)/(\/l +x2), i.e., the defect indice s of the operator A ar e equal to (1 , 1). EXAMPLE 1.1 6 [58 , 96]. Growth of derivatives of a symbol Le t / b e a symbol i n on e variable an d g(£ ; f ;£ ) = /(( £ ~ £ ) f ) • Conside r the operators ^4 , 5 satisfyin g relatio n (1.46 ) an d all the conditions o f 1. 1 (for

2 3 example, A = id/dx, B = x i n L (R)) . Putting the operator A i n the first 1 2 3 place in the function g(A\ B\ A) an d using the formula (1.15) , we get 123 12 1 4 S 2 a 1 2 6 3 5

Since £({ * ; g; f 1) = /(0) an d M, 5] = zl, we obtain 12 3 / ^ Sg\ 1 2 4 3 g(A;B-A) = /(0) + i (^3^) ^'>B,B;A). (1.48 )

By the definition o f difference derivative , we have Ai*V ^ 2 ^ 2' .3, a (f({e 3-zl)z2)-f({z3-z1)*2')

= f\{?-£> x)t;\ FORMULAS OF NONCOMMUTATIVE ANALYSI S 295

So formula (1.48 ) yield s

f({A - A)B) = /(0)1 + if'((A - A)B). (1.49 ) For example, suppose /({ ) = exp(-if); then if' = /, /(0) = 1, and (1.49 ) yields the contradiction 0 = 1 . This erro r i s cause d b y th e fac t tha t althoug h th e symbo l / itsel f lie s in th e clas s ^(R), th e symbo l g = el{^ "^ )< r use d i n (1.48 ) doe s not lie in S°°(R ) (th e derivative s o f g gro w slowl y a s |{ | -> oo , but th e growt h becomes faster a s the order of derivatives increases). We note that i f three operators A , B, C satisfy th e relations [A,B] = iC, [A,C] = [B,C] = 0 and al l of them ar e bounded (i.e. , the behavior o f the symbols at infinity i s of no consequence for them) , then just a s above w e obtain the formul a fo r the resolvent 3 2 1 -i 1 (i * 1 (A-C) = JOLPIJ{A-A)B\9 A^O .

In this cas e the spectru m o f th e operato r C consists o f a singl e point {0 } [142]. This page intentionally left blank APPENDIX I I

Calculus of Symbols an d Commutation Relations

One of the main ideas of the operator calculu s is to avoid analysis in the space of operators by reformulating al l the problems in the language o f th e symbols of these operators, and then to use analysis in the space of symbols. The situation s i n whic h w e ca n reduc e ou r proble m t o th e analysi s o f functions o f a finite number o f "elementary" generators f(A l, .. . , An) ar e of specia l interest. W e assume that w e know everythin g abou t eac h genera - tor A J: th e propertie s o f it s one-parameter group , spectra l properties , etc . The generators A J ar e model operators (which themselves can be sufficientl y complicated). The n it is a combinatorial problem to rearrange the function s of generators A , ... , An , i.e., the symbols. Th e central role is played by the possibility o f transferring th e associative multiplication fro m th e algebra o f operators to the algebra of symbols, in other words, of constructing a calculus of symbols. The spac e o f symbol s an d th e rule s o f analysi s i n thi s spac e attrac t u s first o f al l b y thei r universality , sinc e the y ar e independen t o f th e specia l form o f the initia l operator s and ar e define d onl y by permutation relation s among them . Bu t no t w e do , fo r arbitrar y permutatio n relation s succee d in constructin g a n effectiv e calculu s o f symbols . W e conside r onl y certai n classes of suc h relations following the methods of [68 , 96, 100 , 101]. 2.1. Generalize d Jacob i condition s an d Poincare-Birkhoff-Wit t property . Suppose a n algebr a M wit h generator s A 1, .. . , An i s given. Suppos e 3° n is the space of polynomials in n variables . Unde r what conditions is the set i n {f{Al, .. . , An) | / e 3° n) a n algebra and hence coincides with Ml Obviously, for this purpose one needs the existence of operators L\ .. . ,L n on the space of symbols such that In I n J l n J l n A [f(A ,...,A )] = (L f)(A ,...,A )9 (2.1 ) as well as operators R l, .. . , Rn suc h that

In I n [f(Al, .. . , An)]AJ = (RJf)(Al, .. . , An) (2.2 )

297 298 CALCULU S O F SYMBOLS AND COMMUTATIO N RELATION S for an y symbo l / an d an y j = 1,...,«. Th e operator s L j an d R J wil l be called lef t an d right operators of ordered regular representation of the se t 1 n Al,...,An.

LEMMA 2.1 . Suppose A , ... , A are generating operators in a Banach scale or in an arbitrary poly-Banach algebra {see IA in Appendix I ) and the left operators LJ of the ordered regular representation (2.1) exist. If L l, .. . , L n are generators in the algebra J?(S°°(R n)), then the composition formula

n n n [g(A\ ... J )]-[f(A\ ... J )] = (g * f)(A\ . ..9A )9 (2.3 ) where

l n g*f=g(L ,...9L )f9 (2.4 ) n holds for any f 9ge S°°(R ). If right the operators RJ (2.2 ) exist and they are generators in ^f(S°°(R n)), n i then (2.3 ) holds, where g*f = f{Rl, .. . , R n)g. n PROOF. Denot e & = S°°(R ) x J?(B ) , where {B^} i s the Banach scal e from th e conditio n o f ou r lemma . W e defin e th e operator s L J, A J, Q o n %f a s follow s H'TXV)' *V)-UT)-

where /G5°°, re^}. The operators L J, A 1 ar e generators in £?{%?), an d f n gd\...9i")( T) =(gd\...,£ )f)9 g(0):

n g(0)f g{A\...J ){^ =( 1 n l n g(A 9...9A )TJ We now use the general quasicommutation formul a (1.16 ) of Appendix I and take into accoun t th e fac t tha t th e quasicommutators vanish : [Q, L J\AJ] = 0. W e obtain 1 n 1 » n l n Q- g(V 9 ... 9 L ) = g(A 9 ... 9 A )-Q. By applying the left- and right-hand sides to an arbitrary element f M e %?, we obtain the desired relaton (2.3) . Th e second statement o f our lemma ca n be established i n the same way. Th e lemma i s proved. CALCULUS O F SYMBOLS AND COMMUTATIO N RELATION S 299

If the initia l se t o f operator s A = (A 1, .. . , An) i s a Wey l se t an d ther e exist operator s L fJ, R u o n S°°(R n) suc h that, instea d o f (2.1) , (2.2) , the following relations hold for the Weyl function s Aj[f(A)] = (L,Jf)(A), [f(A)]A j = (RfJf)(A), (2.1a ) the operator s L ,J an d R ,J wil l be calle d lef t an d righ t operators of Weyl regular representation.

1 ,n n LEMMA 2.2 . If the set L' = (l/ ,..., L ) is a Weyl set in JS?(S°°(R )), then g(A) - f(A) = (g * f)(A), g*f = g(Lf)f (2.5 ) If the set R' = (R fl, .. . , R,n) is a Weyl set, then g*f = f{R')g. The operation * will be called the twisted product in the space of symbols or simply the *-product. Naturally, if the space of symbols coincides with the space of polynomials &n, the n bot h lemma s ar e trivial an d th e *-produc t i n & n i s define d b y formulas (2.4 ) or (2.5 ) without an y additional conditions . We note that the product define d i n (2.5 ) differ s fro m (2.4) . Althoug h we denote these operations by the same sign * , this will not result in confusio n since it will be clear in any case whether we speak about the ordered or Weyl calculus of functions o f operators. We note that i f the symbol s / = f(£ l, .. . , £") i n (2.1 ) ar e independen t of the variables f' +1 , ... ,

AJ[f(Al,...,AJ)] = A Jf(Al,...,AJ). Thus, i t i s natura l t o assum e tha t th e operato r L J act s o n thi s symbo l a s ordinary multiplication b y the variable £? . The n ^(1)=^, L 2(/(«1))={2/«1),.-., 1 n and hence g(L l, .. . , Ln)\ = g{^ , ... , g(L , ... , Ln) i s injective , i.e. , tha t ther e i s n o symbo l g ^ 0 fo r whic h g{Ll,...,Ln) = 0. 300 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

Any equality o f the for m

,...,Al>») = 0 (2.7 ) with a certai n symbo l tp an d indice s i Q fro m th e se t {1 , ... , n} wil l b e called a relation between the operators A , ... , An .

1 n l n LEMMA 2. 3 [101] . The ordered calculus JLL : /— • f(A , .. . , A ) is injective, i.e., the zero operator can be obtained only from the zero symbol, if and 1 w only if the left operators L , ... , L of ordered regular representation satisfy (2.6) and all the relations that the operators A ,.. . , A do. In this case the ^-product defined in Lemma 2. 1 is associative. The same statement holds for the Weyl calculus. PROOF. Jus t as in Lemma 2.1, we can show that

ViA1*, .. . , >)/£(/) = li((p(& , ... , L1"-)/). Thus relation (2.7) and the fact that JJ, i s injective yield the following relation

1 i n „ /(L1,...,Ln) = 0 . But then, by (2.6), we have / = 0, i.e. , / / i s injective. Thi s fact an d the associativity of multiplication in an algebra of operators yield the associativit y of the *-product . Th e lemma i s proved. We now consider a n abstrac t algebr a spanne d b y certai n generator s A l, ... , A n wit h a certai n numbe r o f relation s o f for m (2.7) . A lef t regula r representation o f thi s algebr a wil l be define d just a s in (2.1 ) (o r in (2.1a)) . The existence of operators L l, .. . , Ln mean s that any element of the algebra I , n n can be written in the form of a polynomial in generators A , ... , A ordere d in th e standar d way . I f suc h a polynomial i s unique, w e shal l sa y that th e Poincare-Birkhojf-Witt property (PBW) holds in this algebra . Lemma 2.3 states that the PBW property holds if and only if the operators l) , ... , Ln satisf y (2.6 ) an d al l the initia l relations , i.e. , satisf y (2.8) . Th e statement ca n be formulated i n terms o f right operators R , ... , Rn a s fol - lows: the PBW property is satisfied i f and only if 1 * g = g an d the operators Rl satisf y relation s conjugate t o (2.7) , i.e., (p(Rim,...,Ril) = 0. (2.8a ) Equations (2.8 ) an d (2.8a ) wil l be calle d generalized Jacobi conditions [100, 101]. CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 301

For a classical algebra with defining relation s j k k j k J s A A -A A - ik s A = 0; k, j = 1, ... , n, (2.9 )

J J where X S = const, th e operators L wer e calculate d i n (2.28), (2.28a ) of Appendix I. It is easy to see that these operators L J themselve s satisfy rela - tions (2.9 ) if and onl y if the structure constants kj satisf y identitie s (1.26 ) of Appendix I. Thus, these identities are generalized Jacobi conditions in this case. Here Lemma 2.3 says that the identities mentioned above are necessary and sufficient i n order that any polynomial in the generators A 1, ... , An be uniquely expressibl e i n the form o f a Weyl (o r ordered) polynomial . Thi s statement is the classical Poincare-Birkhoff-Witt theore m [122] . EXAMPLE 2.1. Pauli matrix. Denot e "1 0" "o r 0 — i~ !o -1 °2 = 1 0 <73 = i 0 These matrices are anticommutativ e

0\0"2 = — Gy^\ ' ^1^ 3 = ~ ~ ^3^1 ' °2^ 3 = ~~°3°2 * (2-11 )

Thus, by the quasicommutation formula , w e have a if{a;) = f{-Oj)oi fo r /Yj. Hence ,

1 2 3 3 1 2 4 1 1 2 3 r/7 M /v \ ft \

l i.e., L = ^/2/3, wher e I k i s the operator o f inversion (chang e o f sign) 7 7 3 3 with respect to the A;t h variable. Similarly , L = £ I3 an d L = £ . The operators L l, L 2, L3 satisf y relation s (2.11) . Thu s the generalized Jacob i conditions hold for an algebra with defining relation s (2.11) . However, a specific realization of relation (2.11) in terms of Pauli matrices satisfies som e other relations, for example, [ax, a 2] = 2/cr3, [er 2, a 3] = 2iox, [a 3 , crj = 2/tr2, (2.12 ) which do not hold for the constructed operators L 1, L2 , L3. According to 12 3 Lemma 2.3, this means that the calculus / •- • f(al, o 2, a 3) i s not injective , i.e., th e PBW propert y doe s no t hold i n the algebra wit h relation s (2.11) , (2.12). In the same wa y on e ca n show tha t relation s (2.12 ) ca n b e used a s th e initial one s and operators o f left regula r representatio n ca n be constructed with respect to them. Relation s (2.11 ) wil l not hold for these operators. All of this also concerns the Weyl functions o f Pauli matrices. W e note in passing a useful formul a fo r Wey l functions [149 ]

EXAMPLE 2.2. Cyclic quadratic relations. [A,B] = C 2 [B,C] = A\ [C,A] = B' Each polynomial i n the generators A, B, C ca n be written i n the form o f 1 2 3 a polynomial i n the ordered generator s A, B, C. Fo r example, for polyno - mials of the third degre e we have AB2 = C 2B + CA 2, BC2 = CA 2 B2A, AC2 = -CBL BA' A2C = -B2A C 2B, A2B = C 2A CB2 B2C = C2A + BA 2 and for polynomials of the fourth degre e A2B2 C4 + C 2BA CB A2C2 B C3B BAJ 2C2A2. AC3 = B3A -A 4- C 2B2 - 2CBA 2. B2C2 = A4 + CBA2 - B*A, BC3 = -C3B + B*-C 2A2 BA3 - CB 2A, AB' C3A, and so on by induction. Her e the PBW property holds (but explicit formula s for the operators of regular representation ar e unknown); see [21]. We have a similar picture for the algebra with four generator s [123 ]

[S0, S a] = Ufiy(SpSy + SySfi), [S a, S 0] = i(S 0Sy + SyS0), (2.13 ) where (a, 0 , y) i s a cyclic permutation o f the triple of indices (1, 2, 3), and J» = (J a - Jo)/J , J a = const . Th e simples t representation s o f relation s (2.13) yiel d th e sam e Paul i matrice s S a = a a, S 0 = I o r th e followin g three-dimensional matrice s ro I o i ro -i 0 / S{ - y/2J 2J3 1 0 1 ? 2JiJl i 0 - / [o I o . 0 i 0 ri o o i J, 0 S - 2yJj J 0 0 0 0 0 3 x 2 sn /, + J2 •J* [o o - IJ 0 Infinite serie s of representation s o f this algebr a ar e presented i n [123] . Th e PBW property holds . I f al l the numbers J {, J 2, J 3 ar e different , the n n o explicit formula s fo r operator s o f the regular representation ar e known. Th e degenerate cas e when J { = J 2 i s considered belo w in Example 2.6. EXAMPLE 2.3 . Generalized shifts and relations among their generators. Suppose J£ i s a certai n se t an d fo r eac h x e ^# a linea r operato r U x i s given in the space of functions F(^# ) s o that the following axiom s hold( ) : UxUy y U = I fo r a certain poin t e e ^#, u xu\ (2.14) (U*f)(e) = f(x) fo r an y function / o n Jf.

y (^The lowe r inde x i n operators , e.g. , the inde x x i n notatio n V x show s the variable wit h respect t o which the operator acts . CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 303

Then U is called a generalized shift on J?, an d the mappin g A : V(JT) - > V(Jt) ® F(^f), A(/)(y , x) = (£/*/)O 0 is called the comultiplication in F(^f) . Only smoot h shift s ar e usually considered , i.e. , it is assumed that Jl i s a smooth manifold, U x i s defined on smooth functions, and (U xf)(y) de - pends smoothl y on ije/. I n this cas e F(Jt) = SF{J£) i s the algebr a of smoot h function s o n J? . Th e simplest exampl e o f generalized shif t i s given by the group shif t (U xf)(y) = f(yx), wher e x, y e Jf, Jf a Li e group. Generalize d shifts were first introduced by Delsarte [170 ] precisely as a natural generalization o f group shifts . Here is an example of nongroup shif t o n a line [37] (Uxf)(z) = 2/(z + x) -/(z) -/(*) +/(0) . (2.15 ) Many simila r example s can be found i n works devoted to the theory of hypercomplex structures, hypergroups, generalized shift s an d Hopf algebras [9, 37, 89, 92, 93]. In th e space o f distributions o n J[, eac h generalize d shif t generate s a convolution by the formul a where the asterisk denote s the adjoint operator . I f the actio n o f a distribution on a test function i s written in the form o f an integral, the convolution formula ca n be represented as follows

x (9*V,f)= [ 9(z) [ V(x)(U f)(z)dxdz = (

The convolution is associative and the Dirac function 8 e concentrate d at a point e e J? i s its two-sided unit . If the third axio m in (2.14) i s violated, the operator U wil l be called an almost generalized shift, an d the related operatio n

The generators of group shift ar e (left-invaraint) vecto r fields everywhere on J?. However , for example , the generator o f the shift (2.15 )

u2f(z) = 2f\z)-f'(0) is not a differential operator . The linear operator s

dxi \x=e are called generators of the smooth representation T o f a generalized shift . By differentiating (2.14 ) and (2.14a ) with respect to y a t the point y — e, we obtain the permutation relation s X x i U[U = C/V, T A = i^r*. (2.14b ) Let ^ b e a polynomial on M^ , J V = dim J?. Th e following distributio n on J( u N &(x)=&(u ,...,u *)de(x) will be called its Fourier transformation wit h respect to the generalized shif t U. The n the Weyl polynomial in the generators of the representation T ca n be written i n the form &>(A) = &>{Al, ... , AN) = Yl(^), an d henc e &>(A) • n(^) = n(^ * q>) = n(^(u U , ... , u N*)q>), &>{A)-Q(A) = !!(&> *Q). If a regular representation and the *-produc t exists in the algebra generated by A 1, .. . , AN , we have P * Q = & * Q, wher e the asterisk in the left-han d side is the *-produc t o f symbols, and the convolution of distributions on J£ in the right-hand side . As we saw for Li e groups (se e Example 1. 9 i n Appendix I), a convolutio n of Fourier symbols can exist also in the case when the *-produc t of these symbols is not defined . Thus , it i s more convenien t t o consider the convolutio n and th e generalize d shif t correspondin g t o thi s convolution . It s generator s uJ ar e global analogs of operators o f the right regular representation R J. REMARK 2.1 . The coordinate s x l9...,xN i n a neighbourhoo d o f the poin t e o n J? ar e calle d coordinates of the first kind i f U x = N j exp(Xj u + h x Nu ). I f w e expres s the generator s u i n term s o f thes e 1 2 J coordinates: u x = -ir'iid/dx, x) , the n w e obtai n th e followin g (local! ) expression for the operators of right regula r representation : Rj = rj(l, id)d() , wher e $ = (t\ ... , £N) eR N * T^. CALCULUS OF SYMBOLS AND COMMUTATION RELATIONS 305

Here the £ J ar e the coordinates in the dual spac e on which the symbol s are defined (se e Example 1. 9 i n Appendix I again). Until now we have considered the chain: (operator s A J) — • (commutation relations among A J) — • (regular representation R J) — • (relations among R J (generalized Jacob i identities)). Obviously , instea d o f these latter relations , we can consider relations among the generators u J o f a generalized shif t o n ^.

LEMMA 2.4 [37] . (a ) Any relation among generators of generalized shift is equivalent to the restriction of this relation to the point e . (b) If the generators of generalized shift are vector fields on J[, then Jt is a Lie group or its quotient with respect to a discrete subgroup, and the generalized shift coincides with a right group shift. (c) If polynomials of degree N in the generators determine a jet of finite order at the point e, then the generators satisfy N-linear commutation relations.

PROOF, (a ) B y (2.14b), we have

lk lk k &(u\ ... ,u\=U* -Ptf^ ... ,U )\e i.e., an y polynomial in the generators a t the point x e Jt i s obtained fro m the sam e polynomia l a t th e poin t e b y th e shif t U x. Thus , th e relatio n among generators hold s a t eac h poin t o n Jf i f an d onl y i f i t hold s a t th e point e . (b)If * 4 = a^)d/dz5, the n

j j [u , u ]\e = tfu\ , wher e tf = {da k/dzi - da kldz.)\2=e.

