INSTITUTE OP THEORETICAL AND EXPERIMENTAL PHYSICS
ITEP - 78
F.A.B@rezin
LIE SUPERGROUPS
MOSCOW 197? IHSTITUTE О? ГНЕОНЕТЮАЬ AHD ЕХРЕНИШИЛЬ EHXSICS
ITSP - 78
F.A.Beresln
LIE SOPEROTODTS
•mot 1977 УДК 517.43 M-I6
The detailed theory of Lie supergroups necessary for the development of general theory of representotion of Lie supergroups is described.
1 ИТЭФ. 1977
Ф.А.Березин Супергруппы Ли Работа поступила в 0Н1И 12/1У-1977Г. Подписано к печати 16/У-77г. Т-07357. Формат 70x108 I/I6. Печ.л.З,25.Тираж 295 экз.Заказ 78.Цена 17коп.Индекс 3624. Отдел научно-технической информации ИТЭФ,II7259,Москва 1. General definition. The coneept of Lie supergroup ie the generalisation of the consept of the usual Lie group.
The idea of this generalisation is the following one. Let vro be a Lie £roup, Д (Qr ) oe the algebra of infinitly diffe- rentiable functions on Cow.r*t the usual addition and multip- lication. Denote f the map of A (Qo) in the analogouee ie toe algebra Д (Go / » *&№• Grrt Cartesian product of Deaot e the Qo by itself ; (4{)($h& • {(кЗ*У • ° automorphism Д (Go) '•
Let us forget now that (xo is the Lie group» We will re- member only that Go is smooth manifold and that there are homomorphisms of algebras, if: A (G0J —» A (G-o у. С-fl/^)"*f and О '• A (Qo)—* A (G-o)( (Jo denotes now the Cartesian pro- duct of two copies of the manifold Qo). It tone oat that If homoraorphlsms V, £ > 0 have some natural properties, they define the unique group structure on Q- , To or tain a Lie supergroup Q one ought to replace A (Q,) by a trifle more sofiaticated algebra 0? . ( QQ) (resp. 4 (""о/ ^ provided with analogous of homomorphiems p £ Ъ№О (denoted by the some letters) having the same
properties» the Lie group Go ie called under the circums- tances the basic subgroup of Lie supergroup (j- , Let us cone to the exact definitions. bet Q be a Lie group of dimension p • Qjo Is the Grassmann algebra with Q, generators, СП, (G )ie the algebra of infintly diffe-
rentiable functions on Qo with values in Ojg » Denote
where 0 6 (j-0 » Cv» 1^i^ 9, » are generators of Grass- ваш algebra (Q* , ^^...Cjjj) - infinitly differentiable
functions on Qo
Denote convergence in ЭД - ((r0 у .' /и. (3/1y ~*t(3/ if coefficient functions i.^ -ii ». -t OP °*eleman *a
f. Гоч? ^ approach with all their derivatives to the cor- responding coefficient functions -f-ij.... л,«.(§) °* *be ele- •ent +(Ч(Ь ) * Th* convergence is supposed to be uniform on each compact set К. с G; Q • from the pure altebraic point oi view elements of are like functions of P + 9/ variables, parameters of the group Gr and generators \v having equal rights. In сазе (j"0 being a complex group, it ia useful some-
times to tiacuas besides ^e, (Ge/ algebras Щл (Gyjae well tne latter consisting ot elements having the form (Z.I), the required property ot j^...-С^ (%*) ie n0* emoothaeee but repreaeatability in the form J • • (o\ = ** ЛЗ
(being the holomorphio functions. Later on functions of this kind are called analitical functions on \x •
Together with 4
where defined by the fo.rmula
is sometimes called the diagonal map). Uote that if О Q| f Г ^ ^ is 01 i«e typt i2.2) then C^. ie th* product of ^ 'g ^ elf r T* ' ' * ffc ' Denote J\ the algebra of complex or real numbers, i.e. K-R « 1С- С . ы, * # Ю
maps that are defined by the formulae
j. is the constant function, it is the unity of the algebra 9jt/ ( ^ > ^6 X) e £i 8th eUnit yO fth egrou p Jenote A t Go } C 9(*-? ^ ° ' the algebra of smooth . functions on Q& andd denotdenote U,U,*U**" : ЩкЧ,Щк (G"oG ) "^ П l ""о У