Thus, the generators u l a t the point e , an d hence, everywhere on Jf satisf y the Lie relations with structure constants A^ 7. B y a standard argument , thi s implies [227 ] that J[ i s a Lie group (i f i t i s simply connected), and the u l are left-invariant fields on the group. (c) Suppose polynomials of degree N i n u l generat e a jet o f order M a t the point e. The n the Nt h powe r of the generator can be linearly expresse d in terms o f the derivative s

«"••••<'%= s>.(&)"| •

a Conversely, eac h differentiatio n (d/dz) \z=e i s expressed i n terms o f a cer- m tain polynomial & a{u , ... , u )\e o f degree not greater than N i n the generators u l. B y substituting thes e polynomial s fo r (d/dz) a, w e obtain th e relation 306 CALCULU S OF SYMBOLS AND COMMUTATIO N RELATION S at the point e . Thus , such TV-linea r relations hold everywhere on Jt . Th e lemma is proved. Note that generalized shift s wit h linea r relations between their generator s and analogs of Lie theorems fo r this case were considered i n detail in [89]. If a generalized shift on Jf i s given and its generators satisfy the relations 1 m y>{ul, .. . , u m) = 0, then th e generator s A 1 o f an y representatio n o f thi s shif t als o satisf y th e same relation s

il im p(A ,... 9 A ) = 0. The generalize d Jacob i condition s wil l certainl y hol d i f th e algebr a M is generate d b y thes e (an d onl y these ) relations . Thus , considerin g th e re - verse chain : (convolutio n o n J? ) — • (generalize d shif t o n ^# ) - * (rela - tions among its generators)— • (algebra M generate d by these relations), we shall always obtain "good" algebras in which, for example, the PBW property certainly holds. The manifol d Jf o n whic h th e convolutio n i s define d play s th e sam e role for suc h a n algebr a M a s the Li e group doe s fo r it s Li e algebra. But , naturally, there is no analog of the group law on J£ i n the general case. W e shall call J£ = cospec(M) th e cospectrum of the algebra M . We shall return to permutation relation s amon g generators o f generalize d shifts i n 2.7, 2.8. 2.2. Chang e o f orde r an d *-produc t ove r the Heisenber g algebra . A s w e already mentioned , ou r ai m i s t o presen t thos e permutatio n relation s fo r which we can calculate the *-produc t or the convolution. I t is useful to begin with the simplest case. The famou s uncertaint y principl e o f quantum mechanic s i s based o n th e simple Weyl inequality fo r two operators A an d B :

3r(A)-&(B)>±\([A9B])\. Here 3f(A) = ((A - (A))*(A - (A))) i s dispersion, {A) = (Ay/, y/) i s the averaged valu e o f a n operato r i n th e stat e y/ , \\y/\\ = 1 . I n particular , i f [A, B] = ihl , wher e h > 0, the n 3f{A) • 9f{B) ^ h/2 . I t i s precisely thi s inequality which is the mathematical for m o f the uncertainty principle . We have alread y me t th e Heisenber g relation [A , B] = ihl abov e in Ex - amples 1.8 , 1.10 , 1.1 1 o f Appendix I. We shall use the construction o f 2.1 in order to illustrate this relation. W e consider its multidimensional version o f it at once. Suppose the set s of operators A = = (A1, .. . , An), B = (B {, .. . , Bn) an d the relation s [AJ, A k] = [BJ,Bk] = 0, [A j , Bk] = ihd jkI. CALCULUS OF SYMBOLS AND COMMUTATION RELATIONS 30 7

1 2 are given. Th e operators o f ordered regular representation o f the set (A 9 B) have the following for m LA = q + iH dj> L B=P> R A=4> R B=P + ih-QJ> where (q,p) denote s the arguments of the symbol f = f(q,p) correspond - ing to the sets of operators A, B . CO 10 The operator s o f regula r representatio n o f a Wey l se t (A, B) ar e con - structed in a similar way T, ih d _ , ih d

ih d f ih d ^ = «-T5^Taj' RB=p' R B=P + + ^~df In this case the general formulas (2.3) , (2.5) ma y be made more precise. 2 THEOREM 2.1. For any symbols f,ge S^R "), the following composition formulas hold: • in the ordered case

lf(A,B)] • [g(A,B)] = (f*g)(A 9B), (2.16 ) where (f*g)(q,p) = f(q + ihd/dp,p)g{q,p) = g(q,P + ^d/dq)f{q,p) ; • in the Weyl case

CO CO CO CO CO Q) [f(A,B)]-[g(A,B)] = (f*g)(A,B), where W* g)W = f(z + ^Jd)dz)g(z) = g(z- l pd)dz)f{z), , , , r o /i (2-16a) zS(q,p), J=[_ I 0 J- The operation (2.16) , as well as the operation (2.16a) , define the structure of an associative poly-Banach algebr a with unit 1 on 5°°( R n ). The expansion of the right-hand sides of (2.16), (2.16a) in ordinary Taylor series in powers of h yield s formulas (1.36 ) o f Appendix I. The operatio n o f multiplicatio n (2.16a ) als o ha s the followin g beautifu l geometric expressio n

(f*g)(z)= fff(z")g(z)^p(-T I dpAdq)-^--^-, where the integral of the form dpAdq i n the exponent is taken over the trian- gle with vertices at points z , z , z" el" . This form an d its generalizations to certain nonlinear phase space s are discussed i n [10 , 11]. We now consider the relations between different ordering s of the generators A\ B j. 308 CALCULU S OF SYMBOLS AND COMMUTATION RELATION S

LEMMA 2.5. The operator

dl dpdq jE^: dqjdpj is a generator of the algebra S°°{R ") (seel.I in Appendix I). The corresponding one-parameter group U„ = exp {-<^H~ S°°(R2n) relates different orderings of generators of the Heisenberg algebra

f{A,B) = (U_ l/2f){A,B) = (U l/2f){A,B). These formula s sho w tha t th e Weyl cas e an d the simplest ordere d cas e can b e transformed int o eac h othe r i n the space o f symbol s b y means of the transitio n operator s U £. I t is natural t o try to classify othe r possibl e orderings according to the form o f the transition operator . Let a{W) = (l/2)(QW, W), We R 2n , be a symmetric quadratic form . We define the transition operato r

*2n t/° = expi-ihQ (iJjA\ , Z G and set (by definition )

O 0 c o a > ftA,Bf = (U°f){A,B). (2.17) R S Let Q = , wher e the blocks R — R' an d Q = Q' ar e symmetric. .S Q Then w e have

n / 2 4 A +A 1 » 5 3142\ f(A, Bf = f(S(A -A) + —I- + jQ(B -B),B + ^R(A - A)).

0 / In particular, if Q = e , then / 0

a y f(A,B) = f((j + e )A+ [j-^A, B}=(UJ){A,B).

Thus the general Q-orderin g o f operators A , B denne d by formula (2.17 ) coincides in particular cases with the Weyl case and with the simplest order - ings. Theorem 2. 1 ca n be carried ove r to the case of general Q-orderin g [74]

XA,B)a.g(A,B)a = (f$g)(A,B) a, CALCULUS OF SYMBOLS AND COMMUTATIO N RELATION S 309 where the associativ e produc t * i n the spac e o f symbol s S°°(R n ) i s to b e calculated a s follow s

/ * g = f (z + i (\j + jaA d)dz\ g(z)

= g(z-i (\j - JQj) d)dz\ f(z).

EXAMPLE 2.4 . Unitary group of the oscillator [58, 74] . B y using the for - mulas give n above , w e shal l calculat e th e grou p generate d b y th e operato r 2 2 (A + B )/2 . Le t us construct i t by means of the £2-permutatio n A B -it ! xfm l e < =g t(A9B) (2.18)

We obtain the following Cauch y problem fo r th e symbol g t

dgt{z) l 2 ,n dt *0(*)=1- -iS, Then it is easy to calculate the solution g t = e ' /y/J~t explicitly; for exam - pie, i f £1 = e , then I O j 4eqp(\ cost) + sin/(#2 4 - p2) St(q,p) 2Jt (1/2- 2e2) + (l/2 + 2e 2)cosf. 2 2 We see that whil e |/ | < arccos((e - l/4)/(e + 1/4)) , the function s S t, J t depend smoothl y o n t an d defin e th e one-parameter grou p o f the oscillato r according t o formul a (2.18) . Bu t ther e exist s a point / suc h tha t th e rep - resentation (2.18 ) wit h Q = er doe s no t hol d a t thi s poin t an y more . However, i f w e choose another ordering , fo r example , £1 = l 1^0. , then i n this case, J t = 3/ 2 + sin / — (cos t)/2 i s nonzero for al l t an d the representation (2.18 ) hold s uniformly wit h respect to the parameter / . W e leave as an exercise for the reader the calculation o f the phase S t correspondin g to such a permutation Q. . At thi s poin t w e par t wit h th e Heisenber g algebra , referrin g th e reade r to Chapte r IV , where thi s algebr a i s used a s a loca l mode l whe n nonlinea r commutation relation s are considered i n the semiclassica l approximation . Below we shall be interested precisel y in nonlinear relations . 2.3. Semilinea r commutatio n relations . Conside r th e followin g natura l generalization o f the linear relations (2.9 ) [AJ,Ak] = iAl k(B)A\ j,k=l, n9 [Bl,Ak) = -iii lk(B), (2.19) l s [B ,B ] 0, sj= 1 , m. 310 CALCULU S OF SYMBOLS AND COMMUTATION RELATION S

Here A^, /jfk ar e given function s o f m variables , A - (A 1, .. . , An) an d B = (B , ... , B ) ar e Wey l set s o f operators . Everywher e th e repeate d indices mean summing . As the simplest example, we can take the following:

Aj = iajk(£)-?-r, B k = £k, wher e £ e R n.

If th e matrix a = ((a j ) ) i s not degenerate anywhere , the n thes e operator s satisfy relation s (2.19 ) if we take

Ajk , jm ~ kl km r. jL — 1 Ik kl K = ( a d ma ~ a d ma ) als > I* =a . In this case, the numbers A ^ (£) ar e the structure constants o f a certain Lie l algebra for each £ i f all the forms aj s dt, ar e closed. If the Jacobi matrix a{£) degenerates , but still satisfies the Jacobi condi - tions jk kj ^ jm 0 kl ~ aJ =-a , & a J da =0 , U,k,i) then the operators A = ia(i)d/di, B = £ als o satisf y relation s (2.19) . I n this case , the numbers A ^ ({), generally speakin g (i f the functions a j ar e nonlinear), are not the structure constants for any Lie algebra. Now b y usin g th e method o f [61] , we construct th e operator s o f right regular representation fo r the relations (2.19). Denote by 38 = 38(b, z , t) th e solution o f the Cauchy proble m d_ -3S = IL(38)Z, &\ = = b, (2.20 ) dt t 0 where b € R m, z e R n. Assum e tha t th e solution exist s fo r t € [0 , 1]. Further, construc t the matrix z.A(6) = ((z/f(6))) and defin e

rb{z)=(f Expl-f z»A{^{b,z,t))dt\dT\ . (2.21 )

Here by Ex p w e denote the multiplicative integra l (se e Example 1. 6 of Ap- pendix I), and assume that z i s sufficiently smal l so that the inverse matri x in (2.21 ) exists. N Fix x,y i n a neighborhood o f the origin i n R an d denote by z x (b) the value of the solution o f the Cauchy problem fo r t = 1

b •jjZ = r (z)y9 z\ t=Q = x. (2.22 )

THEOREM 2.2. The following composition formula holds:

2 1 -ix-A -iy-A -iz x (B)A e e = e x -y CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 311

lx lx PROOF [61] . Differentiat e th e operator T(x,x) = e ' e ' wit h respect to the parameters x , x an d represent the result in the for m dT ~ 2 1 dT 2 i ifZ-TyzJB.A), (2.23)

Let us find explicit formula s fo r the vector-functions y^ x an d y~ . W e have i-$=-T = e-^BBje-ix'A = Te ix'ad{A)(Bj), (2.24 ) where ad^) denote s commutatio n wit h A J fro m th e left. Not e tha t the following commutation formul a -ix-ad(A)(f(B)-A + g(B)) s k s j ks = fj(B)A J(B)xsA + / (B)xsdkfj(B)A + fi (B)xsdkg(B). holds for any f x (b), ... , fn{b), g(b) . I f we use the shorter notatio n

(Jgyi,A) = f(B)'A + g(B), then these formulas ca n be written as follows -ix • *&{A)( ("Q (1,V) = (£ ) (l,i), where W) [ o (^),a/a6) J \gj- This yields -(x-A+(/ix,d/db)) >ad( 0 2 1 ' ^((^)(5j)) = 0 < ? -(/ix,d/db) (2.25) We note that e~' {MX'9/db) g(b) = g(&(b,x, -t)) an d moreove r - (,.A (^,^)) |_^ e ( + /(fc) = Exp ,u,r-t))dr\f{^{b,x,-t)), (2.26) where the function 38 is defined in (2.20). By applyin g formul a (2.25 ) t o the right-hand sid e o f (2.24) , w e reduce it to the form (2.23) , wher e y~ x(b, a) = 38{b,x, -1) . Further , b y differentiating T(x, x) wit h respect to x an d usin g formula (2.23 ) again , w e obtain

-t(x-A+{ftx,d/db)) 1 .dT Ux i{A j 2 1 - = T f e '* \A )dt=T fT I dt (B,A). j JO JO o V o J 312 CALCULU S OF SYMBOLS AND COMMUTATION RELATION S

Here th e uni t i n th e colum n stand s a t th e jth place . Thi s fac t an d (2.26 ) yield the following explici t form o f the vector-function y^ i n (2.23 )

j k y~x(b, a) = a f Ex p j- f x • L{3S{b, x , r - 0 ) dx\ dt.

2 1 2 1 Denote u = {x9x)9 F(u) = {y^ x(B, A) 9y~ x (B, ^4)) , and rewrite (2.23 ) in simpler for m l^r = T^r^ = r (")(r(")+liu)Hu)- (2 -27)

By calculating the mixed derivatives d T/du adUn, w e obtain the identities

l l J *«, 9u a ' * ' a which mea n tha t th e operator s V(u) + id/du a (fo r diiferen t a ) commut e with eac h other. Thu s (2.27 ) implie s the permutation formul a

p(u9 id/du)T(u) = T(u)p(u, T{u) -f id/du)l(u) for an y function p(u , T) . Le t us choose the function p — p J s o that

J ^ = p (u9T(u) + id/du)l(u). (2.28 )

2 1 By taking int o account that y~ {B , ^4 ) i s independent o f 5 an d x , as 2 1 well a s that y~ x(B, ^4 ) i s independent o f x an d linea r i n A , w e se e that J J the function p = p (x9 x 9 y 9 y) i s independent o f x an d linear in y , i.e., k J ^ = y o k(x 9 y). B y substituting this expression int o (2.28) , we obtain

j k A = A (f Expj - f x-A(^(B 9x9 r-t))dz\ dt\aj{x 9&(B9 x 9 •1)).

y This yield s th e formul a a(x 9 y) = r (x) (i n th e notatio n o f (2.21)) . So , ^ = y V(JC)£ • Further , by (2.28) ,

j j T(u)A = p (u9 id/du)T(u). (2.29 ) Hence, 2 T(u)e =-iyA -i[yp{u,id/du)]r< ? T(u). By the formul a fo r /? 7 obtaine d above , recallin g tha t r(w ) = £~ /x# e -/* and cancelling e~ ix' fro m th e left an d fro m th e right, we obtain B -ix'A —ivA (r (x)y ,d/dx) , -ix*A s e e = e {e ) . CALCULUS OF SYMBOLS AND COMMUTATION RELATIONS 31 3

This formul a an d th e definitio n o f th e functio n z v (b) (2.22 ) impl y th e required formula. Th e theorem i s proved. By integrating th e produc t o f (2.29 ) b y th e Fourie r transfor m g o f th e symbol g = g{b, a) , w e obtain

J j j [g(B9A)]A = Jg(u)T(u)A du = J[p (u, -id/du)g(u)]T(u)du

= {R Ajg)(B,A), where 1 1 2 J k j RAJ = p (id/db, id/da ,b,a) = a r\id/da) k, (2.30 ) In the same way, from (2.25 ) we get ~ix'ABj =^ j{B,x, l)e~ ix'A, and thus j [g(B,A)]B = (R BJg)(B,A), where J RBj=^ (b, id/da, 1) . (2.30a ) 2 1 So, the operators R B, R A of right regular representation of the sets B , A can be calculated by formulas (2.30) , (2.30a). J j We can now verify under what restrictions imposed on A s , p th e latter satisfy th e initia l condition s (2.19) , i.e. , th e generalize d Jacob i condition s hold. LEMMA 2.6. In the case of semilinear relations (2.19), the generalized Ja- cobi conditions are equivalent to the following relations:

* = s ' (j,f,m) { s r +9 sArV ) = 0 > ms KJk , n. m k sj ^ mj sk ^ J J p A ; +d sp P -d sp p =0 , where & denotes summation over cyclic permutations. 2.4. Strongl y nonlinear and solvable relations. Th e problem o f construct - ing operator s o f regula r representatio n fo r genera l algebra s wit h nonlinea r relations i s extremely complicated . W e no w presen t a clas s o f algebra s fo r which this problem can be solved successfully b y using the calculus of ordered operators [96] . We shall follow the outline proposed i n [100 ] and develope d in [56-58, 101]. Consider the relation s

J k s k j A A = ^A (ajs (A )9 l^k^n, 2^j^n, (2.31 ) 5=1 314 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

J where co s ar e give n function s (fo r example , polynomials) , an d co'J{2) = J zd s . I f these functions ar e nonlinear, then the relations (2.31 ) will be called strongly nonlinear. Let us write (2.31 ) in matrix for m rAJ 0 rAl A" rAl A" 0 0 0 0 0 0 coU)(AJ), 0 0 0 ... 0 0 J where the co (£) ar e the matrice s wit h element s co s (£) . B y applying th e quasicommutation formul a (1.16 ) o f Appendix I , we obtain g(AJ) 0 0 A' A" *AX A" 0 g(0) 0 0 0 0 0 g(coU)(AJ)), 0 0 *(0) 0 or g(AJ)Ak =^A s[g(coU)(Aj))] 5=1

Here [...] 5 i s the (s, k)th elemen t of the matrix in square brackets. B y using the permutation formul a severa l times, we get on the following identit y

J i n J i A gCA ,...,A ) = ±A\Dg) s(A ,...,A"). 5=1 The operato r D transform s scala r function s int o matri x function s n vari - ables according to the rule [2>g]to1,...,0 1 u+ +I w [gtf ,...,{', £ V' ), .. . , V «T))tf fo r j

k=\ Thus, w e obtain the following result .

THEOREM 2.3. If the inverse operator 3~ exists, the operators

5=1 are left operators of the ordered regular representation of the set A , .. satisfying relations (2.31). The formal conditio n fo r the existence o f 2~ ha s the for m CALCULUS OF SYMBOLS AND COMMUTATION RELATIONS 315 for j = 2, ... , n . In particular, it holds if wf ({ ) = *f £ + yf , det((x a})) ^ 0. Anothe r important case arises when all the matrices aft ^ (£) ar e triangular. Here the operator 3J~ X ca n be calculated explicitly . Suc h relation s wil l be called solvable. EXAMPLE 2.5. Cyclic anticommutation relations [100]. A'A2+A2A1=A\ A2A'+A'A2 = A\ A*Al+AlA3 = A2. (2.32 ) These relations are realized, for example , by the matrices 0 0 0 " "0 0 1 " "0 1 0 1 A 0 0 1 ' A2 = 0 0 0 , A' = 1 0 0 0 1 0 .1 0 0 . 0 0 0 In this case the operator & act s according to the formul a

j j (®G) s = J2b*(-id/dZ,hG k(Z), k=\ where ' -2iZlx}-2

We note that de t b(x, £ ) vanishe s for x 2 — n/2 + nk , k — 0, ± 1, ±2, ... . The operator 2J i s invertible on functions o n ^(R3) whos e Fourier transforms i n the variable £ hav e supports not containing the points n/2 + nk 1 7 (for an y £ , { ) . This class of functions wil l be denoted by Sf ; it includes, in particular, al l the polynomials. So, w e have \l 3 3 22 33, 33 [&(Al, A\ A J)]. [f(A l, A\ A 3)] = (&*f)(A l, A for an y / e 8? an d an y polynomial & o n R , where

li 2 ? 3 . and the operators L 1 \%? -+ 3? ar e defined b y the formula s

Lj = l j(-id/d£,h,

/ (x , £ ) = £ LL * . _j _ z£ / sm v + t£ r tanx2 cosx3, cosx

,2 . xir T smx 3 , ,2r K 3 (2*3 -f £ L cos x. - £ L tan x2 sin x3, / = ii LI —cosx^. — /3=«3, 316 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S where I k i s the inversion (chang e of sign) with respect to the variable £ . I t is easy to verify that the operators l) , L2 , L3 satisf y relations (2.32). Thus , the generalized Jacobi conditions hold for (2.32) . We point ou t that the operators obtaine d i n this exampl e are local: thei r symbols posses s a singularit y fo r cosx 2 = 0 . Suc h loca l operator s ca n b e constructed fo r a wide class of graded Lie-Jordan algebra s [68, 107] . Globa l versions o f th e operator s L J, includin g thos e fo r relation s (2.32) , wil l b e considered below . EXAMPLE 2.6 . Degenerate Faddeev-Sklyanin relations. Fro m (2.13 ) fo r

Jx = J 2 = l ? J 3 — l + co , where to ^ 0 , to ^ 1 , w e obtain th e followin g relations connecting the four generator s S 0, S x, S 2, S 3 (se e [124, 127] )

[S0, 5, ] = ia>(S 2S3 + S3S2), [S 0, S 2] = -i(o(S,S 3 + Vi). { [S0,S3] = 0, [S a,Sfi] = i(S0Sy + SyS0). • >

If w e introduc e ne w generator s i n thi s algebr a a ± = S x ± iS 2, b ± = ±S0 + y/toS 3 , then relatio n (2.33 ) ca n be rewritten a s follow s / [b+ , a±] = db v w(b+a± + a±b+), [b _ , a±] = Tv^(b_a± + a±b_),

[b+ , bJ = 0, [a + , a_] = V^(b' - bL) . It is useful t o represent them not as commutation relations, but as permutation relation s

b+a+ = a a+b+ , b_a + = - a_> _ , b +a_ = - a_b +,

2 b_a_ = aa_b_ , [b + , b_] = 0, [a +, a_] = ~(b* - b _), (2.34) where a - ( 1 + y/co)/{l - y/to) . Obviously, these relations are solvable. It is easy to calculate the left regula r representation o f the set

4 3 2 1 t>+>b_, a+, a_ . The scala r variable s relate d t o th e element s o f thi s se t wil l b e denote d b y w , z, y , x . The n L b = w , L h = z . Further , b y the formulas o f quasi - commutation an d by the firs t two relations (2.34) , we have

a+ [/(C b_ , 1+, aj] = a+/^b+, b_ , l+, a_ J =a+/^b+, ab_ , 1+, a_J .

a /a a Thus, L a = y • 7z • 7^ , where 7 z denote s the operator o f a-dilatio n wit h respect to the variable z I°f{w , z, j;, x ) = f /(tu , QZ , j;, x). Similarly, we obtain the chain o f permutations fo r the element a _

53 4321 4 21 3 / 5 1 4 2 l \ a_[/(b+ , b_ , a+ , a_)] = a_/(ab+ , b_ , a+ , aj = a_/l ab + , -b_ , a+ , a_ I , CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 317 and then apply the commutation formula taking into account the last relations (2.34)

43 2 1 i / 4 1 3 2 l \ a_[/(b+, b _ , a+ , a J] = a_/l ab +, -b_ , a+, a _ J

3 + ^TT(b- " b+)8yf V ab+' a h-> a +'a+'a-J•

(2.35)

We now use the permutation relation s

2 2 2 2 a+b - bi(a a+), a +b = b (^i«+) •

This yields

2 2 2 (b - b +)Syf(ab+, -b _ , a+, a +, a _ J = bi^/Ub+, -b _ , a l+, a +, a _ J 2 2 2 K -bi+*

We calculate the difference derivative s in this formul a

V(...,aV,y,...) = ^^-a2j''-)-/(-'J''-) (a - \)y 2 f(...,y,...)-/(...,(l/a 2)y,...) */ (•••.-T>'.J'.---) = -T LT \ a / a — 1 Thus, by substituting al l this into (2.35) , we obtain

\1 a 1/ a X + 9 —( 4 -l ) + a (/ / - 1 ) J w z {a+\)2\y y y y

It is easy to verify that the operators of left regular representation L h , L h ,

La , L a constructe d above satisfy relations (2.8), i.e., the generalized Jacobi formulas an d the PBW property hold . We now make the inverse linear substitutio n

52 = 1(L.+ -L._) S 1

We obtai n th e followin g representatio n o f th e algebr a (2.33 ) i n term s o f 318 CALCULU S OF SYMBOLS AND COMMUTATION RELATION S difference operator s

Sxf = ~^-f{aw,z/a 9y9 x) 2 f{aw, zja, *)- f(aw, z/a, j>> X) + U+i/ 2y aw \ f{aw, z/a, j/a2 - f(aw , z/a ,j> , x) a~TTj 5 / = -^-/(aio ,z/a,y,x) 2 2/ z \ 2f(aw, z/a,a 2y,x)-f{aw, z/a,y,x) a + 1 / 2/ y ( aw \ 2 /(aw , z/a, y/a2 , x) - f(aw , z/a,y,x) ~ Va+1/ 2iy a+ 1 z w w *V = ~—2(^~17^() + )f( > *,y,x), w - z sof = —y-/(w ,z,y,x).

EXAMPLE 2.7 . Quadratic-linear relations. Conside r on e more simpl e example of solvable relations [100] BA = aAB + aC, BC = yCB + vB, CA = pAC + juA, (2.36 ) where a ^ 0, P ^ 0, y^O, fi, v , a ar e certain constants. It is easy to find conditions on the constants guaranteeing the PBW property for polynomials of degree not greater than three in the generators A , B, C. Fo r this purpose, w e permute the neighboring factor s i n the monomial ABC accordin g to the diagram ABC • CAB y N . ABC CBA. N y BAC • BCA The upper chain yield s ABC = A(yCB + vB) = y (icA - ^Aj B + vAB

= L C(±BA-°c) + (v-^)AB. P The lower chain will be

ABC = (iBA -°-C)C= l -B (ICA - %A) - -C \a a J a \p p J a = \{yCB + vB)A - -^BA - -C 2. ap ap a CALCULUS OF SYMBOLS AND COMMUTATION RELATIONS 31 9

By setting these two expressions equa l and taking into accoun t the relation between BA and AB, we obtain

fi afi ap This yields

either a = 0, i/( l - fi) = //(l - 7), o r fi = y9u = /i. (2.37 ) Actually, thes e condition s ar e generalized Jacobi identities for relations (2.36), i.e. , they ensur e tha t th e PBW property hold s i n the whole algebr a generated by these relations. W e shall prove this by constructing the operators of regular representation . We introduce the dilation operato r /7«f) = /(*), /"''/« ) = f{aP~{Lm = W, flf), (a - p) c 7 /(of) + f(PZ) + /(tf) /«) = ? (j8-a)(y-a) (a - fi){y - fi) ( a - y)(j 9 - y). = (5 2/K,^,^) . in the space of polynomials in one variable £ . Such operators in the space of polynomials in three variables f(^ x, f , £3) actin g with respect to the /t h a,p variable will be denoted by I" , /. , 7 f. By the quasicommutation formulas , w e see that (2.36 ) give s

4123 3124 3 1233 Cf(C9B9A) = Cf(C,B,fiA) + iiA8zf{C,B9A9 fiA). The first summand in the right-hand side will be transformed agai n by means of the quasicommutation formul a

3 12 4 1/112 3\ v 2 / 1 2 1 2 3\

Cf(C9B9fiA) = Cf [C9 -B 9 fiA) - -BS 2f[C9B9 -B 9 fiA) .

By using the dilation operators given above, we write this relation as follows 4 1 2 3 1 2 3 Cf(C9 B 9 A) = fx{C9B9A)9 f x = L cf9 where l y P /y l 1 Lc = e^I 2' --ei j2 ' +^' . (2.38a ) In the same way, for the generator B w e have

4123 212 3 4 1235 Bf(C 9 B9A) = Bf{C ,B 9aA) + aCd3f(C 9 B\A9 aA). The second summand wil l be transformed b y means of the permutation for - mula obtained abov e for the generator C . We get

4 12 3 12 3 Bf(C9 B 9 A) = f2(C9B9A)9 f 2 = LBf9 320 CALCULU S OF SYMBOLS AND COMMUTATION RELATION S where 1 e X LB = a^I-'Oiy + fil" - Hi^iy* ) + afifl^ ' . (2.38b )

And of course the operator L A i s trivial for A: 4 12 3 12 3

Af{C9 5 , A) = /3(C, B 9A)9 f 3 = L Af9 where LA = £?. (2.38c ) Formulas (2.38a,b,c ) defin e th e operator s o f lef t regula r representatio n fo r 1 2 3 the ordered se t of elements C , 5 , A o f the algebra (2.36) . We now calculate the permutation relations among the operators L c , L B , LA b y using the following relation s among the dilation operator s Q 7 c >e* = a£ 0/", a a p a a I '"c >£ == # oT ' + I = a£o 7 •' + /', p,P,rc ,/«./». r y..t >£ == yf' + t COlT ,fi = ,/Q-

2 Tco^a, B .y _ a /»,77» ) I I = f ' 5 a JP,QJ<*,/> _ = 21/ <*P7P,P (we note that these relations are of "quadratic" character; i t is precisely this fact that implies that the operators of regular representation o f the quadratic relations (2.36 ) ca n be written in terms o f the dilation operators) . By straightforward calculatio n w e can prove

LEMMA 2.7. The operators Lc, L B, L A satisfy the initial relations (2.36), i.e.,

LBLA = aL ALB + oLc , L BLC = yLQLB + vLB ,

LCLA = PL ALC + HL A if and only if conditions (2.37 ) hold. COROLLARY 2.1. The conditions (2.37) are necessary and sufficient in order that each element of the algebra generated by the generators A, B, C and 3 2 1 the relations (2.36) be uniquely representable as a polynomial in A, B, C. The product of two elements is given by the formula

[&(A9 B 9 C) ] • [*(A9B9 C)] = {&*a)(A9B9 C). Here * is the associative operation of multiplication in the algebra of polynomials in three variables defined as follows

d &*& =&>{LA,LB,Lc)@. Moreover, 3? * 1 = = 1 * & = 3°. CALCULUS OF SYMBOLS AND COMMUTATION RELATIONS 321

2.5. Quantu m Yang-Baxte r equation . I t i s ofte n difficul t t o investigat e the permutatio n relation s directly . I n suc h case s i t ma y b e usefu l t o con - sider different mapping s of the generators of the initial algebra, for example , mappings suc h as the chang e of variables in Exampl e 2.6. Th e general tran- formation o f suc h type may be sought in the for m

Aa = Y^°s®aS> ( 2-4°) where the A a ar e the initia l generators , the a s ar e the ne w generators, th e a* ar e auxiliary factors . We shal l conside r th e specifi c cas e whe n th e se t o f initia l generator s i s ordered a s a square (n x «)-matrix A = ((A^)), an d instea d o f the index a in (2.40) , we shall write the pair of indices ( M wher e a i s the matrix ro w number, / ? i s the column number . So we have the following chang e of variables

o r s ^f = £(*,)£ ®* * A = os®a . (2.41 ) s

We want to se e how the permutation relation s amon g the elements A^ an d the permutation relation s among the elements a s ar e related. The logi c of ou r consideration s wil l be reversed: w e begin wit h a certai n algebra generated by a s an d conside r the relations obtained amon g the elements (2.41 ) under a special change of the numerical matrice s o s. SupposeT V is an algebr a wit h uni t / . A n elemen t R eT V ®T V will be called a Yang-Baxter element i f we have the identit y

R12R13R23 = R 23R13R12^ ^ ^

12 s Here R , • • eT VT VT V are define d a s follows : i f R = a s ® b , the n 12 s 13 s 23 s R = a s ® b /, R = a s / 0 b , R = / 0 a s<8 > b . We note that eac h element Re N ® N define s a linear mapping TV *— • N by the formul a s TV* 3 / - ( / 0 id) R = l{a s)b e N. (2.43 ) We no w assum e tha tT V c Mat(w , C ) i s a subalgebr a i n th e algebr a o f (n x Az)-matrices and that a nondegenerate linear mapping TV *— •T V is defined s by means of a certain Yang-Baxter element R = a s<8 > b . W e can assume that {as} i s a basis in TV . The nondegenerac y conditio n mean s that th e matrice s {b s} als o for m a basis i n TV . Suc h a Yang-Baxter elemen t wil l be calle d nondegenerate, and the subalgebraT V will be called a Yang-Baxter algebra or a YB-algebra (ther e is still no generally accepted terminology here; see [49, 175]) . 322 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

We transport th e multiplication fro m th e algebra N int o M = N* an d consider the functionals f e M define d a s matrix element s l P(a) d = a^, a eN .By R fifi', w e denote the matrix elements (l fi /^)(R).

LEMMA 2.8. The following permutation relations

<<»=<<«• (2-44 ) hold in the algebra dual to a YB-algebra.

y PROOF. Th e element R ^ = (l fi ® id)(R) = ((R^O ) e N correspond s to the functiona l ll unde r th e isomorphism M —> N. Relation s (2.44 ) ar e equivalent to the following relations in N R^R^=R";X<' (2.42a ) which, in its turn, coincide with (2.42) . The lemma is proved. REMARK 2.2. Equatio n (2.42 ) is called the quantum Yang-Baxter equation [154, 266]. It is considered in the method of the quantum inverse problem [85, 125, 203]. In this theory, the relations (2.44 ) are the basic ones. A n abstract algebra M R with these relations is called the Faddeev-Zamolodchikov algebra (or ZF-algebra) [131,267] . Condition (2.42 ) implie s that the PBW-property hold s in this algebr a on polynomials of degree not more than 3. Lemma 2.8 states that an algebra M adjoint t o a YB-algebra is a representation o f the ZF-algebra M R. s s COROLLARY 2.2. To each representation of a YB-algebra N — • N 0, b — • a there corresponds a representation of the FZ-algebra M R — • M— • NQ. In particular, the elements (2.41 ) satisfy the Faddeev-Zamolodchikov relations y A M®M, A,(/)( a ® a) = f l{aa').

Since A ^ preserves the permutation relations (2.44), A + i s a homomorphism. And this automatically implie s that A is a homomorphism. Thus , the algebras N an d M becom e Hopf algebras. A detailed discussio n o f the Yang- Baxter equation s fro m th e point o f view o f the theory o f Hopf algebra s is given in [49, 94]. In term s o f the structure constant s o f the algebras N an d M a sam = a s m s m r *sm r' b b = V r b the comultiplication A can be calculated by means of the formul a CALCULUS OF SYMBOLS AND COMMUTATION RELATIONS 323 and sinc e A i s a homomorphism, th e following relations hold

m kk' rr it' k k' ,~ A ~^

We note that relations (2.44 ) i n terms of comultiplication hav e the for m R - A( TV , i.e., a®b ^ b <8>a). REMARK 2.4. Suppos e the element R €T VT V is degenerate. The n (2.43 ) is not injective. Nevertheless , let the multiplication be defined on TV * so that (2.45) i s an algebr a homomorphism an d moreove r relations (2.44 ) hold . I n this cas eT V will also be called a Yang-Baxter algebra. (Obviously , equatio n (2.42) an d relation s (2.44a ) hol d automatically. ) I f th e mappin g adjoin t t o the homomorphis m (2.43 ) i s a n anitihomomorphism ,T V wil l b e calle d a Yang-Baxter bialgebra. EXAMPLE 2.8. Quantum groups [49, 120 , 146 , 175 , 196 , 263-265]. Sup - pose JV i s a Lie group, an d n i s its Li e algebra. W e assume that Jf an d n ar e realized in the algebr a o f matrice s Mat(n , C). W e denote by % th e universal envelopin g algebr a o f n . Suppos e /c^ an d JV i s defined i n ^ b y a set of polynomial relation s ^'(...) = 0 ( / = 1 , ... , k ) among the matrix element s (fo r example , the group SL(2 , C) i s defined i n Mat(2 , C ) by the relation det(... ) - 1 = 0). The algebra %* i s abelian; the corresponding comultiplication i n ^ ha s the for m

A (a k) = I®ak + (J k®I. Obviously % i s the Yang-Baxter bialgebra correspondin g to the identity element R = I <8> I. A Yang-Baxte r bialgebr aT V = 2 ^ c Mat(/i , C ) wil l b e calle d a formal quantization of the algebra % i f th e structur e o f thi s algebr a depend s smoothly o n the parameter h — • 0, coincides with % i n the limit , and the limit o f the commutato r lirn^...,...]^ defines a Poisson algebra structure in %* . We note that th e function s o n ^ ar e functions o f matri x elements , i.e. , of linea r functiona l fro m %* . Sinc e th e bracke t betwee n functiona l i s defined, th e Poisson bracket i s defined o n ^ an d on Jf . We no w suppos e tha t ther e exis t polynomial s ^ i n generator s o f th e algebra 2 £ whic h li e i n th e cente r o f thi s algebr a an d i n th e limi t h - > 0 coincide wit h the polynomia l relation s & l amon g th e matri x element s o f l the group Jf . The n the algebr a M = 2^ * wit h additional relations 3P h = 0 (/ = 1 , ... , k ) is called the algebra of functions on the quantum group jV h . The quantum group JVh itsel f is defined as the spectrum (the set of irreducible representations) o f this algebra: JV h = spec(M). 324 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

In orde r to se e the structur e o f this algebr a mor e precisely , conside r th e first approximatio n wit h respec t t o h i n th e deformatio n Jf — > Jf h . W e have R = / + ihr + 0(h 2), A = A ° + zftA 1 + 0(h 2), (2.47 ) where Al/ x sm ^ ~ ink ^ A (** ) = " * G s®Gm> r = r G m®°k' In the firs t approximatio n i n h relation s (2.44a ) amon g the generators o f the algebra M R hav e the for m [Al, Z] = -ih^A\A\, - AlA(,r%,) + 0(h 2) or in matrix notatio n [A®A] = -ih[7, A®A] + 0(h2). (2.48 ) From (2.44 ) or , which is the same, from (2.46) , we have in the first order in h mn A , nl r m dm e\ ACW r fns + r fns=K > ( 2-49) where we have denoted

r\ , I / N . Am Im ml Note that the increments A 1 an d 7 i n (2.47) depend in an effective wa y only on th e basis element s o s e n fro m th e Li e algebra. Thu s th e number s f ns coincide wit h th e structur e constant s o f thi s Li e algebra , an d th e number s m A5 defin e the structure o f a dual Lie algebra o n n * (se e (1.48), Chapter I). Because o f (2.42) , the elemen t 7 satisfie s th e specia l Yang-Baxte r equation , and hence the Jacobi identity holds for X 1™ . We note that th e Hopf condition s (2.45 ) ar e the sam e a s relations (1.28 ) of Chapte r I i n the first approximation wit h respec t t o h . B y (2.49 ) thes e relations hold automatically . So we have obtained infinitesimall y th e structure of a Lie bialgebra on n . Actually the Poisson bracket on % an d on JV i s not only infinitesimally bu t also globally multiplicativ e wit h respec t t o matri x multiplicatio n (se e 1. 3 i n Chapter I) . This easil y follow s fro m th e fac t tha t th e algebr a 2 ^ i s a Hop f algebra. Incidentally, let us point out an important property o f relation (2.48) : th e 7 determinant det((^) ) commute s with eac h generator A a, to mo d 0(h ) i f n c sl{n) (i.e. , i f th e trace s Xra k vanis h fo r al l a k en) . Simila r Casimi r 7 functions ar e related to the minors o f the matrix {(A a)). A quantum analo g of this property is considered in works dealing with the method of the inverse problem. 2.6. Reductio n to triangular form. W e now consider algebras with genera l relations i i i m k i s A A = u> ikA A + n >A . (2.50 ) CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 325

Summation is taken over repeated indices from 1 to n . The complex coeffi - cients co and n i n (2.50) (w e call them scale and structure constants) satisf y the following relation s

^mXp^r = ^L^Tq^fp ( the Yang-Baxte r equation), l J co™ fi s = -fi™ (antisymmetricity) ,

VkV? ~ ^k^ - ^miH^r = ° ( the Jacobi identity) , il jr jr Ik im Ir ij , jr im Ik ij mr kl ij kr ~ ffl %/i + mt%/; - %„<" / + °>mkc>ql Hp - co kmco!p nq - co qkfip = 0 (the compatibility condition) . (2.51) These relations imply that the PBW-property hold s in the algebra (2.50 ) on polynomials of degree 3. The first relation in (2.51) coincides with (2.42a) if we choose R^ = co^m . Besides the relations given above, it is sometimes convenient to requir e mk ij ?m ?k mm tJ ^SjSt w'L = <°Li tf = < (the Hermitian property) , where th e bar mean s comple x conjugation . Th e unitarit y conditio n mean s that relation s (2.50 ) ar e equivalent to the Lie-Jordan conditions , sinc e co = /, i.e. , the spectru m o f the tensor co consists onl y of ±1. Th e Hermitia n property means that the relations (2.50) admit a representation by selfadjoint operators A 1. We no w conside r a n important versio n o f the permutatio n relation s in which we have a//' = 3s Si + csJ , c\ = csJ = -cJ\ , u j = -ujs. (2.52 ) km m k km ' mk km mk ' ^r ^r \*-.~>t~) In this case (2.50 ) has the for m m k k m J r H'. ^1 = jC(A A + A A ) + M'r A . (2.53 ) Here all the commutators between the generators can be linearly expresse d in terms of anticommutators an d of the generators themselves. Suc h relations are calle d quadratic commutational. The y ar e interestin g fro m th e poin t of view o f the quantizatio n procedur e (se e 4. 1 i n Chapter IV ) an d from the purely algebraic point o f view because of the followin g property . s THEOREM 2.4 . If the tensor c ^k in the quadratic commutation relations (2.53) is nondegenerate (i.e., all anticommutators between the generators can be expressed linearly in terms of commutators and of the generators themselves), then relations (2.53) can be reduced to the "triangular" form

} m k k m J [A\A } = \ £ c'J k{A A +A A ) + K A', (2.54 ) m,k^m'm{s ,j) 326 CALCULUS OF SYMBOL S AND COMMUTATIO N RELATION S by a linear transformation, and the coefficients c, pi satisfy the same relations as c, ft.

k 3k PROOF. Denot e L m = ((oj m))i j=u n . The n th e Yang-Baxte r identit y q J J in (2.51 ) mean s tha t a>™ LgL m = co mlLrL™ or , takin g int o accoun t th e k J q k j k l structure o f the tensor co in (2.53) , that [L , L r] + c™ L qL m = c\ mL rL™ . k i J Interchange the indices n <—• j, s «— • r and , by (2.52), obtain —[L s , L r]- q J J c™s L mLq = -c lmL™Lr. B y adding this to the previous relation, we get C^'^^O (2-55 ) Then transforming th e right-hand sid e cjk[L', L m] = -d k[L', L m] = -c m\Lk ,L j] = -c mq\Lk , Lj 1 (here w e used (2.55 ) i n the secon d equality) . B y comparing this expressio n J with (2.55) , we see that c lm[Lr, L™] = 0, o r since the tensor c i s invertible, l [L r,L™] = 0 fo r al l / , r , m, s. So we have a set of commuting matrices. Accordin g to the Schur theorem, there exist s an orthogonal matri x U whic h reduce s al l the L r t o triangula r form - 1 / ° / ° / O c / U LU = L , L = ((co )) r r ' r "> mr"m ,5=1,..., « » s where o)^ r = 5^5 r - h c mr, c^ r = 0 for m > s. I t remains to choos e s 5 5 sl lr il s lr lp I = t/ ^', ? i = °c uiu7 = c uiu ur u~ , j ' m/ : mr I k pr I i k m ' ~5y kl TT s TT j TT —\m v — v U, U, U r m k I r The theorem i s proved. J COROLLARY 2.3. If the tensor c ~ ((c im)) is nondegenerate, and all the ma- jk trices {(c n))i • j n are normal with respect to a certain scalar product (i.e., they commute with their adjoints), then the relations (2.53 ) can be reduced to linear commutation relations (i.e., to a Lie algebra) by a linear change of variables.

PROOF. Sinc e all the matrices L r ar e normal, they can be put in diagona l s form, i.e. , c ^r = 0 fo r m / s . Thus , c^ k = 0 fo r m / s an d m / j, which means that these coefficient s ar e identically zero : c^ k = 0 fo r al l s , j , m , k . The corollary i s proved. Sklyanin's relation s (2.13 ) giv e a n exampl e i n whic h th e nondegenerac y condition i n Theore m 2. 4 i s fulfilled . Thus , the y reduc e t o th e triangula r form (2.54) . Now let us classify the relations (2.54 ) i n the simplest case of three generators, i.e., n = 3 . CALCULUS OF SYMBOL S AND COMMUTATION RELATION S 327

EXAMPLE 2.9. Triangular commutation relations with three generators 2 [A, B] = aA + kA + lB + gC 9 2 2 [B,C] = PA + yAB + oB + qA + rB + nC9 (2.56 ) [4, C] = SA2 + tA + sB+pC. Here relations (2.54) are written(2) for the case n = 3. The coefficients mus t be subject to identities of form (2.51) . In particular, the Yang-Baxter condition give s ay = ao = 0 . Obviously , without los s of generality, we can se t a = 0. Actually , if a ^ 0 , the n y = a = 0. No w if (5 = 0, then, by changing the notation B —• C, C — • J 5 , we arrive at relations (2.56) wit h a = 0. Bu t if 8 ^ 0 , then , changin g B to B f = B - (a/S)C , we again reduce (2.56) to the case a = 0 . So let a = 0. Th e compatibility conditon in (2.51 ) yields #(7 = 0, #( y + 2(5 ) = 0, goy+py = 2ka + 2l8, 2 2 go + po =pa 9 g{

[A,B] = kA + lB + gC 9 2 [B9 C] = A + qA + rB + nC 9

[A9C] = tA + sB- IC. The Jacob i identitie s (2.51 ) fo r th e structur e constant s ar e reduce d t o th e following types (la) g = I, k = / = nr = nt — ns — 0, r = —t\ (lb) g = 0, l{k-n) = 09 l(t + s) = s(k-n), l(q - t) = tn + kr . Case II. c r = 0 , y , * (5. Then , b y the chang e B 1 = B + Qff/( y - 5))^ , we can exclud e the term i n A i n the secon d lin e o f (2.56) . Thus , w e can assume here that / ? = 0. W e obtain [A,B] = kA + lB + gC, [B, C] = yAB + qA + rB + nC, [A,C] = 5A2 + tA + sB+pC. The compatibility conditions and the Jacobi identities give the following versions (Ha) g = 1 , y = -2, r = -t, n = S=l, p = 0;

( )Th e fact that the second line of (2.56) contains AB instea d of the anticommutator AB + BA a s it was in (2.54) does not play any role since, by the first relation in (2.56), AB + BA ca n be expressed as a linear combination of AB , A an d A , B , C . 328 CALCULUS OF SYMBOLS AND COMMUTATIO N RELATION S

(lib) £ = 0, py = 2lS, k(S + y) + n6 = 0, kp + ln = 0, l2S-lt = {n-p- k)s , l{q + kS) + t{p - n) = kr . Case III. 7 = 5^0. B y changin g th e normalization , w e ca n obtai n y = S = 1. Her e we get the relations [A,B] = kA, [B,C] = A2 + AB + qA + rB- 2kC , [A,C] = A2 + tA + s£, and the Jacobi identities yield ks = k(r - 2t) — 0. Case IV. g = 0, c r ^ 0 , ft ^ 0. B y changing the normalization, w e can obtain c r = 1 and by the change B f = B + (y/2a)A, w e can exclude the term AB i n the second line of (2.56) . So , we can assume here that y = 0. Henc e we obtain

[A9B] = kA + lB, [B,C] = PA2 + B2 + qA + rB + nC, [A, C] = SA2 + tA + sB+pC. The compatibility condition s an d th e Jacobi identitie s giv e the followin g versions (IVa) k = l = n=p = 0; (IVb) k = l=p = s = t = d = 0, n = l. Case V. g = / ? = 0 , cr/0 . Precisel y a s above , w e reduc e th e initia l relations to the case a = 1, y = 0. W e obtain [^, B] = fcy4 + /£,