the hofflomorphiem defined by the formula '
в SupposO e thatlsOBOrphls there arem0 give: n homomorphiem such that the following diagrams are commutative
flote, that each homomorphism of 414 ( ^ 0 / in К has the fo» f^,l)-» /f^e> °) • "here Jo 6 QQ is the fixed element that defines the homomorphism. • definition. 2.1. Ihe algebra ^o ( (roj together the шогрМввв f^E.,0 ie celled the (global) Lie supergroup, Q.is called the base of this Lie supergroup* Commutative diagrams (2.4), (2.5)» (2.6) are analogue of associativity, right unit and right inverse eleaent »rl^m» of the usual Lie group theory respectivelyч Tbe homomorphism У® I is understood a* follows. If fC2)o£ tfae fore (2.2) then One defines the action of У® J on the arbitrary elements 7 of ЩгЧ/ чСго' *"* linearity and continue» on Oj bjr continuity. ЬовоюогрйАвшв I * V^ I ® £ > * are analoguosly. Denote Q = vx (v0} ^ 0^ t ) the We supergroup de- flood by the Lie group Or, and homomorphisms V, О t£ . c Let it Go be symmetric neighbourhood of the unit о (i.e. such a neioourhood that if 0 £ <* then so doeo Q }. 1'he notion of tne local iie supergroup СтГИл^.О) is introduced like the notion of the global cne. In tnis case, the role, played by the algebras Ц| к» i.M"o / k k = / vl above., is played by algeoras ^ f, ^{ ^ ) , P • ^ Qo } (X being the Cartesian product of ъ copies of" j algeb- К гав 01кр,кц,Ш ) are defined like OjTко, (Qo ) . Let r be a homomorphiera of Wo, (troy into . £vidently, г generates the homomorphism F of *l\ jl) ^Л generalize to any element by co^tinui- ty. Definition 2.2. Call the homomorphism of Lie supergroups ne F» \* {\xo^^ О, £ / > Сг ( GO/ fy O/&) * algebra homomor- provided with The following general notion is often usea in what follows. K Let 4> fx, • •.? 3C ) be infinitly differentiable in ft, real variables and U1/1/=Q.=Q. + t & Uj /s are even elements of Orassmann algebra U/^/, where Q. are numbers and t are nil- potent element», flefine У(% )'" ;% ) € 4A/ using Taylor decompoeition The sum at the right hand aide of (2.7) is a finite one becau- se of nilpoteney of t . She passing^101" the function *РСХ) in real variables to the element f f£+£/ of the Grassmann algebra is in many ways analogous to the usual analytical con- tinuation. . Definition 2.3. The element ffu + ^) of the Qrasamann algebra of the form (2.7) ie called the Grasemann analytical continuation (G.a.c.) of f (й) • Ihe_ example of §_ Lje_su2ergrqup. Let us oonsider the set в жпвг в of sellular matrices X 6\у6,С<^") (с ,5В У » ^7^ are square matrices of the size inXhi end fb* И, reepeotiTely and D and ~ are rectangular matrices. She elements of mat- rices о and С are generators of the Graeaaaaa algotea while elements of A and JS are real or complex nuBbere, such that det A # 0 , det Ъ Ф 0 , Bue'to thij condition X is invertible. X "^ (^ Z? ) *** **» toxm analogoaee to that of A » though A^ Bt oonsist of even elements of О.л*пп = Let us consider Go Ql (™) * ** L t 'O • Elements of 8i. C4 &Z С a. are natu^aily written in the form Note tnat wnere А-ДА*61Сг,е-^Вя+в^г,С»Сг^+?