The following condition s o n the coefficient s ar e possibl e

(Va) k = q = n=p = S = 0> r = t9 /=l ; (Vb) fc = -l, l = n=p = S =1, 5 = -r, r = -$; (Vc) k = l = n=p = s = 0, (Vd) fc = / = f = .s=p = < 5 = 0, n=l . So al l possibl e type s o f triangula r relation s (2.56 ) ar e given . Th e type s (II) an d (V ) wher e / ? = 0 belon g t o th e clas s o f relation s (2.31) . W e ca n construct a regular representation fo r them b y using the method give n in 2. 4 3 2 1 3 2 1 (for example , fo r th e ordering s ^ , B , C o r 5, ^ , C) . W e leave this a s an exercise to the reader . 2.7. Spectru m an d cospectrum o f quadratic-linear relations. W e now present a n approac h t o the problem o f constructing a regular representation o f the general quadratic-linear relation s (2.50 ) base d o n their inne r symmetry . CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 329

Precisely a s in the previou s section , w e shal l us e change s o f variables of type (2.40) , but not restrict ourselve s to scalar coefficietn s o" . No w assume that the a " are matrices or, in general, noncommuting linea r operators. W e impose conditions o n the m s o that th e clas s of relation s (2.50 ) remain s in - variant wit h respec t t o th e chang e (2.40 ) an d whic h lead t o relation s o f a simpler form . The ide a o f introducin g suc h noncommutin g (fo r example , anticommut - ing) factor s originate d i n th e theor y o f superalgebras an d supergroup s [12 , 218] an d i s actively use d in "noncommutative geometry " [164 , 175 , 264]. The effectivenes s o f such operato r change s o f variable s wa s show n i n [68], where an example of certain classes of quadratic-linear relations was considered. Her e w e shall present the method give n i n that pape r directl y fo r th e general case. Let M b e th e algebr a wit h generator s {A 1} an d permutatio n relation s (2.50). Denot e th e linea r envelop e o f generators b y 3* . Th e scal e an d structure constants in (2.50 ) ca n be regarded a s linear mappings. co e Hom(^ 0«5 * -> & ®&), tie Hom(^ ® .2* -> &), co(Al 0 Aj) = co^A™ 0 Ak, ii{A l 0 Aj) = $A k. Let M b e anothe r algebr a wit h generator s {A 1} and correspondin g scal e and structur e constants co , Ji . Als o let P be a certain algebr a with unit /. An element deHom(J?^5?)®P (2.57 ) will be called a P-homomorphism o f relations (2.50 ) if (co 01)d®2 = d®2(cb 0 /), (/i 01)d®2 = d(Ji 0 /). (2.58 ) The tensor squar e d® 2 e Horn {&%>&->&<&&) ®P i s define d i n th e natural way: 2 l j d® (a ®b) = asbk{A 0 A) 0 d\ • d) s for an y a = asA , b = b kA fro m i ? ; the d\ e P are matrix element s o f d. The conditions (2.58 ) have the following inde x for m

wrpdi • dj = comldr • dp, n mdt • dj = ii, d m. (2.58a ) These conditions immediately imply that the element s l s l B = A ®d s€5?®P (2.59 ) satisfy th e "tilde" relations i J i m k s B B = 5> ikB B + ^B , (iT0 ) i.e., they defin e a representation o f the algebr a M. S o we obtain a homomorphism o f algebra s 330 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

If d e Hom(J?— • JS?)®P i s a certain new P-homomorphism o f relations (2.50), then the composition d ® d i s a P P-homomorphism o f the initial relations (2.50). In particular, if deliom(^f — •«57)®P( , where P(_) is antihomomorphic to P, then th e element d <8> d e Hom(^ -^yj^P® P {_) generate s the composition o f algebra homomorphism s M-+M®P{~] -+ M®P®P{~\ (2.60 ) i.e., the elements B l = Bl ® d\ = As d\ 0 d\ satisf y th e initial relation s i i i m k i j s (2.50) B B = (D tllkB B + n s B . If the elements d an d d ar e right-invertible, the homomorphism (2.60 ) is injective. The n the functions o f generators of the algebra M ca n be expressed in terms of functions o f generators of the algebra M an d of matrix elements of d . An d vice versa. Th e algebras M and M ar e equivalent in this sense. Two algebras with quadratic-linear relations will be called equivalent if they are related by a right-invertible P-homomorphism . Th e relations themselves will also be called equivalent. We point ou t that equivalen t algebra s are not necessarily isomorphic, but if P i s one-dimensional (i.e. , consists of scalars), then this is true. In a given clas s of equivalent algebra s an y right-invertible P-homomor - phism d wil l be called a symmetry, an d the corresponding algebra P wil l be called algebra of symmetries. Here w e assume tha t th e algebr a o f symmetries i s the minimal algebr a generated by matrix element s of the symmetry d l.. (_) REMARK 2.5 . In the role of the algebra P in (2.60), it is convenient to tak e th e subalgebra i n Hom( P— • P) generate d b y operators o f right multiplication r(q) by elements q e P, and as the symmetry d , t o take the operator d = r(d ) . The adjoint operator s fro m Hom( P— > P ) wil l be denoted by n = r(d~1)*. So , we set d) = r(djU), n'^r(d; U)\

~~Here d~ ] i s right invers e to d , i.e. , d s.d~u = d\ -1. s J j ^ REMARK 2.6. Th e set of elements of the form d®d generates a subalgebra (_) MR c Hom(^ -> &) 0 P 0 P ,~~

7 which i s the ZF-algebra correspondin g t o the Yang-Baxter elemen t R[ W = ac 1 s l ^Ink ( tually, th e matrix element s A . = d . 0 d s satisfy , a s can be easily seen, the relations (2.44a)). It follows from (2.60 ) that the elements M R ca n be identified wit h automorphisms o f the algebra M , i.e. , M R «-• Aut(Af). Now we shall consider operators of regular representation an d generalized shift. CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 331

So, suppose M an d M ar e equivalent algebras with quadratic-linear relations. W e shall show that i f a regular representation i s known for M , i t can also be constructed fo r th e algebr a M . Bu t first of al l we change our poin t of view on the notion o f symbol . Denote by B = A®d th e set of elements (2.59). B y f(B) = f(A d) w e denote the Weyl function o f these elements with polynomial symbo l / . The element f(A ® d) belong s to M ® P. Usin g the secon d componen t of this tensor product, we can define a pairing with any functional fro m P* . In particular , w e ca n assum e tha t th e symbol f itself takes values in P* . We denote the space of suc h P*-value d polynomials in n variable s by £? n . Then a n element f(A) e M i s defined fo r an y / e & n . Fo r example, if / a can be expanded with respect to the basis f = Y1 X fa > where x° € P*, an d the f a ar e ordinary scala r symbols, then f(A) = J2(Xa>f«(A®d»2> ( 2-61) where the functional x" ac t via the second component of the tensor product M P (thi s is reflected b y the subscript 2 at the angle brackets). Further, / wil l be called the vector-symbol of the element f(A) .

LEMMA 2.9. Suppose the relations (2.50 ) possess right regular representation in terms of operators Rl on the class of scalar symbols. Then the relations (2.50) possess a right regular representation on vector-symbols; more precisely f{A)Al = (R'fiiA) , R ( = RS® 7 £ (2.62 ) And in this case, the algebra M coincides with the set of elements {f(A)\f e £Pn). If the generalized Jacobi conditions hold for (2.50) , they also hold for the relations (2.50). Further f(A). g{A) = (/* g)(A) , / * g d ±f g(R)f, (2.62a ) and the almost ^-product of vector-symbols defines on & n the structure of an associative algebra with right unit.

~~PROOF. Fo r the sake of clarity, we shall use definition (2.61) . W e have J s U U s f(A)A = J2(x\f a{B)-B {I^d; )) = Y,{r{d; )*x\fa{B)B )~~

~~= E<<^ • (^/Q)W) = £~~

~~Further, we see that the generalized Jacobi conditions for (2.50 ) i n terms of operators o f right regular representation hav e the for m~~

~~R R = comiR R + fil R (these relations are dual to (2.50)) . O n the other hand, the matrix element s l n s satisf y th e relation s~~

~~ij p r ~sk j i ij s ~sk j i , ~ r0 i_\~~

~~This follows fro m th e definition o f it an d (2.58a) . By direct calculations , w e obtain the followin g relation s for the operator s Rl fro m th e identities written abov e~~

J i i J m r J l R R ^oj r mR R + M i R , dual t o (2.50) ; thi s mean s tha t th e generalize d Jacob i condition s hol d fo r (2.50). The formula fo r th e *-produc t follow s immediatel y fro m (2.62) , and th e fact that this product i s associative follow s fro m th e generalized Jacobi con - ditions (se e 2.1). Q Any vector-functio n o f th e for m c = {c a}, wher e 2c QX (/) = 1 an d ca = cons t ca n b e a righ t uni t fo r th e *-product . Actually , the n w e hav e c(R) — id. Th e lemma i s proved. In th e algebr a £P n w e no w conside r a two-side d idea l / consistin g o f vector-symbols annihilatin g the *-produc t fro m th e right: J = {g\f * g = 0 V/e#"}. Obviously , w e have g G J <=> g(R) = 0 , an d thu s g(A) = 0 also holds, since g(A) = c(A)g(A) = (g(R)c)(A). Therefore, symbol s fro m / ar e nonessentia l i n an y representatio n o f th e algebra M . The subse t Spec(M) = {({ , p) e C " x p\(g(t), p) = 0 Vg e J} will be calle d th e spectrum of the algebra M . A n elemen t o f th e quotient - algebra o f symbol s Sm b = ^ n/J i s a "function o n the spectrum". Fo r an y such function / e Sm b th e element f(A) e M i s well defined .

COROLLARY 2.4 . Suppose the algebra M (see (2.50)) is equivalent to the algebra M for which the PBW-property holds. Then in the class Sm b of functions on the spectrum Spec(M) , the ^-product is defined', it determines an algebra structure with two-sided unit, and the mapping f *-* f(A) is an isomorphism of algebras Smb -> M. However, i t i s not clea r whethe r th e PBW-propert y hold s i n th e algebr a M. CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 333

We now describe the dua l picture, i.e., instead o f the *-product , w e consider the convolution o r the generalized shif t (se e Example 2.3). Assume that the operator of generalized shift U correspondin g to (2.50) is defined o n the manifold J£ . Its generators will be denoted by u l. Suppos e 7~ i s the representation o f this shift wit h generators B = A®d (2.59) , and let peP*. ,P We set T A = (p, T A(S>d)2, x e J£ , where the inde x 2 means that w e take the pairing with respect to the second component o f the tensor product. Thus, the mapping T A : J£ x P* —• M i s defined . The set of functions on Jf x P*, linea r in the second argument and smooth in the first one, will be denote d b y B"(Jt) 0 P , an d the spac e adjoin t t o i t by g*(Jt)®P*. W e defin e n» = (2.63 ) for an y q> e %*{jt ) P*. Obviously, i f / e &n , then the Fourier transformation i s

&-f = f(u*)8{x)eg*{jr)®P* an d U A{^f) = f(A) e M. Thus, the initial algebra M belong s to a set of elements o f the form (2.63) . This wider se t also forms a n algebr a (th e analo g o f the grou p algebra) , an d its law of composition i s given by the followin g

~~LEMMA 2.10 . (a) UA((p)nA(y/) = UA((p *vf)9 9*V = (V,U)*q>. (2.64 ) Here the family U = {U x,p\x eJ?,peP*} of operators in %{Jt)® P is defined by the formula~~

uX P f l ' = (P> ^d)3> d = r(d~ ). (2.65 ) (b) The family U is an almost generalized shift on Jt x P*, i.e., the first two axioms of (2.14) hold for it [if we take (e , c) as the identity element in ixf*, where c e P*, (c,I) = l). (c) The convolution (2.64) is associative on <&*(jt)®P* and possesses the right unit cd(x). The proof o f these statements is the same as that o f Lemma 2.9. Let us clarify the structure of formula (2.65) . The elements d%d generat e 1 a ZF-algebr a relate d to th e solutio n o f R^ m = co ^ o f th e quantu m Yang - Baxter equation (se e Remark 2.6) . The y define th e automorphis m

u i- > u =u ®d s®dk of relation s (2.50) . Th e operato r 7 ~ i n (2.65 ) i s a representatio n o f th e generalized shif t U x wit h generators u l. 334 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

~~There is one more formula for the almost convolution (2.64 ) generated by the almost shif t U~~

~~(9*V)(y) = (J„*(x,y,x)dx9I®l} . (2.66 )~~

Here cp , y/ e %*(J£ )<8 > P*, th e elemen t 0(J C J,Z)GP*0P* belong s t o %?(jfr) with respect to the variable x , an d to %*(jfr ®J?) wit h respect t o y, z; th e inne r integra l i n (2.66 ) mean s th e pairin g o f a distributio n wit h a function o n J? , while the external pairing with the unit take s place with respect t o the secon d componen t o f th e tenso r product . Her e th e elemen t 4>(x, y, z ) i n (2.66 ) i s the solution o f the followin g Cauch y proble m

~~u[{I ® ef )d> = S£(7r f ® /)*, 0| ^ = pO O 0 ^(z), (2.67 )~~

lk lk where n\ = r(d~ )*, 0 f* = l(d~ )* ar e operators on P* . However, th e almos t convolutio n constructe d abov e doe s no t posses s a unit, since the third axioms in (2.14 ) is violated for U . In order to get a true convolution, a true generalized shift an d a genuine cospectrum o f the algebra M, i t is necessary to take the quotient o f the se t /x?* wit h respect to a certain equivalenc e relation . Denote b y / ' th e idea l i n the algebr a <^*(J?) ® P* consistin g o f func - tional annihilatin g the convolution fro m th e right : j' - {vl? * ¥ = 0 Vp}.

Then i f tp e f , we have Tl A(i//) = 0. Thus , the nonessential "directions" in /xP* ar e those along which w e "differentiate" th e distributions fro m /' . Denote by %(J£) th e subset o f &{J?) P annihilatin g / ' &{Jt) = {Fi ® p.| {y , Fi ®p. ) = 0 V^ € /}. Two points from jfc xP* wil l be called equivalent if any element from 8^(^# ) takes the same values in them. Th e quotient o f J£ x P* b y this equivalence relation wil l be denote d b y J£ . Obviousl y W{J?) ca n b e regarde d a s th e space o f "smooth " function s o n J£, an d th e quotien t %>*(Jfc) ® P*/J* d = %*{J£) a s a space of distributions o n J? .

~~LEMMA 2.11 . The set %{Jf) is invariant with respect to the operators Ux,p. If the points (x, p) ~ {x , //) ar e equivalent, then U x'p = U x ,/ ? .~~

PROOF. Th e last statemen t follow s fro m th e fac t tha t (y/ , U) = 0 fo r al l l !^G7' du e t o (2.64) . Further , le t F (S>p t e ^{^) . W e must prov e tha t x p l ^^{F^p.) e &{Jt), i.e. , tha t (y/ , U ' {F ® pt)) = 0 fo r al l x e JT , p e P*, y/ e f . For this purpose i t is sufficient t o prove that

, <^^'V '®Pl-)>> = 0 (2.68 ) CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 335 for an y cp e ^*(^# ) P*. Her e th e lowe r indice s x, p denot e variable s with respect to which the distribution (p acts. By (2.64) the left-hand sid e of (2.68 ) i s equal to

((p, 17)V , F l ®p. ) = {y/*p.) . Since /' i s an ideal, we have i// * cp e j\ and hence , by the definition o f ^(Jt), (y/ * cp, F l 0 p.) = 0. Thi s yields (2.68). The lemma is proved. By this lemma, a generalized shif t o n J? i s well defined b y the operator s U (2.65 ) (i t wil l be denoted b y the sam e letter U). Th e equivalenc e clas s of the element (e 9 c) fro m Lemm a 2.10(b ) serve s as the uni t o n J(, Th e convolution (2.64) , (2.66 ) define s th e structur e o f associativ e algebr a wit h unit in %*{Jt) . S o we have proved

THEOREM 2.5 . Suppose the algebra M (2.50 ) is equivalent to the algebra M for which the generalized shift U exists. Then the generalized shift U (2.65) exists for M on the cospectrum J£ = cospec(Af) . Moreover-. (a) A representation of the generalized shift U on Jf is well defined by the family T A. The mapping H A (2.63 ) is well defined on &*(^) and determines a representation of this algebra. (b) The Fourier image & f e <§**(•# ) is defined for any f e Sm b and

nA{Ff) = f(A), (p*Ff = f{u)*

~~Here u = (ul, .. . , un) is the set of operators~~

u[ = uk ® r{d~U) : g(JT) - + g(J!) , where the u are the generators of generalized shift U. (c) The operators ul are generators of the generalized shift U on the cospectrum l Uu^u'iU), T A-A = u\T A) and satisfy the initial relations (2.50) i" / / / m k , ij k u u - u> mku u + ju^u .

2.8. Transformatio n o f scale an d structure constants . Her e w e present a certain specifi c versio n o f th e genera l approac h explaine d above , usin g th e constructions o f th e wor k [37] . W e obtai n formula s (explici t i n a certai n sense) relating the scale and structure constants of equivalent quadratic-linea r relations, and simple r formulas fo r the generalized shif t (2.65) . Let T b e a semigrou p wit h uni t E , an d le t F = F(T ) b e a n algebr a o f functions o n it . W e do no t specif y th e topologica l structur e o f T an d th e structure i n the spac e o f function s F ; fo r example , on e ca n assum e that T is compact an d F = L 2(T). 336 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

Suppose that i n F(r) , beside s the ordinar y pointwis e multiplication, an - other multiplicatio n o i s define d wit h respec t t o whic h F(r ) i s a n algebr a with unit 1 . Suppos e this multiplication i s right-invariant, i.e. , (i?(a)*) o (i?(a)*¥) = R(a)*(& o ¥) for an y O, * F e F(r) an d any a e T , wher e R(a) i s a right shift o n T . W e shall denote< D © ¥ = f (O o ¥)(£). Further, w e assume tha t th e automorphis m H: F — • GL(« , C ) i s give n and H i s related to the scal e and structur e constant s o f th e algebr a (2.50 ) by the following equalitie s w • (H(a) 0 H(a)) = (H(a) 0 H(a)) . 5 , ]i'{H{a)®H{a)) = H{a)-Ji V a € T.

-1 ls J LEMMA 2.12. Suppose the matrix A = // ©# (/.& , A ^ = H~ ®H S) is r invertible. Then the matrix M ml = // ^ 0#^ w fl/so invertible (the formula for the inverse matrix H _1 is given below). Then relations (2.50 ) are equivalent to the quadratic-linear relations (2.50) w/7/ z £ca/e constants co = H_15H

~~PROOF. B y the right-invarianc e o f th e multiplicatio n o an d b y th e fac t that H i s an antihomomorphism, w e have~~

~~, r r H loH m = {H' k®H t)H}H'm, (2.70 )~~

u r xk r u s and similarl y H~ oH m = {H~ © H s)H~ H m . I n particular , H~U o H'= H~ U ® H') = Ar, o r rf ° H\ = <$[, where J V = A~ iH~l. Thi s relation and (2.70) , after invertin g //, giv e i fc [7vrfoArio(^/// )]H;; = j;,4, i.e., the matrix H i s invertible and lsk m H- = (Nr oN!) 0 (//>/) . By left multiplications , w e imbed th e algebr a F int o Hom( F— • F), i.e. , FBOH (OO ) E Hom( F — • F). Further , b y th e lef t multiplicatio n / , w e imbed th e subalgebr a obtaine d (Fo ) c Hom( F— • F) int o th e algebr a o f operators ove r Hom( F -+ F), i.e., T-*l{T), l(T)Q=TQ. Th e algebra of operators obtained wil l be denoted by P* . Thus , F « P* . Further, to each element T e Hom(F— • F) ther e corresponds the element T* G Hom(F*— • F*), an d i n it s turn , t o thi s elemen t on e ca n associat e the operator o f multiplication fro m th e right , i.e. , T* —> r(T*) , r(T*)Q* = Q*T*. Th e subalgebr a o f operator s {r(T*)\T e (Fo) } wil l be denote d b y P. S o we identify (Fo) * « P . An d in this case P* wil l be the space adjoin t to P i f we identify /(P) * « r(P) . CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 33 7

Now w e denot e dj = (JV/o) * e P . The n th e operator s o f multiplica - tion fro m th e lef t b y Nfo ar e identifie d wit h r(dj)* , an d th e operator s o f multiplication fro m th e left b y HJo ar e identified wit h r(d~ 7 )* = n\. We wan t t o prov e tha t th e elemen t d define s a P-homomorphis m o f relations (2.50) . Fo r this purpos e w e must verif y (2.58a ) o r chec k that th e elements n\ = Hjo satisf y (2.58b) . S o we must prove that r and \i indicate d in the statement of our lemma. Th e lemma i s proved. So w e obtai n a metho d fo r enumeratin g th e quadratic-linea r relation s equivalent t o th e give n ones . I n thi s cas e th e spac e o f function s F o n a certain subgrou p T serve s a s th e initia l object . I n F th e right-invarian t multiplication, a n antihomomorphism H wit h properties (2.69) , and an invertible matrix A mus t b e given. Suc h relations will be called T-equivalenL We not e tha t condition s (2.69 ) hol d automatically , fo r example , i n th e following cases : l J (A) relations (2.50 ) ar e abelian, i.e., 5^ = 8 kd m , // ^ = 0 ; (B) relation s (2.50 ) ar e linear , an d th e semigrou p T act s b y automor - phisms on the corresponding Li e group Jt ; (C) for relations (2.50) , a generalized shift U i s defined o n the subbmani- fold Jf , the semigroup T act s by diffeomorphisms o n Jt , th e point e e J£ is fixed, and the shif t U i s T-invariant with respect to this action, i.e. , x x U K{a) = (K(a)x 0 K{a))U . (2.71 ) Here K denote s the representation o f T i n

uxi \x=e Then propertie s (2.69 ) ar e consequence s o f th e commutatio n relation s be - tween th e generator s u l o f th e generalize d shif t U an d th e identitie s ob - tained fro m (2.71 ) b y differentiation wit h respect to x t fo r x = e :

i i J u K{a) = Hj K{a)u . (2.71a ) In al l three cases (A) , (B), (C ) Lemma 2.1 2 works, and thus Theorem 2. 5 works too , i.e. , fo r th e quadratic-linea r relation s (2.50 ) on e ca n construc t 338 CALCULU S OF SYMBOLS AND COMMUTATION RELATION S a generalize d shif t b y usin g th e shif t U . Moreover , th e constructio n o f Theorem 2. 5 i s simplified here . Consider the embedding s

~~l l l K~ : giJT) - ?(i)0P* , K~ (F)(x,a) = (K(a)~ F)(x)9~~

K*: g*(JT) - &*(JT)®P\ K\X)(x 9a) = (K(a)*X)(x). Here we identify F«P * (se e the proof o f Lemm a 2.12) . No w we consider the compositio n o f thes e embedding s wit h th e mappin g (p ~~y/ -^ O fro m (2.67):~~

~~4 I -~~

~~(2.72) Here the mapping K~~ l in the lower arrow acts on the first component, i.e. , on %{J[ x), an d the new mappin g Q :<§?*(. # xi)->r(i)^*(i xJt) appears in the left vertica l arrow .~~

~~LEMMA 2.13 . We set xd x U =(KxeK)U . (2.73 ) Then the mapping Q defined by the formula n(l®p)(x,y,z) = (U x*l)(y)p(z)~~

~~(where k , p e <£*(jt ), x 9y, z e JZ ) makes the diagram (2.72 ) commutative.~~

PROOF. Th e mapping x i n (2.72 ) is defined b y the solution of the Cauchy problem (2.67) . Th e statement o f the lemma means that the solution o f this problem for special initial data of the form {x9y,z) = K*y®K; K*z[a(l®p){x9y9z)]. (2.74 ) Let us prove this formula . Indeed, in the case considered, equation (2.67 ) ha s the for m K(I ® (HJo)*)Q = Uy((HJo) ® 7)0. (2.75 ) Since the multiplication o i s right-invariant, w e have the identity Hj o K* = s j H tK\H s © K*). Further , b y (2.71a) , we have u^rfK* = #*£** . B y com- bining this with the previous identity, w e get J TH[ O K* = K*u*(H s 0 K*) = K*ul*, (2.76 ) CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 33 9 where j J °u = {H sQ)K)u. (2.77 ) J s After conjugation , (2.76 ) yields K&iHJo)* = (H s®K)u K o r l j U\HIO)*K~X = K~u . (2.76a ) We now substitute (2.74 ) into (2.75 ) and , by (2.76) and (2.76a), obtain the following equation for Q

~~j u[a{k ®p) = u y£l{X 0 p) (2.78 ) with initial condition S1(X © p)\x=e = X(y)p{z) . o Now let us study the properties of the operator U .~~

~~LEMMA 2.14 . Formula (2.73 ) defines a generalized shift with generators (2.77) on J" .~~

~~PROOF. Sinc e the point e i s a stable point of the action of the semigrou p T, w e have (KF)(e) = F(e) fo r any F e &(JT). Hence ,~~

~~x x And moreover , {U F)(e) = (Kx © I)(U F){e) = {Kx © I)F(x) = F(x). I t o remains to verify th e associativity axiom (2.14 ) for U . B y the T-invarianc e of (2.71) , we have~~

x y x y x y U l/ = (K x(DK)U (Ky®K)U = (KxQ K)(Ky® K xK)U U . (2.79 ) Further, since the multiplication in F is right-invariant, the following relation (a o x)(a) = {a © x)a(a)x(a) V a G T. holds for any two commutin g homomorphisms a an d x o f the semigrou p r. A s such homomorphisms, w e choose two copies of the representation K , one of which act s wit h respec t to x an d the other act s with respec t to the moving variable on J£ . W e obtain K xo K = (Kx © K)KxK o r

y x y x U xU = (K yGKx)(KyKxQK)U U Now we must only take into account the fact that the generalized shift U an d the multiplication in F ar e associative, i.e., K © (Kx o K) = (Ky o Kx) © K. o o o o o 3; y x Thus we get C/*C/ = U xU , i.e., [ / i s a generalized shift . It s generators can be calculated by differentiating (2.73 ) wit h respec t to x- at x = e; this yields (2.77). Lemm a 2.1 4 is proved. 340 CALCULUS OF SYMBOLS AN 9 COMMUTATIO N RELATION S

~~As a corollary we immediately obtain the following equation s~~

J uxU = U u o r u xU =u U . Thus, the solution (2.78 ) has the form mentioned in the statement of Lemma 2.13. Lemm a 2.1 3 is proved. We now substitute (2.74) into (2.66) and get a formula fo r the convolution generated b y the generalized shif t U , namely , (K*X) * (K*p) = K*(A * p), where **P = (f~(Kx[U***){K*p){x)dx,l\ = (p, U)*L

o Thus the convolution A*/ ? corresponds to the generalized shift U . Moreover , (2.76) yield s th e commutation relation s fo r generator s o f tw o generalize d shifts U and U, namely , u jK~l =K~ luj.

COROLLARY 2.5 . Suppose the quadratic-linear relations (2.50) and (2.50 ) are T-equivalent, and for relations (2.50 ) there exists a T-invariant generalized shift U on the manifold Jt . Then the generalized shifts U (2.65) o and U (2.73 ) corresponding to (2.50) are related by the intertwining operator K~ {and their convolution algebras are related by the homomorphism K*). In particular, the cospectrum of the algebra (2.50) coincides with J£ . The transition fro m relation s (2.50 ) t o (2.50 ) an d from th e shift U t o o o the shift U can be repeated if U turns out to be T-invariant .

COROLLARY 2.6 . If the multiplication in F(T ) is a two-sided invariant, for o example, if the semigroup Y is abelian, then the shift U (see (2.73)) is T- oo invariant. In this case one more generalized shift U exists (also T-invariant) corresponding to quadratic-linear relations with scale constants M coB. and structure constants juW 2 (see Lemma 2.12) , and so on. The descriptio n o f all scale and structure constant s whic h appea r i n this way remains a n open problem . I t is also unknow n whethe r on e can always choose a semigroup T and multiplication i n F(T ) fo r given scale and structure constants s o that the given relations wil l be T-equivalen t t o some stan - dard, sa y linear, relations . Th e important ste p her e i s to describe al l right- invariant multiplication s i n F(T). EXAMPLE 2.10 . Shifts generated by abelian groups (se e also [37]) . Th e simplest exampl e i s when T is a compact abelia n group . W e expand eac h function 4 > € F(T) wit h respect to the orthogonal basis of characters {x a} c a r*, O = ^2^ aX an d construct the required product in F(T ) by the formula (•o^w^j^ViA), CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 341 where Q a = x" © x • Th e numerical matri x Q ca n b e define d arbitraril y with a single restriction: the associativity condition {x a°X )°X C — Xa°(X °X C) must hold. After th e matri x Q i s chosen , on e ca n writ e th e produc t o i n a mor e convenient for m (

{fi*y¥{ya) dh(fi) dh(y) , rxr where the matrix SB i s defined b y the formul a 3S = X~XQX'\ X = U\m (2.80 ) (the prim e mean s transposition) , h i s th e Haa r measur e o n Y wit h uni t density at the point E . In terms of the "integral kernel" & , the operator (2.73 ) can be written a s follows (UxF)(y) = jj<%{P, y)(U mF)(y(y))dh{P) dh(y) , (2.81) rxr where F e %>{J?) . We see that the group Y generate s a lattice on ^ # ove r which we sum in (2.81). The simples t lattic e (one-dimensional ) correspond s t o a grou p wit h on e generator. W e consider, fo r example , the cyclic group with generator a an d relation a 2 = E . W e have the following basis of character s 1 for/3 = E, for ji = a. and multiplication tabl e 1 1 Q 1 l+4 e Here e i s an arbitrar y numbe r characterisin g th e deviation o f the multipli - cation fro m th e usua l pointwis e multiplication . I n thi s cas e the matri x SS (2.80) has the follwing for m 1 1 1 1 1 -he -e 1 - 1 1 l+4 e - e e The generalized shif t (2.81 ) correspondin g to suc h a group Y i s defined b y the formul a Ux = ( 1 + e)U x - e{U a{x) 4 - K{a)Ux - K{a)U a{x)) (2.82) Exactly the same shift i s obtained i n the case when Y i s a semigroup with one generator a 2 = a . oo o We note that the shift U constructe d fro m U b y Corollary 2. 6 doe s not o contribute anythin g new ; i t simpl y coincide s wit h U afte r w e substitute e for 2e( l + 2e). Th e shift doe s not chang e at al l when e = -1/4. 342 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

The case e = -1/4 i s a special case; it corresponds to the trivial multiplication table Q= \ l I ^ L 1 o In th e genera l case , w e hav e th e followin g multiplicatio n i n F(T ) wit h trivial table (

(U*xF)(y) = j[{U xF)(P{y)) + (U PMF)(y)]^- (2.83) -II (U F)(y(y)) h{r) h{r) . TxT Such a shif t doe s not vary when the operation # i s applied repeatedly . W e call it a shift ofDelsarte type. By using other multiplication tables, we arrive at generalized shifts of more complicated structur e than Desarte shifts . Consider, for example, the abelian group with three generators a , fi , a/ ? 2 2 and relations a/i = Pa , a = / ? = is. Her e w e have the followin g matri x of characters x (se e (2.80)) 1 1 1 fl = 0 l - 1 -1 ll 1 1 1 - 1 -1 2 ' -1 - 1 1 3i a P ap ,ab The following three types of multiplication tables Q — ((Q )x ) are possible 1 1 1 1 1 1 1 0 P 0 0 s Type I Type II st = mr\ 1 Q 0 0 0 t 1 0 0 0 m st 1 1 1 ps s Type III p ±P ±pt t ±s ±t st Here p , q , s , t, m , r ar e arbitrar y comple x number s (onl y on e restrictio n is imposed o n them i n Type II). Each of these tables defines a bilinear right-invariant associative product on F(T) wit h two-sided uni t 1 . Th e matri x = X QX ha s ^e followin g form i n each of the three cases. CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

~~Type I:~~

~~1 0 0 -1 0 1 -1 0 -1 1 0 (P-Q) 1 0 0 16 0 1 -1 0 + 16 1 0 0 1 0 0 1 0 -1 1 + «~~

~~Type II:~~

~~1 1 1 0 0 0 (s + t) 0 0 0 (s~t) 1 1 1 16 0 0 0 + 16 1 1 - 1 1 1 1 0 0 0~~

~~1 0 0 -1 0 -1 1 0 (m + r) -10 0 1 (m -r) 0 1 -1 0 + 16 -10 0 1 16 0 1 -1 0 1 0 0 -1 0 -1 1 0~~

~~1 -1 -1 -1 1 1 st where st = mr. + 16 -1 1 1 + . *£>' 1 -1 -1~~

~~1 0 0 - -r 1 - 1 0 0 2/7 0 - -1 1 0 2s - 1 1 0 0 16 0 1 - -1 0 + 1 6 0 0 -1 1 . - 1 0 0 I. 0 0 1 -1~~

~~r j • 1 0 -1 0" 1 It 0 -1 0 I st — 1 + T 6 -1 0 1 0 + 1 6 0 1 0 -1. - ] - 1~~

~~" 1 -1 1 -r — 1 + E1 -1 1 -1 I — 1 16 1 -1 1 -l 16 .-1 1 -1 1. — 1 — ] 344 CALCULUS OF SYMBOLS AND COMMUTATIO N RELATION S~~

~~For the lower sign e w e have~~

~~0 1 - -1 0' 0 0 -1 1 2p -1 0 0 1 2s 0 0 1 -1 16 1 0 0 - -1 + T 6 1 -1 0 0 .0-1 1 0. . - 1 1 0 0~~

~~•o-i 0 1" 2t 1 0 -1 0 St + T 6 0 1 0 -1 + T 6 .-1 0 1 0.~~

~~ps_ EL + 16 16 + .~~

~~Here~~

~~•7 3 3 3 1 3 - 1 - 1 - 1 16 3 -1 - 1 - 1 3 - 1 - 1 - 1 and the empt y matrice s i n the las t formul a fo r Typ e III e denot e th e sam e matrices as those at st, ps, pt i n Type III 0 .~~

~~In al l three case s the summand 3S D generate s a Delsarte component U of the generalized shif t~~

(U*xF)(y) = ^U(U xF)(y) + 3(UxF)(a(y)) + 3{UX F){fi(y)) + 3(U xF)(a/}(y)) + 3(U a{x)F)(y) + 3(U PMF)(y) + 3(U afi{x)F)(y) - (U aix)F)(a(y)) - (U a{x)F)(fi(y)) - (U a(x)F)(a0(y)) - (U fi{x)F)(a(y)) - (U fi{x)F)(P(y)) - (U Pix)F)(aP(y)) - (U a,3{x)F)(a(y)) - (U aP{x)F)(^(y)) - (U ap{x)F)(af}(y))).

~~The other summands in the matrix 38 (wher e the numbers p , q , s, t, m, r ac t a s factors ) defin e th e non-Delsarte component o f th e generalize d shift b y formula (2.81 )~~

~~U = U + (non-Delsarte) . CALCULUS OF SYMBOLS AND COMMUTATION RELATIONS 34 5~~

~~For example, we have for Type I (non-Delsarte)*.FO>) = ^j^[{U*F)ty) - (U xF)(aP(y)) - (U a(x) F)(a(y))~~

~~+ (U a{x)F)(p(y)) + (Ufi{x)F)(a(y)) - {U P(x)F)(P{y)) -(Uafi{x)F)(y) + (U al){x)F)(aP(y))} + { -^^-[(UxF)(a(y)) - (U xF)(/3(y)) - (U a{x)F)(y)~~

+ (U a{x)F)(aP(y)) + (U mF){y) - (U fi{x)F)(afi(y)) - (U aP{x)F)(a(y)) + (U aP{x)F)(P(y))]. We shall not write a similar formul a fo r Type II, and consider onl y one particular version for Type III e (fo r p = s — I, t = -1): [1 1 0 0 " HO 0 0 0 2 1 -1 0 0 Lo ooo . The following operato r o f generalized shif t correspond s to this version (UxF)(y) = ^[(U xF)(y) + (UxF)(a{y)) + (UP{x)F)(y) - (U p(x)F)(a(y))]. (2.84) Its generators have the form u = ^ [«' ' + K(a)u' + Hifi)^ - tf(a)#(£)jZr /] . The matrix A fro m Lemm a 2.1 2 is the following A=hl + a + b-ab), wher e a = H{a), b = H{p).

~~Since a = b 2 = /, [a , b] = 0, it follows that A i s invertible.~~

COROLLARY 2.7 . If a, P are two diffeomorphisms of the manifold jfc, and a = ft = id , a ft = Pa , and moreover, if the generalized shift U defined on J? is invariant with respect to a and /? , i.e., X a{x) U yF{a{y)) = (U F)(a(y)) (and the same for p ), then formula (2.84 ) defines a generalized shift on J? corresponding to (2.50) with scale and structure constants co = H~ SH, JJ, = JiM. Here W=-[I®I + I<8>b + a<8>I-a<8>b], a = da, b = dB. 2 e e Below we calculate the constants co , ju explicitly in the case when J? i s a Li e group, U i s a right group shift , a an d b ar e automorphisms o f the Lie algebra. 346 CALCULU S OF SYMBOLS AND COMMUTATION RELATION S

2.9. Algebra s equivalen t t o Li e algebras . A n algebr a M wit h relation s (2.50) is called (in accordance with 2.7 ) an equivalent Lie algebra with structure constants ^ , if there exists an algebra P an d a set of elements d l- e P such that « • dj = dr * < > 0m < * 4 = Af ' dL > ( 2'58C) and the matrix d — ((d lj)) i s right invertible, i.e., there exists a matrix d~ l — ({d~u)) c P suc h that d] • d~u = 5)1. j j * j Let us see what these conditions look like for the Faddeev-Zamolodchiko v algebras. EXAMPLE 2.11 . ZF-algebras and equations for the L-operator. Le t R = (R^/) b e a nondegenerat e solutio n o f a quantu m Yang-Baxte r equation . Consider an algebra generated by the generators A% (a,/?=1,...,«) an d the relations (2.44a ) R^4Z = R7^,V. aa p p pp a a It i s sometimes convenient to write these relations without indice s

,f Tfc A' j' ' A Vw * j ' A T A" T A RAA=AAR, A =A®I, A = I ® A. In the quantum metho d o f the inverse problem [125] , the so-called equation for the L-operator RL(w)'L{vf = L{V)"L{W)'R (2.85 ) is associated to these relations. Her e L(w) = ((Lf(tu))) i s a matrix consisting of element s o f a certain algebr a o f operator s P dependin g o n a paramete r w (whic h is called the spectral parameter) . Suppose th e paramete r w run s ove r a squar e n x /2-lattic e w = (£) , H,v = I, ... , n. Denot e

~~(;) Then the equation (2.85 ) ca n be rewritten a s follow s~~

~~(::)•(:)(;)(;•)(::)(:)• where~~

00 (:•)•(:)""" The equation just obtained coincides with the first condition o f (2.58c). Th e second condition holds trivially since the structure constants pt and A vanish in this case. CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 34 7

So if there exists a solution of the equation for the L-operator (2.85) with spectral parameter varying on a lattice and this solution is invertible ^K" (•)-<:'*•' • then a ZF-algebra with quadratic relations (2.44a ) is equivalent to a commutative algebra and al l th e statement s o f Theore m 2. 5 hol d fo r relation s (2.44a). EXAMPLE 2.12. Projective representations of graded Lie algebras. We now consider grade d quadratic-linea r relation s fo r whic h th e constructio n give n in 2.7 was first developed in [68 ] (see also [107, 236, 248]). Let T b e a finite abelian group (i.e. , the direct su m of cycli c groups) an d let x : T x r - • C\{0} b e its 2-cocycl e x(a, P)x{ap , y) = x(a, J?y)x(j8 , y). (2.86 ) Without loss of generality, w e can assume that x{a,E) = x{E,a)= 1 . (2.87 ) Denote co(a , /? ) = (x(a , /?))/(*(/? , a)). No w consider the algebra with generators {A a\a e T } an d commutation relation s AaAfi = a>{fi , a)ApAa + JI0'V. (2.88 ) Here th e structur e constant s /z a'^ ar e connecte d wit h th e scal e constant s co(f}, a) b y the following identitie s

~~co(a, fi)/i a = -ju (antisymmetry) , jia'V'7 - /' V ,/?y - co(y , fi)f' V" ,/? = 0 (th e Jacobi identity). Moreover, by definition, th e constants co(fi , a) satisf y th e relation s~~

w a = ( ' P) VTFR V > tw(a , a) = 1 (unitarity) , Q)(p, a ) co{fi9 a)co(y, a) = co(py, a) . These relations impl y th e compatibilit y condition s fro m (2.51) . Th e Yang - Baxter equation fro m (2.51 ) i s trivial i n this case. We note that the relation (2.88 ) ca n be written i n the for m

P aJ aP [A\A ]x = X A , (2.89 ) where X a,p = x(a , fi)fi a , and the "quasicommutator" in the left-hand sid e is defined b y the formul a [Aa , A\ = x(a, fi)A QAfi - x(fi , a)A fiAa. Obviously, the quasicommutator define s a Lie algebra structure on the given span & o f the generators {A a} . Its structure constants are: k a^S^ wher e 348 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

S"' i s the Kronecke r symbol . W e denote thi s Li e agebr a b y g r. I t i s sai d to b e graded b y th e grou p T , an d ou r linea r mappin g g r - > 2? wil l b e called a projective representation of g r wit h multiplicator x. T o construc t a regula r representatio n o f th e relation s (2.88 ) mean s t o establis h th e rela - tionship between the multiplication in the enveloping algebra U(Q T) and the multiplication i n an algebra M wit h generators {A a} .

~~LEMMA 2.15. The relations (2.88) are equivalent to the linear commutation relations (2.89) in the algebra U{g r).~~

~~PROOF. B y using the group T an d its cocycle * , w e construct a new group~~

Tx wit h multiplicatio n a/ ? = x(a , P)OLP (b y the hat a w e denote the ele - ment from T x correspondin g to the element a eT). Suppos e P = U(r K) i s the enveloping algebra. W e set d% = S^a. The n the element d define s a P- homomorphism o f the relations (2.88) , i.e., satisfie s (2.58c) . Obviously , this P-homomorphism i s invertibl e (th e invers e ha s the for m d~ la = ^/T 1). The lemma is proved. Thus, the results of 2.7 can be applied to relations (2.88) . Le t us calculate the *-produc t fro m Lemm a 2. 9 an d th e almos t convolutio n fro m Lemm a 2.10. We identify th e spac e P* = [/(TJ * wit h the space F(T ) o f functions o n the group T an d introduce two shifts o n F(T )

~~Then the operator s d , n , 6 whic h appea r i n (2.61) , (2.62) , (2.65 ) ar e defined by the formula s d* = r{d~xp) = S's'l\ n' = d's'1, /(~~

/* g = g(P)f= £(*/, > */>( * ® * ® WW3/ = £<*,(* ® 5~! ® r)(/?0 ^ 0 *'> ' 7)3- ( 2-9°) Here the R a ar e operator s o f righ t regula r representatio n o f the linea r re - lations (2.89) , an d / = {/(£)}, g = {gp(€)} ar e symbol s polynomia l in { E &y with value s i n P* = F(T) . W e denot e b y {x P} th e basi s fi X (a) = £ f i n F(T) , an d b y (.. . , I}3 th e pairin g wit h uni t I e P wit h respect t o th e third componen t o f th e tenso r product , i.e. , (% , /) = z(£) > where ^ e P* = F(T), and £ " i s the unit i n T .