>,С SB * С,&д + 8,2^, • JLLenents of g,4 ; C1; в4/ Сг. ere genera- tors of Щьтл » elements of О, С are ezpreeeed in terms of 6t-,C^ they are generators of ^„h С ^/f). A consist of even elements of Ucmomorphisme О ^ ^ are defined by I (X1) j-(Xi^i) toYe the form analoe0"8 to *bat of the coefficient functions iitu 5 • • • у \ lt ( A ) ® ' being understood in the aence of G.a.c. The commutatiylty of (2.5)- -(2.7) follows from the aseociativity of matrix product'and the properties of unit and inverse elements. The constructed Lie supergroup is denoted by group Q (Cro "f, 0.6/ is its homomorphism in QL fft), И/.Th e linear representation is faithful if the image of ть ft ^.) " C//«j)coinsidee with Ot% (Go). Влша^Хе. Let us consider \x [\jroy ¥ > v, £ / , Let Д- Д"^)с C^fvn), £> = %($)eGl{*)£o™ the linear repre- sentation of Po and 6= О Г^)^ С- С(^) consist of odd elements of the Graesmarm algebra O7e >^= ( E> 3'• " ' ^ J 10 are generators of Of/j,. X (A}}B,CjS>) , } are cell»- lar matrices of the same form ae earlier. Denote ' tho The restriction of I ' on I «i.e. the substitution of ,,C.S) instead of Atf!),Cj£) t providee ae with Ьш- 1 (C morphiem ?''. 0Ль ЛС// ») * CtM) ~* °}t *) ' r' - V r'z ю Г* In case г6 = О г aad г у - У ' tbe linear repre- is sentation of С ((ro Ф,&, £ ) obtained. We shall see below that this situation is a mlvermal one, all linear representations of Lie supergroups hare that form. 2. Recovering a Lie supergroup from a Lie superalgabra> Let Cf = %® tyi be a Lie superalgebra. iJenote ^(-^jSv^ Qo^(M)} К/Ю • "e «roupe with Lie algebras (j(*~\ ^0^) Cjo> X (M)j JtJM) respectively (see ofa.I, n" 3). Denote set of eleaents of the form 9 = &Xf> cc , «here Z*^ el '^> ^ being; odd elemente of ®$Af She algebra Jt(Af) is a nilpotent one, so each elenent A t3i.(Al) could be represented in the form $ - Z*p(Z-Q. X^ * L J ^Z.^^\ where O. t ot are respectively even and odd elemente of the algebra QJV . It follows from this remark that one can «tract the unique square root of each elemente of J(.(Nj : А if cj=expf;?a Xt- + Z^Sp then |' . 4/2 £<£t.(*~), too. £arlier, we noted that the au.tomorphiem (1.4) of the Lie eupera3.gebra 4 generates the automorphiem of the Lie algeb- ras jt^) «ad therefore, it t;enerates the automorphism of «ha Lite group GlCv),g —* £ • •*•* АЭ evident that acinsi&ea with the set of fixed points of thie automorphism 4 ^-VJio the eet of all tfeoee elements 9. that eatiefy *- . 16 £«Uova -Cm» bere «be important propertyi 4 09 theorem 2,1. I) §^д1еош^о£ G rcpreaeatedlntbefone ?,) Beah eleaeat of QO(M) ls^jmlquely is tho foxse 6 <{Ъ в « ^ d в > t 01/ (2<9) 3) Suggoaie e € Go ia jjepreffented jp the for» ? • а'Хг] /Aj,1 J3e4jPg,tue base of tbo JUie algebra Qo . Ihen j ^ are -flTea^nllPOteat. elements of ^. esd £«i ere__gdd elemente of Фд, . Ц t * ^v» (A/) , ttea ,|^' • 0 . T 0 where 3tl ауе_пиддЬага, { . ЯЪе first Btateaent.Sappoes . (Beeall «bat for each 0 e 9 «here ie uniqaely determined element в «^>4(.лГ)|. On the othc;^ •'I, 0 ti» i.e. f e GB f/A • *bue, tae first decomposition of toe element в. ie obtained. Let ue prove the uniqueness. Let Jf$-f~3.tGi» f<£.€Gr0 • € obtained from thie ono: ^ г f Вл --^ Ihe second statement, for the elements from the suffici- ently small neighbourhood]/£ GJ^~)ot the unit elements the eacond decomposition (2.9) follows froa the enalogous decom- position of the Lie algebra $0 My . Let now 4 be an ar- bitrary element, ? *2»'... I-H , f/ -^ ^ 6 ^ /; € £ к- е К (Л/) .. ЫЮ is normal diTisor, so where Q- Ji...3^ Go, k*^.-'J».) Цл. The uniqueness is proved in the nnmn wwy as in the previ- ous case, so is obtained the second decomposition. She third statement. According *» tftf i?