We no w not e tha t th e operator s s a defin e th e projectiv e representatio n of the grou p T wit h multiplicato r x , an d th e operator s t a defin e a representation wit h adjoin t multiplicator . Thus , the operators s~ 0 t a actin g i n (2.90) define an ordinary representation of the (abelian) group T i n the space CALCULUS OF SYMBOLS AND COMMUTATION RELATION S 34 9

~~F(r x T). W e expand it into irreducible component s~~

where T * is the group of characters, A £(a) ar e the eigenvalues of the matrices e s~ ®t a, an d X ar e the eigenprojectors. Thi s expansion define s a n actio n of the group of characters T* o n the coalgebra j £ b y the formul a Q £->«(£), e(£) = Ae(a)

~~y / * 8 = E g fMR))fy(X*(X ® /), /> 2.~~

~~e y The pairing (...,.. . )2 give s the matrix elemen t (X ) JE , and moreove r~~

~~gf{e(R))f7 = f 7*(e*gf), where on the right we have the *-produc t for the linear commutation relations (2.89). S o the almost *-produc t fo r relations (2.88 ) has the for m~~

eer* P,yeT According to the definitions o f 2.7 , the spectru m o f relation s (2.88 ) ha s the form Spec = {K,p)K€8;, p = {p'}€F(r)\ E^)» /, = ov ^€jr} where the ideal / consist s of vector-symbols g = {gA fo r whic h E E^fe^^ = ° V a e I\ V { G fl^

~~An interesting problem i s to describ e this spectru m explicitl y a s well as the cospectrum fro m Theore m 2.5. In this case Lemma 2.1 0 implies~~

~~COROLLARY 2.8 . Suppose G Y is a Lie group corresponding to the Lie algebra Q T (2.89) . The following almost convolution on ?*(G r)xF(r) corresponds to relations (2.88)~~

eer p,yer where V e is a representation of the group G T in $?{G T) with generators a a a e(3f) = A £(a)£? , the 3$ being left vector fields on G T. The generators of the almost generalized shift related to (2.88) have the form u°^={«h)3°F*}- < 2-91» 350 CALCULUS OF SYMBOLS AND COMMUTATION RELATION S

The spectrum an d cospectru m o f (2.88 ) a s well as the real *-produc t an d generalized shif t ca n be easil y obtaine d unde r a n additiona l restrictio n im - posed upon the cocycle H . Assume that the function *(<*,- ) is a character of the group T for each fixed a e T and, moreover, assume the invertibility of the matrix

We define the multiplication i n F(r ) b y formula (2.80 ) and choos e •' i.,y-^4W . (2.92 ) y y) So under the assumption impose d above , the spectrum o f relations (2.88 ) i s 0* w R" an d the cospectrum is the group G r. EXAMPLE 2.1 3 [68] . Cospectrum of cyclic anticommutation relations. W e again consider the relations (2.32). The y can be written i n the form (2.88 ) i f we take for Y th e group with three generators a, 0, y suc h that y — a/3 — Pa, a 2 — p2 = E, an d assum e that A =A L A =A' A3 = A y. 1 - 1 x= ((x(a,/?))) • 1 - 1 - 1 CALCULUS OF SYMBOLS AND COMMUTATION RELATIONS 351

~~The Li e algebr a $ r (2.89 ) i n thi s cas e i s th e algebr a R x su(2) wit h th e following relation s [XE, X a] = [X E, X f] = [XE,Xy] = 0, [Xa, X ] = Xy ( + cyclic permutations).~~

The antirepresentation H o f the group T i n g r ha s the following for m H(E) = id H(a)(XE,Xa,X^,Xy) = (XE,-Xa,Xfi, -X y), H{p){XE,Xa,Xfi,Xr) = (XE,-Xa, -X 1*, X 7), E f y E P 7 H(y)(X 9 X \ X 9 X ) = (X 9X°,-X 9-X ). It correspond s to the antirepresentatio n o f the group T b y automorphism s of the Lie group G T = SU(2). The multiplicatio n (2.92 ) i n F(T ) correspond s t o the type III @ fo r t = p = 1, s = — 1 (se e Example 2.11). The generalize d shif t (2.93 ) o n SU(2 ) correspondin g t o (2.32 ) ha s th e following for m

(UxF)(y) = \[F(yx) + F(ya(x)) + F(yfi(x)) + F(yy(x)) + F(a(y)x) - F(a(y)a(x)) + F(a(y)fi(x)) - F{a{y)y(x)) + F(jJ(y)x) - F(/3(y)a(x)) - F(fi{y)fi(x)) + F(0(y)y(x)) + F(y(yw + F(yOOaM) - ^OWM) + ^yOOK*))]. Its generators (2.94 ) ar e (uaF)(y) = {2aF){P{y)), (S^O O = (#')(y00), y y (5 F)(y) = (^ F)(a(y))> where .S^ , 2* , .2^ i s the basi s o f lef t fields o n SU(2) . Th e generator s it, u p, w r satisf y th e relation s (2.32 ) (her e w e di d no t lis t th e a trivia l generator u E correspondin g t o the cente r o f th e algebr a (2.95)) . Thu s th e sphere SU(2 ) « S 3 is the cospectrum of relations (2.32). In conclusion w e note that i n this case the generators (2.91 ) will have the form (se e [68]) 0 /' ia3 0 y 0 ia 3 u = -/ 0 u 3\ 0 ia 3 j ia3 0 where / i s the identity 2 x 2-matrix, a 3 i s one of the Pauli matrices (2.10), and / i s the imaginary unit . This page intentionally left blank References

1. D. V. Alekseevskii, A . M. Vinogradov, and V . V. Lychagin, Main ideas and concepts in differential geometry, Itogi Nauki i Tekhniki: Sovremenny e Problem y Mat.: Fundamen - tal'nye Napravleniya, vol. 28, VINITI, Moscow, 1988 ; English transl. in Encyclopedia of Math. Sci., vol. 28 (Geometry, I), Springer-Verlag, Berlin and New York, 1991. 2. V . I . Arnold, Mathematical methods of classical mechanics, 3r d ed., "Nauka" , Moscow, 1989; English transl. of 2nd ed., Springer-Verlag, Berlin and New York, 1989 . 3. , On a characteristic class entering into conditions of quantization, Funktsional. Anal, i Prilozhen . 1 (1967) , no . 1 , 1-14 ; Englis h transl . i n Functiona l Anal . Appl . 1 (1967). 4. V. I. Arnol d an d A . B. Givental', Symplectic geometry, Itog i Nauk i i Tekhniki: Sovre - mennye Problemy Mat.: Fundamental'nye Napravleniya, vol. 4, VINITI, Moscow, 1985, pp. 7-139; English transl. in Encyclopedia of Math. Sci., vol. 4 (Dynamical Systems, IV), Springer-Verlag, Berlin and New York, 1990 . 5. V . I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical aspects of classical and ce- lestial mechanics, Itogi Nauki i Tekhniki: Sovremenny e Problemy Mat.: Fundamental'ny e Napravleniya, vol. 3, VINITI, Moscow, 1985 , pp. 5-303; English transl. in Encyclopedi a of Math . Sci. , vol. 3 (Dynamical Systems , III), Springer-Verlag , Berli n an d Ne w York , 1988. 6. V . M. Babich, The multidimensional WKB method or the ray method. Its analogues and generalizations, Itogi Nauki i Tekhniki: Sovremenny e Problemy Mat. : Fundamental'ny e Napravleniya, vol. 34, VINITI, Moscow, 1988 , pp. 93-134; English transl. in Encyclope- dia of Math. Sci., vol. 34 (Partial Differential Equations , V), Springer-Verlag, Berlin and New York (t o appear). 7. I . A. Batalin, Quasigroup construction and first class constraints, J. Math. Phys. 22 (1981), 1837-1850. 8. V . V . Belov an d S . Yu. Dobrokhotov, The Maslov canonical operator on isotropic manifolds with a complex germ, and its applications to spectral problems, Dokl. Akad . Nauk SSSR 298 (1988), 1037-1042 ; English transl. in Soviet Math. Dokl. 37 (1988). 9. Yu. M . Berezanskii an d A. A . Kalyuzhnii , Spectral decompositions of representations of hypercomplex systems, Spectral Theory o f Operators and Infinite-Dimensiona l Analysis , Akad. Nauk Ukrain. SSR , Inst. Mat., Kiev, 1984 , pp. 4-19. (Russian ) 10. F. A . Berezin, Quantization, Izv . Akad . Nau k SSS R Ser . Mat. 3 8 (1974) , no. 5 , 1116 - 1175; English transl. in Math. USSR-Izv. 8 (1974). 11. , Quantization in complex symmetric spaces, Izv. Akad . Nau k SSS R Ser . Ma t 39 (1975), no. 2, 363-402; English transl. in Math. USSR-Izv. 9 (1975). 12. F . A. Berezin and G. I. Kats, Lie groups with commuting and anticommuting parameters, Mat. Sb . 82 (1970), no. 3, 343-359; English transl. in Math. USSR-Sb. 1 1 (1970). 13. Arthur L . Besse, Manifolds all of whose geodesies are closed, Springer-Verlag , Berli n and New York, 1978 .

~~353 354 REFERENCES~~

14. P . A. Braun, Discrete quasiclassical method in problems of quantum mechanics for molecules, Question s o f Quantum Theor y o f Atoms and Molecules , vol. 2, Leningrad, 1981 , pp. 240-252. (Russian ) 15. Gle n E . Bredon, Sheaf theory, McGraw-Hill, New York, 1967 . 16. N. Bourbaki, Topologie generate, Chaps. 1 , 2, 4th ed. , Actualites Sci . Indust., no. 1142 , Hermann, Paris , 1965 ; English transl. , Hermann , Paris , and Addison-Wesley , Reading , MA, 1966 . 17. , Groupes et algebres de Lie, Chaps . 1-3 , Actualite s Sci . Indust., nos. 1285 , 1349, Hermann, Paris, 1971, 1972; English transl., Hermann, Paris, and Addison-Wesley, Read- ing, MA, 1975. 18. V . S . Buslaev , The generating integral and the Maslov canonical operator in the WKB method, Funktsional. Anal, i Prilozhen. 3 (1969), no. 3, 17-31 ; Englis h transl. in Func- tional Anal. Appl. 3 (1969). 19 , Quantization and the WKB method, Trudy Mat. Inst. Steklov. 11 0 (1970), 5-28; English transl. in Proc. Steklov Inst. Math. 1972 , no. 110 . 20. L . L . Vaksma n an d Ya . S . Soibelman , An algebra of functions on the quantum group SU{2), Funktsional . Anal , i Prilozhen. 2 2 (1988), no. 3 , 1-14 ; Englis h transl. i n Func- tional Anal. Appl. 22 (1988). 21. A . M. Vershik, Algebras with quadratic relations, Spectral Theory o f Operators and Inf - inite-Dimensional Analysis, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1984 , pp. 32-57. (Russian) 22. A . M. Vinogradov and I. C. Krasilshchik, What is Hamiltonian formalism?, Uspekhi Mat. Nauk 30 (1975), no. 1 , 173-198; English transl. in Russian Math . Surveys 30 (1975). 23. Yu. M . Vorob'ev , S . Yu . Dobrokhotov , an d V . P . Maslov , Quasiclassical approximation for models of spin-spin interaction on a one-dimensional lattice, Zap. Nauchn. Sem. Leningrad. Otdel . Mat . Inst . Steklov . (LOMI ) 13 3 (1984) , 63-76 ; Englis h transl . i n J . Soviet Math. 31 (1985), no. 6. 24. Yu . M . Vorob'e v an d S . Yu . Dobrokhotov , Quasiclassical quantization of the periodic Toda chain from the point of view of Lie algebras, Teoret . Mat . Fiz . 5 4 (1983) , no . 3, 477-480; English transl. in Theoret. and Math. Phys. 54 (1983). 25. Yu . M. Vorob ev and M. V. Karasev, Corrections to classical dynamics and quantization conditions arising in the deformation of Poisson brackets, Dokl. Akad . Nau k SSS R 29 7 (1987), 1294-1298 ; English transl. in Soviet Math. Dokl. 36 (1988). 26. , About Poisson manifolds and Schouten brackets, Funktsional . Anal , i Prilozhen. 22 (1988), no. 1 , 1-11; English transl. in Functional Anal . Appl. 22 (1988). 27. , Deformation and cohomology of Poisson brackets, Topological an d Geometri - cal Methods of Analysi s (Novo e v Global. Anal., vup. 9), Izdat. Voronezh. Gos. Univ., Voronezh, 1989 , pp. 75-89; English transl. in Global Analysis-Studies and Applications , Lecture Notes in Math., vol. 1453 , Springer-Verlag, Berlin and Ne w York, 1990 . 28. R . V. Gamkrelidze, Chern's cycles of complex algebraic manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 20 (1956), no. 5, 685-706. (Russian ) 29. I . M . Gel'fan d an d Iren e Ya . Dorfman , Schouten bracket and Hamiltonian operators, Funktsional. Anal , i Prilozhen . 1 4 (1980) , no . 3 , 71-74 ; Englis h transl . i n Functiona l Anal. Appl. 14 (1980). 30. , Hamiltonian operators and classical Yang-Baxter equation, Funktsional. Anal , i Prilozhen. 1 6 (1982), no. 4, 1-9 ; Englis h transl. in Functional Anal . Appl. 1 6 (1982). 31. I . M . Gel'fand an d I. V. Cherednik, Abstract Hamiltonian formalism for classical Yang- Baxter sheaves, Uspekhi Mat . Nau k 3 8 (1983) , no . 3 , 3-21; Englis h transl . i n Russia n Math. Surveys 38 (1983). 32. Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Amer. Math. Soc, Prov- idence, RI, 1977 . 33. I . M . Glazman an d Yu. I. Lyubich, Finite-dimensional linear analysis: A systematic pre- sentation in problem form, "Nauka" , Moscow , 1969 ; Englis h transl. , MI T Press , Cam - bridge, MA, 1974 . 34. Claud e Godbillon , Geometrie differentiate et mecanique analytique, Hermann , Paris , 1969. REFERENCES 355

35. K. K. Golovkin, Parametric-normed spaces and norm mass, Trudy Mat. Inst. Steklov. 106 (1969); English transl., Proc. Steklov Inst. Math. 1972, no. 106 . 36. R. Ya . Grabovskay a an d S . G . Krein , The formula for the permutation of functions of operators that represent a Lie algebra, Funktsional. Anal, i Prilozhen. 7 (1973), no. 3, 81; English transl. in Functional Anal. Appl. 7 (1973). 37. D. I. Gurevich, Operators of generalized shift on Lie groups, Izv. Akad. Nauk Armyan . SSR Ser . Mat. 1 8 (1983) , no. 4 , 305-317; Englis h transl . i n Sovie t J. Contemp . Math . Anal. 18(1983). 38 , Poisson brackets associated with the classical Yang-Baxter equation, Funktsional. Anal, i Prilozhen. 2 3 (1989), no. 1 , 68-69; English transl. i n Functiona l Anal . Appl. 23 (1989). 39. Yu. L. Daletskii and S . G. Krein, Formulas of differentiation with respect to a parameter for functions in Hermitian operators, Dokl. Akad. Nauk SSSR 76 (1951), 13-16. (Russian) 40. Yu. L. Daletskii, Lie superalgebras in the theory of Hamiltonian operators, International Working Group II: "Problems of Nonlinear and Turbulent Processe s in Physics", vol. 1 , Kiev, 1985 , pp. 34-38. (Russian ) 41. V . G. Danilov, Estimates for a pseudodifferential canonical operator with a complex phase, Dokl. Akad. Nauk SSS R 24 4 (1979), 800-804; English transl. in Sovie t Math. Dokl. 20 (1979). 42. V. G. Danilov and V. P. Maslov, Pontryagin's duality principle for calculation of an effect ofCherenkov's type in crystals and difference schemes. I, II, Trudy Mat. Inst. Steklov. 166 (1984), 130-160 ; 167 (1985), 96-107; English transl. in Proc. Steklov Inst. Math. 1986 , nos. 1 , 2. 43. P . M. Dirac, Hamiltonian methods and quantum mechanics, Proc. Roy. Irish Acad. Sect. A 63 (1964), 49-59. 44. , Lectures on quantum mechanics, Befer Graduate School Sci., Yeshiva Univ., New York, 1964 . 45. Iren e Ya. Dorfman, Deformation of Hamiltonian structures and integrable systems, Inter- national Working Group II: "Problems of Nonlinear and Turbulent Processes in Physics", vol. 1 , Kiev, 1985 , pp. 39-41. (Russian ) 46. V. G . Drinfe l d , Hamiltonian structures on Lie groups, Lie bialgebras, and geometric meaning of classical Yang-Baxter equations, Dokl. Akad. Nauk SSS R 26 8 (1983) , 285- 287; English transl. in Soviet Math. Dokl. 27 (1983). 47. , Constant quasiclassical solutions of the Yang-Baxter quantum equation, Dokl. Akad. Nauk SSS R 273 (1983), 531-535; English transl. in Soviet Math. Dokl. 28 (1983). 48. , On quadratic commutation relations in the quasiclassical case, Mathematica l Physics and Functional Analysis, "Naukova Dumka", Kiev, 1986 , pp. 25-34. (Russian ) 49. , Quantum groups, Zap . Nauchn . Sem . Leningrad . Otdel . Mat . Inst . Steklov . (LOMI) 15 5 (1986), 18-49 ; English transl. in J. Soviet Math. 41 (1988), no. 2. 50. B . A . Dubrovin, S . P . Novikov , an d A . T . Fomenko , Modern geometry. Theory and applications, "Nauka", Moscow , 1979 ; English transl., Parts I, II, Springer-Verlag, Berlin and New York, 1984 , 1985. 51. B . A. Dubrovin, I. M. Krichever, and S . P. Novikov, Integrable systems. I, Itogi Nauki i Tekhniki: Sovremenny e Problemy Mat.: Fundamental'ny e Napravleniya, vol. 4, VINITI, Moscow, 1985 , pp. 179-284 ; Englis h transl. in Encyclopedi a o f Math . Sci. , vol. 4 (Dy- namical Systems , IV), Springer-Verlag, Berli n and New York, 1990 . 52. Yu . V . Egorov , The canonical transformations of pseudodifferential operators, Uspekhi Mat. Nauk 24 (1969), no. 5, 235-236. (Russian ) 53. M. V . Karasev, Expansion of functions of noncommuting operators, Dokl. Akad . Nau k SSSR 214 (1974), 1254-1257 ; English transl. in Soviet Math. Dokl. 1 5 (1974). 54. , Certain formulas for functions in ordered operators, Mat. Zametk i 1 8 (1975) , no. 2, 267-277; English transl. in Math. Notes 1 8 (1975). 55. , Asymptotic spectrum and oscillation front for operators with nonlinear commutation relations, Dokl. Akad. Nauk SSSR 243 (1978), 15-18; English transl. in Soviet Math. Dokl. 19(1978). 356 REFERENCES

56. , Operators of the regular representation for a class of non-Lie commutation relations, Funktsional. Anal, i Prilozhen. 13 (1979), no. 3, 89-90; English transl. in Functional Anal. Appl. 13 (1979). 57. , Weyl calculus and the ordered calculus of noncommuting operators, Mat . Zametki 26 (1979), no. 6, 885-907; English transl. in Math. Notes 26 (1979). 58. , Collection of problems in operator methods, Moskov. Inst . Elektron . Mashino - stroeniya, Moscow , 1979 . (Russian) 59. , Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds. I , II , Preprint, Moskov. Inst . Elektro n Mashinostroeniya , Moscow , 198 1 = Manuscrip t No . 1092-82 , deposited at VINITI, 1982 ; English transl. i n Selecta Math. Soviet. 8 (1989), no. 3. 60. , Asymptotic behavior of the spectrum of mixed states for self-consistent field equations, Teoret. Mat. Fiz. 61 (1984), no. 1 , 118-127; English transl. in Theoret. and Math. Phys. 61 (1984). 61. , Quantization of nonlinear Lie-Poisson brackets in semiclassical approximation, Preprint No . ITF-85-72P , Akad . Nau k Ukrain . SSR , Inst . Theoret . Phys. , Kiev , 1985 . (Russian) 62. , Poisson algebras of symmetries and asymptotic behavior of spectral series, Funk- tsional. Anal , i Prilozhen. 2 0 (1986) , no. 1 , 21-32; Englis h transl . i n Functiona l Anal . Appl. 20(1986). 63. , Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Izv. Akad . Nau k SSS R Ser . Mat . 5 0 (1986) , no . 3 , 508-538 ; Englis h transl . i n Math . USSR-Izv. 28 (1987). 64. , Hidden symmetry of a system of equations of nonlinear optics. Hysteresis, Dokl. Akad. Nauk SSS R 286 (1986), 852-856; English transl. in Soviet Math. Dokl. 31 (1986). 65. , Quantum reduction to orbits of symmetry algebras and the Ehrenfest problem, Preprint No . ITF-87-157P, Akad. Nauk Ukrain . SSR , Inst. Theoret. Phys. , Kiev , 1987 . (Russian) 66. , Supercommutator of forms and generalized Dirac brackets, Theory of Group Rep- resentations and It s Applications i n Physic s (Tambov , 1989) , "Nauka", Moscow , 1990 . (Russian) 67. , Flat Poisson manifolds and finite-dimensional pseudogroups, Mat . Zametk i 4 5 (1989), no. 3, 53-65; English transl. in Math. Notes 45 (1989). 68. M . V. Karasev an d V. P. Maslov, Algebras with general commutation relations and their applications, Itog i Nauk i i Tekhniki : Sovremenny e Problem y Mat. , vol . 13 , VINITI , Moscow, 1979 , pp. 145-267 ; English transl. in J. Soviet Math. 1 5 (1981), no. 3. 69. , Global asymptotic operators of regular representation, Dokl . Akad . Nau k SSS R 257 (1981), 33-37; English transl. in Sovie t Math. Dokl. 23 (1981). 70. , Quantization of symplectic manifolds with conical points, Teoret . Mat . Fiz . 5 3 (1982), no. 3, 374-387; English transl. in Theoret. and Math . Phys. 53 (1982). 71. , Pseudodifferential operators and the canonical operator on general symplectic manifolds, Izv. Akad . Nau k SSS R Ser . Mat. 4 7 (1983) , no. 5 , 999-1029 ; Englis h transl . i n Math. USSR-Izv. 23(1984). 72. , Asymptotic and geometric quantization, Uspekh i Mat . Nau k 3 9 (1984) , no . 6 , 115-173; English transl. in Russian Math . Survey s 39 (1984). 73. M . V. Karasev and M. V. Mosolova, Infinite products and T-products of exponents, Teoret. Mat. Fiz . 2 8 (1976) , no . 2 , 189-200 ; Englis h transl . i n Theoret . an d Math . Phys . 2 8 (1976). 74. M . V . Karase v an d V . E . Nazaikinskii , On quantization of rapidly oscillating symbols, Mat. Sb. 106 (1978), no. 2, 183-213; English transl. in Math. USSR-Sb. 34 (1978). 75. Eli e Cartan , La theorie des groupes finis et continus et la geometrie differentielle traitees par la methode du repere mobile, Gauthier-Villars, Paris, 1937 . 76. G . I . Kats , Ring groups and duality principle. I , II , Trud y Moskov . Mat . Obshch . 1 2 (1963), 259-301 ; 1 3 (1965) , 84-113 ; Englis h transl . i n Trans . Mosco w Math . Soc . 1 2 (1963); 13 (1965). REFERENCES 357

77. A. A. Kirillov, Geometric quantization, Itog i Nauki i Tekhniki: Sovremenny e Problem y Mat.: Fundamental'ny e Napravleniya , vol. 4, VINITI, Moscow , 1985 , pp. 141-178 ; En- glish transl . i n Encyclopedi a o f Math . Sci. , vol . 4 (Dynamica l Systems , IV) , Springer - Verlag, Berlin and New York, 1990 . 78. , Elements of the theory of representation, "Nauka", Moscow, 1972 ; English transl., Springer-Verlag, Berlin and Ne w York, 1976 . 79. , Constructions of unitary irreducible representations of Lie groups, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1970, no. 2, 41-51; English transl. in Mosco w Univ. Math. Bull. 25(1970). 80. , Local Lie algebras, Uspekh i Mat. Nauk 3 1 (1976), no. 4, 57-76; English transl . in Russian Math. Surveys 31 (1976). 81. Shoshich i Kobayashi an d Katsumi Nomizu, Foundations of differential geometry. Vol. I, Interscience, New York, 1963. 82. I . S . Krasilshchik , Hamiltonian cohomology of canonical algebras, Dokl. Akad . Nau k SSSR 251 (1980), 1306-1309 ; English transl. in Soviet Math. Dokl. 21 (1980). 83. S . G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of linear operators, "Nauka", Moscow, 1978 ; English transl., Amer. Math. Soc, Providence, RI , 1984 . 84. I. M . Krichever , The Baxter equations and algebraic geometry, Funktsional . Anal , i Prilozhen. 15 (1981), no. 2, 22-35; English transl. in Functional Anal . Appl. 1 5 (1981). 85. P . P. Kulis h an d E . K . Sklyanin , Solutions of the Yang-Baxter equation, Zap. Nauchn . Sem. Leningrad. Otdel . Mat. Inst. Steklov . (LOMI ) 9 5 (1980), 129-160 ; Englis h transl. in J. Soviet Math 1 9 (1982), no. 5. 86. V. V. Kucherenko, The quasiclassical asymptotic behavior of a point source function for a stationary Schrodinger equation, Teoret. Mat . Fiz . 1 (1969) , no . 3 , 384-406 ; Englis h transl. in Theoret. and Math. Phys. 1 (1969). 87. V . F. Lazutkin, The complex billiard, Probl . Mat. Fiz., vol. 11 , Izdat. Leningrad. Univ. , Leningrad, 1986 , pp. 138-164 . (Russian ) 88. , Quasiclassical asymptotics of eigenfunctions, Itogi Nauk i i Tekhniki: Sovremen - nye Problem y Mat. : Fundamenta l ny e Napravleniya , vol . 34 , VINITI , Moscow , 1988 , pp. 135-174 ; Englis h transl. i n Encyclopedi a o f Math . Sci. , vol. 3 4 (Partia l Differentia l Equations, V), Springer-Verlag, Berlin and Ne w York (t o appear). 89. B . M. Levitan, Theory of generalized shift operators, rev . ed., "Nauka" , Moscow , 1973 ; English transl. of 1s t ed., Israel Program Sci . Transl., Jerusalem, an d Davey , Ne w York, 1964. 90. Jean Leray , Analyse lagrangienne et mecanique quantique, Seminaire su r le s Equation s aux Derivees Partielles (1976-1977), I, Expose 1 , College de France, Paris, 1977 ; English transl., MIT Press, Cambridge, MA, 1981. 91. Gerar d Lion and Michele Vergne, The Weil representation, Maslov index and theta series, Progress in Mathematics, vol. 6, Birkhauser, Boston , MA , 1980 . 92. G . L . Litvinov , Dual topological algebras and topological Hopf algebras, Trudy Sem . Vektor Tenzor . Anal . 1 8 (1978) , 372-375 ; Englis h transl . i n Select a Math . Soviet . 1 0 (1991). 93. , Hypergroups and hypergroup algebras, Itogi Nauk i i Tekhniki : Sovremenny e Problemy Mat. : Noveishi e Dostizheniya , vol . 26 , VINITI, Moscow , 1985 , pp. 57—106 ; English transl. in J. Sovie t Math. 38 (1987), no. 2. 94. V . V. Lyubashenko, Hopf algebras and vector symmetries, Uspekh i Mat . Nauk 41 (1986), no. 5 , 185-186 ; English transl. i n Russian Math . Surveys 41 (1986). 95. A . I. Mal'tsev, Analytic loops, Mat. Sb. 36 (1955), no. 3, 569-576. (Russian ) 96. V. P . Maslov , Operational methods, "Nauka" , Moscow , 1973 ; Englis h transl. , "Mir" , Moscow, 1976 . 97. , Perturbation theory and asymptotic methods, Izdat. Moskov. Gos. Univ., Moscow, 1965; French transl., Dunod, Paris, 1972 . 98. , Complex Markov chains and the Feynman path integral for nonlinear equations, "Nauka", Moscow , 1976 . (Russian) 99. , The complex WKB method in nonlinear equations, "Nauka" , Moscow , 1977 . (Russian) 358 REFERENCES

100. , Application of ordered operators method for obtaining exact solutions, Theoret. Mat. Fiz. 33 (1977), 185-209 ; English transl. in Theoret. and Math. Phys. 33 (1977). 101. V . P. Maslo v an d V. E . Nazaikinskii, Algebras with general commutation relations and their applicaions. I, Itogi Nauki i Tekhniki: Sovremenny e Problemy Mat., vol. 13, VINITI, Moscow, 1979 , pp. 5-144; English transl. in J. Soviet Math. 1 5 (1981), no. 3. 102. V . P. Maslov and M. V. Fedoryuk, Quasiclassical approximation for the equations of quantum mechanics, "Nauka" , Moscow , 1976 ; Englis h transl. , Semi-classical approximation in quantum mechanics, Reidel, Dordrecht, 1981 . 103. P. O . Michee v an d L . V. Sabinin, Quasigroups and differential geometry, Itog i Nauk i i Tekhniki: Problem y Geometrii , vol . 20 , VINITI , Moscow , 1988 , pp. 75-110 ; Englis h transl. in J. Soviet Math. 51 (1990), no. 6. 104. A . S . Mishchenko, B . Yu. Stemin , an d V . E . Shatalov , Lagrangian manifolds and the method of the canonical operator, "Nauka" , Moscow , 1978 ; English transl. , Lagrangian manifolds and the Maslov operator, Springer-Verlag , Berlin and New York, 1990 . 105. A . S . Mishchenk o an d A . T . Fomenko , Generalized Liouville integration method for Hamiltonian systems, Funktsional . Anal , i Prilozhen . 1 2 (1978) , no . 2, 46-56; Englis h transl. in Functional Anal . Appl. 1 2 (1978). 106. , Integration of Hamiltonian systems with noncommuting symmetries, Trudy Sem. Vektor. Tenzor. Anal. 20 (1981), 5-54. (Russian ) 107. M . V. Mosolova, Functions of noncommutative operators generating a graded Lie algebra, Mat. Zametki 2 9 (1981), no. 1 , 35-44; English transl. in Math. Notes 29 (1981). 108. Edwar d Nelson, Analytic vectors, Ann . of Math. (2 ) 70 (1959), 572-615. 109. A . I. Nesterov an d V. A. Stepanenko. On methods of nonassociative algebra in geometry and physics, Preprint No. 400F, Sibirsk. Otdel. Akad. Nauk SSSR, Inst. Fiz., Krasnoyarsk, 1986. (Russian ) 110. N. N. Nekhoroshev, Action-angle variables and their generalizations, Trudy Moskov. Mat. Obshch. 26 (1972), 181-198; English transl. in Trans. Moscow Math. Soc. 26 (1974). 111. S . P . Novikov, Hamiltonian formalism and multivalue analog of the Morse theory, Us- pekhi Mat . Nau k 3 7 (1982) , no . 5 , 3-49 ; Englis h transl . i n Russia n Math . Survey s 3 7 (1982). 112. , Algebraic construction and properties ofHermitian analogs of K-theory over rings with involution from the viewpoint of Hamiltonian formalism. I, II, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), no. 2, 253-288; no. 3, 475-500; English transl. in Math. USSR-Izv. 4(1970). 113. A . M. Perelomov , Generalized coherent states and their applications, "Nauka" , Moscow, 1987; English version, Springer-Verlag, Berlin and New York, 1986 . 114. F . F . Pomarret , Systems of partial differential equations and Lie pseudogroups, Gordo n and Breach , New York, 1983 . 115. L . S. Pontryagin, Continuous groups, 3rd ed., "Nauka", Moscow, 1973 ; English transl. of 2nd ed., Gordon an d Breach, New York, 1966 . 116. V . N. Popov and L. D. Faddeev, Perturbation theory for gauge invariant fields, Preprint, Akad. Nauk Ukrain. SSR , Inst. Theor. Phys., Kiev, 1967 . (Russian) 117. M . M. Postnikov, Smooth manifolds. Lectures on geometry. Semester III, "Nauka", Mos- cow, 1987 ; English transl., "Mir", Moscow , 1989 . 118. L . V. Sabinin, Methods of nonassociative algebra in differential geometry, appendix to the Russian transl . o f S . Kobayashi an d K . Nomizu , Foundations of Differential Geometry. Vol. I, "Mir", Moscow, 1981 , pp. 293-339. (Russian ) 119. M. A. Semenov-Tyan-Shanskii, What is a classical R-matrix?, Funktsional. Anal, i Prilo- zhen. 1 3 (1983), no. 4, 17-33 ; English transl. in Functional Anal. Appl. 1 3 (1983). 120. , Classical R-matrices and quantization, Zap . Nauchn . Sem . Leningrad . Otdel . Mat. Inst . Steklov . (LOMI ) 13 3 (1984) , 228-236 ; Englis h transl . i n J . Sovie t Math . 3 1 (1985), no. 6. 121. , Poisson groups and dressing transformations, Zap . Nauchn. Sem. Leningrad. Ot- del. Mat. Inst. Steklov . (LOMI ) 15 0 (1986), 119-141 ; Englis h transl. in J . Sovie t Math . 46(1989), no. 1 . REFERENCES 359

122. Jean-Pierre Serre , Lie algebras and Lie groups, Benjamin, Ne w York, 1965 ; Algebres de Lie semi-simples complexes, Benjamin , New York, 1966 . 123. E . K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation, Funk- tsional. Anal , i Prilozhen. 1 6 (1982), no. 4, 27-34; Some algebraic structures connected with the Yang-Baxter equation. Representations of a quantum algebra, Funktsional . Anal, i Prilozhen. 1 7 (1983), no. 4, 34-48; English transl. in Functional Anal. Appl. 16 (1982); 17(1983). 124. , On an algebra generated by quadratic relations, Uspekh i Mat . Nauk 40 (1985), no. 2, 214. (Russian ) 125. E. K. Sklyanin and L. A. Takhtadzhyan, Quantum method of the inverse problem, Theoret. Mat. Fiz. 40 (1979), 194-220 ; English transl. in Theoret. and Math. Phys. 40 (1979). 126. A . A . Slavno v an d L . D. Faddeev , Introduction to the quantum theory of gauge fields, "Nauka", Moscow, 1978; English transl., Frontiers in Physics, vol. 50, Benjamin, Reading, MA, 1980 . 127. V. O. Tarasov, L. A. Takhtadzhyan, and L. D. Faddeev, Local Hamiltonians for integrable quantum models on a lattice, Teoret. Mat. Fiz. 57 (1983), no. 2, 163-181; English transl. in Theoret. and Math. Phys. 57 (1983). 128. V . V. Trofimov an d A. T. Fomenko, Geometry ofPoisson brackets and methods for integration, in the sense ofLiouville, of systems in symmetric spaces, Itog i Nauki i Tekhniki: Sovremennye Problemy Mat.: Fundamental'ny e Napravleniya, vol. 29, VINITI, Moscow, 1986, pp. 3-108; Englis h transl. i n Encyclopedi a o f Math . Sci. , vol. 2 9 (Geometry , II), Springer-Verlag, Berlin and New York (to appear). 129. V . G . Turaev , Cocycle for symplectic first Chern class and Maslov indices, Funktsional. Anal, i Prilozhen. 1 8 (1984), no. 1 , 43-48; English transl. in Functional Anal . Appl. 18 (1984). 130. L. D. Faddeev, Feynman integral for singular Lagrangians, Teoret. Mat . Fiz . 1 (1969), no. 1 , 3-18; English transl. in Theoret. and Math. Phys. 1 (1969). 131. , Completely integrable quantum models of field theory, Problems o f Quantu m Field Theory, Dubna, 1979 , pp. 249-299. (Russian ) 132 , Operator anomaly for the Gauss law, Phys. Lett. B 145 (1984), 81-84. 133. L . D. Faddee v an d S . D. Shatashvili , Algebraic and Hamiltonian methods in the theory ofnonabelian anomalies, Teoret. Mat. Fiz. 60 (1984), no. 2, 206-217; English transl. in Theoret. and Math. Phys. 60 (1984). 134. B. V . Fedosov , Quantization and index, Dokl . Akad . Nau k SSS R 29 1 (1986) , 82-86 ; English transl. in Soviet Phys. Dokl. 31 (1986). 135. B. L. Feigi n an d D . B . Fuchs , Cohomology of groups and Lie algebras, Itogi Nauk i i Tekhniki: Sovremenny e Problemy Mat.: Fundamental'nye Napravleniya, vol. 21, VINITI, Moscow, 1988 , pp. 121-209 ; English transl. in Encyclopedi a o f Math. Sci. , vol. 21 (Li e Groups and Li e Algebras, II), Springer-Verlag, Berli n and New York (t o appear). 136. Richar d P . Feynman , Space-time approach to non-relativistic quantum mechanics, Rev . Modern Phys. 20 (1948), 367-387. 137. , An operator calculus having applications in quantum electrodynamics, Phys . Rev. (2)84(1951), 108-128 . 138. V . A. Fok, On canonical transformation in classical and quantum mechanics, Act a Phys. Acad. Sci. Hungar. 27 (1969), 219-224. 139. A . T . Fomenko , On symplectic structures and integrable systems on symmetric spaces, Mat. Sb. 115 (1981), no. 2, 263-280; English transl. in Math. USSR-Sb. 43 (1982). 140. , Differential geometry and topology. Additional chapters, Izdat. Moskov . Gos . Univ., Moscow, 1983 . (Russian) 141. , Symplectic geometry. Methods and applications, Izdat. Moskov. Gos. Univ., Mos- cow, 1988 ; English transl . o f a first draft, i n tw o halves , Symplectic geometry, Gordo n and Breach , Ne w York , 1988, ; an d Integrability and nonintegrability in geometry and mechanics, Kluwer, Dordrecht, 1988 . 142. Paul R. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, NJ and Toronto, 1967. 143. N. E. Hurt, Geometric quantization in action, Reidel, Dordrecht, 1985 . 360 REFERENCES

144. Lars Hormander, Fourier integral operators. I , Acta Math. 12 7 (1971) , 79-183; The calculus of Fourier integral operators, Prospect s i n Mathematic s (Proc . Sympos. Princeto n Univ., Princeton, NJ, 1970) , Ann. of Math. Stud., vol. 70, Princeton Univ. Press, Prince- ton, NJ, 1971 , pp. 33-57. 145. A . M . Chebotarev, On representation of the Schrodinger equation solution in the form of expectation value of functionals of jump processes, Mat. Zametk i 2 4 (1978) , no . 5 , 699-706; English transl. in Math. Notes 24 (1978). 146. I . V. Cherednik, "Quantum" deformations of irreducible finite-dimensional representations of QI N , Dokl. Akad. Nauk SSS R 287 (1986), 1076-1079 ; English transl. in Soviet Math. Dokl. 33(1986). 147. I. A . Shereshevskii , Quantization in cotangent bundles, Dokl . Akad . Nau k SSS R 24 5 (1979), 1057-1060 ; English transl. in Soviet Math. Dokl. 20 (1979). 148. Luther P. Eisenhart, Continuous groups of transformation, Princeton Univ. Press, Prince- ton, NJ, 1933 . 149. R. F. V. Anderson, The WeyI functional calculus, J. Funct. Anal. 4 (1969), 240-267. 150. M. Atiyah, Convexity and commuting Hamiltonians, Bull . London Math. Soc. 14 (1982), 1-15. 151. M . Atiya h an d R . Bott , The moment map and equivariant cohomology, Topology 2 3 (1984), 1-23 . 152. R. Baer, Nets and groups, Trans. Amer. Math. Soc. 46 (1939), 119-122 . 153. L. Baulieu and B . Grossman, Constrained systems and Grassmannians, Nuclear Phys . B 264(1986), 317-336. 154. R. J. Baxter , Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193-228. 155. F. Bayen, C. Fronsdal, Andre Lichnerowicz, and D. Stemheimer, Deformation theory and quantization. I , II, Ann. Physics 11 1 (1978), 61-110; 111-151 . 156. F. Bayen and C. Fronsdal, Quantization on a sphere, J. Math. Phys. 22 (1981), 1345-1349. 157. D. I . Blokhintzev , The Gibbs quantum ensemble and its connection with the classical ensemble, Acad. Sci. USSR J. Phys. 2 (1940), 71-74. 158. A . Borel and F . Hirzebruch, Characteristic and homogeneous spaces II, Amer. J. Math . 81 (1959), 315-382. 159. L . Boutet de Monvel and Victor Guillemin, The spectral theory ofToeplitz operators, Ann. of Math. Stud., vol. 99, Princeton Univ . Press, Princeton, NJ, 1981. 160. R. Brown, G. Danesh-Naruie, and J. P. L. Hardy, Topological groupoids. II, Math. Nachr. 74(1976), 143-156 . 161. Eugeni o Calabi, On the group automorphisms of a symplectic manifold, Problems in Anal- ysis, ( A Symposium i n Hono r o f Salomo n Bochner) , Princeto n Univ . Press , Princeton , NJ, 1970 , pp. 1-26 . 162. Eli e Cartan , Les groupes de transformations continus infinis simples, Ann . Sci . Ecol e Norm. Sup. 26 (1909), 93-161. 163. Jack F. Conn, Normal forms for smooth Poisson structures, Ann. of Math. (2) 12 1 (1985), 565-593. 164. Alain Connes , Noncommutative differential geometry, Inst . Haute s Etude s Sci . Publ . Math., No. 62 (1985), 257-360. 165. Alain Coste , Pierre Dazord , an d Ala n Weinstein , Groupoides symplectiques, Publ. Dep. Math. Nouvelle Ser . A, vol. 2, Univ. Claude-Bernard, Lyon , 1987 , pp. 1-62 . 166. Albert Crumeyrolle, Le cocycle d'inertie trilatere d'une variete a structure presque symplectique et la premiere classe de Chern, C. R. Acad. Sci. Paris Ser. A 284 (1977), 1507-1509 . 167. , Algebre de Clifford sympleciique revetements du groupe symplectique indices de Maslov et spineurs symplectiques, J . Math. Pures Appl. (9) 5 6 (1977), 205-230. 168. J. Czyz, On geometric quantization and its connections with the Maslov theory, Rep. Math. Phys. 15(1979), 57-97 . 169. Bryc e S . DeWitt, Dynamical theory of groups and fields, Gordon an d Breach, New York, 1965. 170. Jean Delsarte , Sur une extension de la formule de Taylor, J. Math . Pure s Appl . (9 ) 1 7 (1938), 213-230. REFERENCES 361

171. J. Dixmier, Algebres quasi-unitares, Comment. Math. Helv. 26 (1952), 275-322. 172. , Les algebres d'operateurs dans Vespace Hilbertien (Algebres de von Neumann), Gauthier-Villars, Paris, 1957 . 173. Miche l Duflo, Sur les representations unitaires des groupes de Lie contenant un sous-groupe invariant nilpotenU C. R. Acad. Sci. Paris Ser. A 270 (1970), 578-581. 174. J . J. Duistermaat an d H. J. Heckman , On the variation in cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259-268. 175. L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtadzhyan, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989), no. 1 , 178-206; English transl. in Leningrad Math. J. 1 (1990). 176. D. Finkelstein, On relations between commutators, Comm . Pur e Appl . Math. 8 (1955), 245-250. 177. M. Flato, Andr e Lichnerowicz , an d D . Sternheimer , Deformations of Poisson brackets, Dirac brackets and applications, J. Math. Phys. 17 (1976), 1794-1762 . 178. C . Fronsdal, Some ideas about quantization, Rep. Math. Phys. 15 (1979), 111-145. 179. Hidenori Fujiwara , Gerar d Lion , an d Bernar d Magneron , Operateurs d'entrelacement. Calcul d"obstructions sur des groupes de Lie resolubles, Noncommutative Harmonic Anal- ysis and Lie Groups (Marseille, 1980 ) Lecture Notes in Math., vol. 880, Springer-Verlag, Berlin and Ne w York, 1981 , pp. 102-137 . 180. Hidenor i Fujiwara, Certain operateurs d'entrelacement pour des groupes de Lie resolubles exponentiels et leurs applications, Mem. Fac. Sci. Kyushu Univ. Ser. A 36 (1982), 13-72. 181. V . L . Golo, Nonlinear regimes in spin dynamics of superfluid He , Lett. Math. Phys. 5 (1981), 155-159 . 182. M . J. Gotay and Jedrzej Sniatycki, On the quantization ofpresymplectic dymanical systems via coisotropic imbedding, Comm. Math. Phys. 82 (1981), 377-389. 183. Alexande r Grossman n an d P . Huguenin , Group-theoretical aspects of the Wigner-Weyl isomorphism, Helv. Phys. Acta 51 (1978), 252-261. 184. Alexander Grossmann , Gu y Loupias , an d E . M. Stein , An algebra of pseudodifferential operators and quantum mechanics in phase, Ann. Inst. Fourier (Grenoble) 1 8 (1986), fasc. 2, 343-368. 185. Victor Guillemin an d Shlom o Sternberg , Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math. 101 (1979), 915-955. 186. , The moment map and collective motion, Ann. Physics 12 7 (1980), 220-253. 187. , The metaplectic representation, Weyl operators and spectral theory, J. Funct. Anal. 42(1981), 128-225 . 188. , Homogeneous quantization and multiplicities of group representations, J . Funct. Anal. 47(1982), 344-380. 189. P. Hahn , Haar measure for measure groupoids, Trans. Amer . Math . Soc . 24 2 (1978) , 1-33. 190. R . Hermann, Quantum mechanics and geometric analysis on manifolds, Internat . J. The- oret. Phys. 21 (1982), 803-821. 191. Haral d Hess, On a geometric quantization scheme generalizing those of Kostant-Souriau and Czyz, Differentia l Geometri c Method s i n Mathematica l Physic s (Proc . Internat . Conf., Clausthal-Zellerfeld , 1978 ) Lectur e Note s i n Phys. , vol . 139 , Springer-Verlag , Berlin and New York, 1981 , pp. 1-35 . 192. L . Hormander , The Weyl calculus of pseudo-differential operators, Comm. Pur e Appl . Math. 32 (1979), 359-443. 193. R . Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), 188-254. 194. P . Huguenin , Expression explicite de Vexponentielle gauche pour les elements finis du groupe symplectique inhomogene, Lett. Math. Phys. 2 (1978), 321-324. 195. T . V . Huynh , Star-polarization: a natural link between phase space representation of quantum mechanics, Lett. Math. Phys. 4 (1980), 201-208. 196. M . Jimbo , A q-difference analogue of UQ and the Yang-Baxter equation, Lett. Math . Phys. 10(1985), 63-69. 362 REFERENCES

197. M. V. Karasev, Index of two-dimensional films and Maslov's quantization, Bak u Interna - tional Topology Conference (Baku , 1987) . 198. Joseph B . Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Ann. Physics 4 (1958), 180-188 . 199. M. Kikkawa, Geometry of homogeneous Lie loops, Hiroshima Math. J. 5 (1975), 141-179. 200. B . Kostant, Quantization and unitary representations. I, Lectures in Modern Analysis and Applications. Ill (C . T. Tamm, ed.) , Lecture Notes i n Math., vol. 170 , Springer-Verlag , Berlin and Ne w York, 1970 , pp. 87-208. 201. , Quantization and representation theory, Representation Theor y o f Li e Group s (Proc. SRC/LM S Res. Sympos., Oxford , 1977) , London Math . Soc . Lecture Note Ser. , vol. 34, Cambridge Univ. Press, Cambridge, 1979 , pp. 287-316. 202. Jean Loui s Koszul , Crochet de Schouten-Nijenhuis et cohomologie, Eli e Carta n e t le s mathematiques d'adjourd'hui, Asterisque , Numero Hors Serie, Soc. Math. France, Paris, 1985, pp. 257-271. 203. P. P. Kulish , N. Yu. Reshetikhin, an d E . K. Sklyanin , Yang-Baxter equation and representation theory, Lett. Math. Phys. 5 (1985), 393-403. 204. K . Kumar, Expansion of function of noncommuting operators, J . Math . Phys . 6 (1965) , 1923-1927. 205. Jean Leray , Solutions asymptotiques et groupe symplectique, Fourie r Integra l Operator s and Partial Differential Equation s (Colloq. Internat., Nice, 1974) , Lecture Notes in Math., vol. 459, Springer-Verlag, Berlin and Ne w York, 1975 , pp. 73-97. 206. Andre Lichnerowicz , New geometrical dynamics, Differentia l Geometrica l Method s i n Mathematical Physic s (Proc . Sympos. , Bonn , 1975) , Lectur e Note s i n Math. , vol . 570 , Springer-Verlag, Berlin and New York, 1977 , pp. 377-395. 207. , Les varietes de Poisson et leurs algebres de Lie associees, J . Differentia l Geom . 12(1977), 253-300. 208. , Deformations et quantification, Feynman Pat h Integral s (Proc . Internat. Colloq. , Marseille, 1978) , Lecture Notes in Phys., vol. 106 , Springer-Verlag, Berlin and New York, 1979, pp. 209-219. 209. S . Lie (unter Mitwirkung von F.Engel), Theorieder Transformationsgruppen. Vol. 2, Teub- ner, Leipzig , 1890 . 210. Gerard Lion , Extensions de representations de groupes de Lie nilpotents et indice de Maslov, C. R. Acad. Sci. Paris Ser. A 288 (1979), 615-618. 211. Jiang-Hu a L u an d Ala n Weinstein , Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom . 31 (1990), 501-526. 212. G . W . Mackey, Ergodic theory, group theory and differential geometry, Proc. Nat. Acad . Sci. U.S.A. 50(1963), 1184-1191 . 213. Bernar d Magneron , Une extension de la notion d 'indice de Maslov, C. R. Acad. Sci. Paris Ser. A 289 (1979), 683-686. 214. , Operateurs d'entrelacement des representations unitaires irreductibles des groupes de Lie nilpotents et indice de Maslov, C. R. Acad. Sci. Paris Ser. A 290 (1980), 943-946. 215. W . Magnus , On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. 7 (1954), 649-673. 216. Jerrold E. Marsden and Alan Weinstein, Reduction ofsymplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-131 . 217. Jerrold E . Marsden, Tudor Ratiu, and Alan Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc. 281 (1984) , 147-177 . 218. J. L. Martin, Generalized classical dynamics and the "classical analogue" of a Fermi oscillator, Proc. Roy. Soc. London Ser . A 251 (1959), 536-542. 219. P. A. Mello and M . Moshinsky, Nonlinear canonical transformations and their representation in quantum mechanics, J. Math. Phys . 1 6 (1975), 2017-2028. 220. Kentar o Mikam i an d Ala n Weinstein , Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci. 24 (1988), 121-140 . 221. Y . Moser, Hidden symmetries in dynamical systems, Amer. Sci . 67 (1979), 689-695. 222. Ole g M. Neroslavskii an d Anatol e T. Vlasov , Sur les deformations de I'algebre des fonc- tions d'une variete symplectique, C. R. Acad. Sci . Paris Ser. I Math. 29 2 (1981), 71-73. REFERENCES 363

223. Takayuk i Nono , Sur les families triples locales de transformations locales de Lie, Hi - roshima Math. J. 2 5 (1961), 357-366. 224. Hideki Omori, Yoshiak i Maeda , and Akir a Yoshioka , On regular Frechet-Lie groups. I, Tokyo. J. Math. 3 (1980), 353-390; II, III, Tokyo. J. Math. 4 (1981), 221-277. 225. E . Onofri an d M. Pauri, Analycity and quantization, Lett . Nuovo Cimento (2 ) 3 (1972), 35-42. 226 , Dynamical quantization, J. Math. Phys. 13 (1972), 533-543. 227. Richar d S. Palais, A global formulation of the Lie theory of transformation groups, Mem . Amer. Math. Soc. 1957, no. 22. 228. T. Renault , A groupoid approach to C* -algebras, Lecture Note s i n Math. , vol . 793 , Springer-Verlag, Berlin and New York, 1980 . 229. P. Rodles, Quantum spheres, Lett. Math. Phys. 14 (1987), 193-202 . 230. S . Poisson, Traite de mecanique, 2nd ed., Bachelier, Paris, 1833. 231. Joh n H. Rawnsley, On the cohomology groups of a polarization and diagonal quantization, Trans. Amer. Math. Soc. 230 (1977), 235-255. 232 , On the pairing of polarization, Comm. Math. Phys. 58 (1978), 1-8 . 233. A . G. Reyma n an d M . A . Semenov-Tyan-Shanskil , Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. I , II, Invent. Math. 54 (1979), 81-100; 63 (1981), 423-432. 234. Kennet h A. Ross, Hypergroups and centers of measure algebras, Sympos . Math., vol. 22, Academic Press, London, 1977 , pp. 189-203. 235. I . Satake , The Gauss-Bonnet theorem for V-manifolds, J. Math . Soc . Japan 9 (1957) , 464-492. 236. M . Scheunert, Generalized Lie algebras, Lecture Notes in Phys., vol. 94, Springer-Verlag, Berlin and Ne w York, 1979 . 237. J . A. Schouten, Uber differentialkomitanten zweier kontravarianter Grossen, Nederl. Akad. Wetensch. Proc. 43 (1940), 449-452. 238. D . J. Simms, Metalinear structures and geometric quantisation of the harmonic oscillator, Geometrie Symplectiqu e e t Physiqu e Mathematique (Colloq . Internat. CNRS , No. 237, Aix en Provence, 1974) , Editions Centre Nat. Recherch e Sci., Paris, 1975 , pp. 163-174 . 239. D. J. Simms and Nicholas Woodhouse, Lectures on geometric quantization, Lecture Notes in Phys., vol. 53, Springer-Verlag, Berlin and Ne w York, 1976 . 240. Jedrze j Sniatycki, Geometric quantization and quantum mechanics, Applied Mathematical Sciences, vol. 30, Springer-Verlag, Berlin and Ne w York, 1980 . 241. Jedrze j Sniatycki and Alan Weinstein, Reduction and quantization for singular momentum mappings, Lett. Math. Phys. 7 (1983), 155-161 . 242. Jean-Marie Souriau, Quantification geometrique, Comm. Math. Phys. 1 (1966), 374-398. 243. , Structure des systemes dynamiques, Dunod , Paris , 1970 . 244. , Mecanique statistique, groupes de Lie et cosmologie, Geometrie Symplectiqu e et Physiqu e Mathematique (Colloq . Internat. CNRS , No. 237, Ai x e n Provence, 1974) , Editions Centre Nat. Recherche Sci. , Paris, 1975 , pp. 59-113. 245. , Construction explicite de Vindice de Maslov. Applications, Lectur e Notes in Phys., vol. 50, Springer-Verlag, Berlin and New York, 1976 , pp. 117-148 . 246. , Interpretation geometrique des etats quantiques, Differentia l Geometrica l Meth - ods i n Mathematica l Physic s (Proc . Sympos. , Bonn , 1975) , Lectur e Note s i n Math. , vol. 570, Springer-Verlag, Berli n and Ne w York, 1977 , pp. 76-96. 247. Masamich i Takesaki, Duality and von Neumann algebras, Lectures on Operator Algebras: Tulane Univ. Ring and Operator Theory Year, 1970-1971 , Vol. II, Lecture Notes in Math., vol. 247, Springer-Verlag, Berli n and Ne w York, 1972 , pp. 665-786. 248. Rober t Trostel, Color analysis, variational selfadjointness, and color Poisson {super)algebras, J. Math. Phys. 25 (1984), 3183-3189. 249. Leo n Va n Hove , Sur le probleme des relations entre les transformations unitaires de la mecanique et les transformations canoniques de la mecanique classique, Acad . Roy. Belg. Bull. CI. Sci. (5) 37 (1951), 610-620. 250. Jacque s Vey, Deformation du crochet de Poisson sur une variete symplectique, Comment . Math. Helv. 50 (1975), 421-454. 364 REFERENCES

251. Ala n Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Adv. in Math. 6(1971), 329-346. 252. , Lagrangian submanifolds and Hamiltonian systems, Ann. of Math. (2) 98 (1973), 377-410. 253. , Symplectic V-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math. 30 (1977), 265-271. 254. , On Maslov's quantization conditions, Fourier Integral Operators and Partial Dif - ferential Equation s (Colloq . Internat. , Nice , 1974) , Lectur e Note s i n Math. , vol . 459, Springer-Verlag, Berli n and New York. 1975 , pp. 341-372. 255. , Lectures on symplectic manifolds, Conf . Boar d Math . Sci . Regional Conf . Ser . Math., vol. 29, Amer. Math. Soc, Providence, RI, 1979. 256. , The local structure of Poisson manifolds, J. Differential Geom . 1 8 (1983), 523- 557. 257. , Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc . 16 (1987), 101-104. 258. , Some remarks on dressing transformations, J. Fac . Sci. Univ. Toky o Sect . I A Math. 35(1988), 163-167 . 259. , Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), 705- 727. 260. Hermann Weyl , Gruppentheorie und Quantenmechanik, Hirzel , Leipzig , 1928 ; English transl., Methuen, London, 1931. 261. H . Widom, A complete symbolic calculus for pseudo-differential operators, Bull. Sci. Math. (2) 104(1980), 19-63 . 262. Nicholas Woodhouse, Geometric quantization and Bogoliubov transformation, Proc. Roy. Soc. London Ser . A 378 (1981), 119-139 . 263. S . L. Woronowicz, Pseudospaces, pseudogroups and Pontriagin duality, Lectur e Notes in Phys., vol. 116, Springer-Verlag, Berlin and New York, 1980. 264. , Compact matrix pseudogroups, Comm. Math. Phys. Ill (1987) , 613-666. 265. , Twisted SU(2) group. An example of non-commutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117-181 . 266. C. N. Yang, Some exact results for the many-body problem in one-dimension with repulsive delta-function interaction, Phys. Rev. Lett. 1 9 (1967), 1312-1314 . 267. A . B. Zamolodchikov and Al. B. Zamolodchikov, Two-dimensional factorizable S-matrices as exact solutions of some quantum field theory models, Ann. Physics 12 0 (1979) , 253-291. After the Russian edition of this book was submitted to the publisher, a number of new papers appeared concernin g its subject matte r and, moreover, the authors got acquainted wit h certai n old papers. We want to draw the reader's attention to them. These additions were introduced in the proofs, s o their numeration continue s the main list of References. First of all, we point out [287], which contains results similar to Lemma 1 , Chapter I. Then we draw the reader's attention to [272, 278] on foliations o f Poisson and symplectic manifold s and abou t infinitesima l structure s relate d t o suc h foliations. Th e important pape r [284 ] deals with comple x Poisso n and Jacobi manifolds. Further , w e came across [283 ] about bifibration s of symplectic manifolds (i n the sense of 2.3, Chapter I), containing some results similar to [59]. An English translation o f [59] appeared in [281]. General theorems on the differentiation o f groupoids and algebroids, but independent of the theory of Poisson brackets, were obtained already in [293]; see also [286]. In [282] pregroupoids (or, in terminology of [299], affinoids) wer e also considered. It was shown in [299] that the symplectic affinoid i s a symplectic manifold wit h the action of a pair of polar symplectic groupoids, i.e., a bifibration (se e 2A i n Chatper II). We also draw the reader's attention to papers [289, 302, 303] about the theory of symplectic groupoids. There ar e interesting collection s o f paper s i n the Seminair e Sud-Rhodanie n d e Geometri e [270], an d paper s o n the quantizatio n o f symplecti c groupoid s an d pseudogroup s [271 , 301 , 304]. The theory of quantum group s and Yang-Baxter equations is being developed intensivel y REFERENCES 365

[275, 292, 295, 296]; its relations to the interacion o f the theory of brackets of hydrodynamical type were discovered [276 , 277]. Theorem on the existence of the *-produc t on an arbitrary symplectic manifold (se e Remark 2.2 in Chapter IV ) were first obtained i n [274] . The equality \[v] — c x (se e (2.35) i n Chapte r IV) was proved i n [273]. New importan t paper s abou t geometri c quantizatio n appeare d [269 , 285 , 288, 291, 294], about deformational quantizatio n [278 , 290, 297], and about the geometry o f Lagrangian man - ifolds [268 , 298]. 268. Michele Audin, Cobordism of Lagrangian immersion into the cotangent bundle of a manifold, Functiona l Anal. Appl. 21 (1987), 223-226. 269. Robert J. Blattner, Some remarks on quantization, Symplectic Geometry and Mathemat- ical Physics, Progress in Mathematics, vol. 99, Birkhauser, Boston, 1991 , pp. 37-47. 270. Alain Coste and Daniel Sondaz , Classification de submersions de Poisson isotropes, Tra- vaux du Seminari e Sud-Rhodanie n d e Geometric I , Publ. Dep. Math. Nouvelle Ser . B, 88-1, Univ. Claude-Bernard, Lyon , 1988 , pp. 91-102. 271. A . Yu. Daletskii, The problem of factorization in a symplectic groupoid and Hamiltonian systems on spaces with nonlinear brackets, Dokl. Akad. Nauk SSS R 30 8 (1989) , 1033 - 1037; English transl. in Soviet Math. Dokl. 40 (1990). 272. Pierre Dazord , Feuillettages a singularity, Nederl . Akad . Wetensch . Indag . Math . 4 7 (1985), 21-39. 273. Pierre Dazor d an d G . Patissier , La premiere classe de Chern comme obsturction a la quantification asymptotique, Symplecti c Geometry , Groupoids , an d Integrabl e System s (Berkeley, CA , 1989) , Math. Sci . Res . Inst. Publ. , vol. 20 , Springer-Verlag , Ne w York , 1991, pp. 73-97. 274. M. De Wilde and P. B. A. Lecompte, Existence of star products and of formal deformations of the Poisson-Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7 (1983), 487-496. 275. V. G. Drinfel'd, Quasi-Hopf algebras and solutions of the Knizhnik-Zamolodchikov equations of some quantum field theory models, Problems of Modern Quantu m Field Theor y (Alushta, 1989) , Res. Rep. Phys., Springer-Verlag, Berlin and New York, 1989 , pp. 1-13 . 276. B. A. Dubrovin, On differential-geometric Poisson brackets on a lattice, Funktsional. Anal, i Prilozhen. 23 (1989), no. 2, 57-59; English transl. in Functional Anal. Appl. 23 (1989). 277. B. A. Dubrovin and S . P. Novikov, Hydrodynamics of weakly deformable soliton lattices. Differential geometry and Hamiltonian theory, Uspekhi Mat. Nauk 44 (1989), no. 6, 29- 98; English transl. in Russian Math. Surveys 44 (1989). 278. C. Duval, J. Elhadad, M. J. Gotay, Jedrzej Sniatycki, and G. M. Tuynman, Quantization and bosonic BRST theory, Ann. Physics 206 (1991), 1-26 . 279. Fouzia Guedira and Andre Lichnerowicz, Geometrie des algebres de Lie locales de Kirillov, J. Math. Pures Appl. (9) 63 (1984), 407-484. 280. N. J . Hitchin , Flat connections and geometric quantization, Comm . Math . Phys . 13 1 (1990), 347-380. 281. M . V. Karasev, The Maslov quantization conditions in higher cohomology and analogues of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds. I, II, Select a Math. Soviet. 8 (1989), 213-258. 282. A . Kock, Generalized fibre bundles, Categorical Algebr a an d It s Applciations (Louvain - La-Neuve, 1987) , Lectur e Note s i n Math. , vol . 1348 , Springer-Verlag , Berli n an d Ne w York, 1988 , pp. 194-207 . 283. Paulett e Libermann , Problemes d'equivalence et geometrie symplectique, Hie Rencontr e de Geometri e d u Schnepfenrie d (1982) , Vol . 1 , Asterisque , no . 107-108 , Soc . Math . France, Paris, 1983 , pp. 43-68. 284. Andre Lichnerowicz , Varietes de Jacobi et espaces homogenes de contact complexes, J. Math. Pures Appl. (9) 67 (1988), 131-173. 285. , Deformations and geometric (KMS)-conditions, Quantum Theories and Geometry (Les Treilles, 1987) , Math. Phys. Stud., vol. 10 , Kluwer, Dordrecht, 1988 , pp. 127-143. 366 REFERENCES

286. K . Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Math . Soc. Lecture Note Ser., vol. 124 , Cambridge Univ. Press, Cambridge, 1987 . 287. Franco Magr i and C . Morosi , A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Univ. di Milano (1984). 288. D. Melotte, Cohomologie de Chevalley associee aux varietes de Poisson, Bull. Soc . Roy. Sci. Liege 58 (1989), 319-413. 289. Kentaro Mikami, Symplectic double groupoids over Poisson (ax+b)-groups, Trans . Amer. Math. Soc . 324 (1991), 447-463. 290. Ivail o M. Mladenov, Reductions and quantization, Internat. J. Theoret. Phys. 28 (1989), 1255-1267. 291. Hidek i Omori , Yoshiak i Maeda , an d Akir a Yoshioka , Weyl manifolds and deformation quantization, Adv. Math. 85 (1991), 224-255. 292. P. Podles and S . L. Woronowicz, Quantum deformation ofLorentz group, Comm. Math. Phys. 130(1990), 381-431 . 293. Jea n Pradines, Theorie de Lie pour les groupoides differentiables. Calcul differentiel dans la categorie des groupoides infinitesimaux, C . R. Acad. Sci. Paris Ser. A 264 (1967), 245- 248; Troisieme theoreme de Lie pour les groupoides differentiables, C . R. Acad. Sci. Paris Ser. A 267 (1968), 21-23. 294. John H. Rawnsley, Deformation quantization ofKdhler manifolds, Symplectic Geometr y and Mathematical Physics. , Progress in Mathematics , vol. 99, Bikhauser, Boston , 1991, pp. 366-373. 295. N. Yu . Reshetikhin an d M . A . Semenov-Tyan-Shanskii, Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990), 133-142 . 296. James D. Stasheff, Differential graded Lie algebras, quasi-Hopf algebras and higher homo- topy algebras, Preprint, Univ . North Carolina, Chape l Hill, NC, 1991. 297. G . M. Tuynman, Reduction, quantization and nonunimodular groups, J. Math . Phys. 31 (1990), 83-90. 298. C . Viterbo, A new obstruction to embedding Lagrangian tori, Invent. Math. 10 0 (1990), 301-320. 299. Ala n Weinstein, Affine Poisson structure, Internat. J. Math. 1 (1990), 343-360. 300. , Noncommutative geometry and geometric quantization, Symplecti c Geometr y and Mathematical Physics , Progress in Mathematics, vol. 99, Birkhauser, Boston , 1991, pp. 446-462. 301. Ala n Weinstei n an d Pin g Xu , Extensions of symplectic groupoids and quantization, J . Reine Angew. Math. 417 (1991), 159-189 . 302. Pin g Xu, Symplectic groupoids of reduced Poisson spaces, C. R . Acad . Sci . Pari s Ser . I Math. 314(1992), 457-461. 303. , Poisson cohomology of regular Poisson manifolds, Preprint, Univ . Pennsylvania , Philadelphia, PA, 1991. 304. Stanisla w Zakrzewski , Quantum and classical pseudogroups. I , II , Comm . Math . Phys . 134(1990), 347-395. Copying an d reprinting . Individua l reader s o f this publication , an d non - profit librarie s actin g fo r them , ar e permitte d t o mak e fai r us e o f th e material , such a s to cop y a chapte r fo r us e i n teachin g o r research . Permissio n i s grante d to quot e brie f passage s fro m thi s publicatio n i n reviews , provide d th e customar y acknowledgment o f the sourc e i s given. Republication, systemati c copying , o r multipl e reproductio n o f an y materia l in thi s publicatio n i s permitte d onl y unde r licens e fro m th e America n Mathe - matical Society . Request s fo r suc h permissio n shoul d b e addresse d t o th e Ac - quisitions Department, America n Mathematica l Society , 20 1 Charles Street , Prov - idence, Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionOams.org. Recent Titles in This Series (Continued from the front of this publication)

84 Da o Trong Thi and A. T. Fomenko, Minima l surfaces, stratifie d multivarifolds , an d the Plateauproblem, 199 1 83 N . I. Portenko, Generalize d diffusio n processes , 199 0 82 Yasutak a Sibuya, Linea r differential equation s in the complex domain: Problem s of analytic continuation, 199 0 81 L M. Gelfand an d S. G . Gindikin, Editors, Mathematica l problem s of tomography, 199 0 80 Junjir o Noguchi and Takushiro Ochiai, Geometri c function theor y in several complex variables, 199 0 79 N . I. Akhiezer, Element s of the theory o f elliptic functions, 199 0 78 A . V. Skorokhod, Asymptoti c methods of the theory of stochastic differential equations , 1989 77 V . M. Filippov, Variationa l principles for nonpotentia l operators, 198 9 76 Philli p A. Griffiths, Introductio n to algebraic curves, 198 9 75 B . S. Kashin and A. A. Saakyan, Orthogona l series , 198 9 74 V . I. Yudovich, Th e linearization metho d i n hydrodynamical stabilit y theory, 198 9 73 Yu . G . Reshetnyak, Spac e mappings with bounded distortion, 198 9 72 A . V. Pogorelev, Bending s of surfaces an d stability of shells, 198 8 71 A . S. Markus, Introductio n to the spectral theory of polynomial operator pencils, 198 8 70 N . I. Akhiezer, Lecture s on integral transforms, 198 8 69 V . N. Salii, Lattice s with unique complements, 198 8 68 A . G. Postnikov, Introductio n to analytic number theory, 198 8 67 A . G. Dragalin, Mathematica l intuitionism: Introductio n t o proof theory , 198 8 66 Y e Yan-Qian, Theor y of limit cycles, 198 6 65 V . M. Zolotarev, One-dimensiona l stabl e distributions, 198 6 64 M . M. Lavrent'ev, V. G. Romanov, and S. P. Shishatskii, Ill-pose d problems of mathematical physics and analysis, 198 6 63 Yu . M . Berezanskii, Sel f adjoint operator s in spaces of functions o f infinitely man y variables, 198 6 62 S . L. Krushkal', B. N. Apanasov, and N. A. Gusevskii, Kleinia n groups and uniformizatio n in examples and problems, 198 6 61 B . V. Shabat, Distributio n o f values of holomorphic mappings, 198 5 60 B . A. Kushner, Lecture s on constructive mathematica l analysis , 198 4 59 G . P. Egorychev, Integra l representation an d the computation o f combinatorial sums, 1984 58 L . A. Aizenberg and A. P. Yuzhakov, Integra l representations and residues in multidimensional comple x analysis, 198 3 57 V . N. Monakhov, Boundary-valu e problems with fre e boundarie s for ellipti c systems of equations, 198 3 56 L . A. Aizenberg and Sh. A . Dautov, Differentia l form s orthogona l to holomorphic functions o r forms, and their properties, 198 3 55 B . L. Rozdesrvenskii an d N. N. Janenko, System s of quasilinear equations and their applications to gas dynamics, 198 3 54 S . G. Krein, Ju. I . Petunin, and E. M. Semenov, Interpolatio n o f linear operators, 198 2 53 N . N. Cencov, Statistica l decisio n rules and optimal inference , 198 1 52 G . I. Eskin, Boundar y value problems fo r ellipti c pseudodifferential equations , 198 1 51 M . M. Smirnov, Equation s o f mixed type, 197 8 (See the AMS catalog for earlie r titles)