*f1n1t1ftnt -each elementjl i« K(tf) has the fora к x e%p (z& %£ + <&•* where OL and +2* Jjj ' • "bere u aro even nilpotent elements of *уд/ A ао£ d are odd onoo. Therefore, |(f ^ «ay one proves'that a = ^aios Icj e •ith respoot to tae^Statemoat of tht> thuort;m I oae could gl»e the following definition. Lot M С (_г„ bu u neighbour- hood of unit that adnitfs the canocioal coordlJaatuu, i.e. g^expfZ^C1 У'^б^, and Jc. bolowgu to /C/4^ . Call Graua» aaaa_canoalcal coordlaateg of ih@ el отел t С'л elemeato definefi fjlom *ne identity c Let If G-o be euch a neighbourhood of unity that the neighbourhood Ij •* О consisting of all products Я-;$х » ^i,3,i £ I' , admitt*o the canonical coordinates. Let X'J^J Ж1(Э2,ч) be the canonical coordinates of elements Q< a. 6 J and Q-^iJ, 6- У respect!viy. Let о '«* tf off* + V х"" кл . 4 + U"i -2 +- /J be Grassmann canonical *' d •* ' , , , coordinates of elements О k^ y A Theorem 2.2. (2.10) vbere 2 ( ЭС+ 4/ ^+ и* J is Grassmann analytical con- tinuation of Z1" r^v^- froof is evident. be Let T (§•/ infinitely differcntiable Xunctio» oa Go «ith values in numbers. Define its Grassmann analiticai 1* continuation on Go [A/) Firstly» let «' €• v and suppose »cffj ie e syaaetr£-> в л я. cal neighbourhood of the unity soon that v adaltjfe canoni- cal coordinates, 4 «^'l , к в К[Ю . bet a:4 be cano- nical coordinates of в' X * A be Graeeaann oanonioal coordinates of О . la euob a oaee ffe'js F(X,•••,&./• Let us define *f(%)*f(i'k)* Pfc** **..., ясМ'Д, e where F fr** I.*, ... , X?*- f. ) 1в м to (2.8). let us fix 0 £ Gn end put 2£r 4 2Г , "У being the ваше neighbourhood of the unit as earlier. In^elde of v» f(£J сЛ are canonical coordinates of Q ° . Let us put fg (£ *vs - ^a fee*^ ,..,*+JiA where 2C +h are Orasanann canonical eo- * » ' • Л? , r / J. I*- <> if") ordinates of the element H and Ь (ОС * * , .«^ЭС *-h У 1в as in (2*в). Index 0, reminds ив of the role of the element 0 in the construction of f. (9 *)• Let ив show that i*(%'4 doee not, in feot, depeni on 0 , Let S € У^Л C/ft-i, . It is sufficient to check the equality ^^^У e where I4,iJ« в wV • denotDenotee XX and H canonical coordi- Khen £1(^ are fixed, the element ^^ la the function of в' , therefore ^^= i* (at1, » .., X ). With the use of Graasmann anelitioal oootinuatiou.ve яь^гпщ the relations between Grasemann canonical coordinates of ele- nento I* t and $x k , I.e. (2.II) The function -f (9 ) is defined in the doeain by the foraula The Gi-aesmanr analytic&l continuation of the function | (ft) is defined in domains ТГ» and *ь£4 by formulae Д (j'k) P j,2 ) respectively. Due to (2.XI) ^4 ty** Л4) ••vJ *гош (2.I2J with the aid of Чтавваава analytical continuation one deduces that does not depend on <У . Hence-lorce, the index Q in (ct (i'k) ie ою" mited. It ia evident that ^ fa' jQ С ОТ, now we are able to introduce a definition. An element f {%'k) € 0\^ ( (?0J ie called Oraaoaann analytical continuation of the function f ( i) onto Qe (Л/). Let ^/3»l^/3 ) € 5<5^^ V Go) • further on we often had to substitute о с Qo(Af) as argument Q in •{•($,£) in- stead of Q e Go • ^ A ^ 1° euoh a case is defined by replacing all coefficient functions r^ • (o\ by thoir Grassmann analytical continuation on Q. fjj}. e Call odd elements 5 5 . • •, S 4// independent if they ienerate a Grasamann algebra 0^ с. ОТ.. le c Call X fey . .. j <£ * ) O(A/J the cheacterietic oet if it is the set of elements of Сму of the foxa the fixed Q -tuple of independent Graeemann runs Qo . ^\ Let j'ijfi6 .A(*)Mt' '/. Due to Theorem 2.1 i ^- .n~,n r ,. - • ^^ eieaentb of ЬУ^ de- pending on ? as on the parameter, 9 -$'(п)^)^^ к ф ^r^i) e К (-W) « I* follows from the ^' c =? tuat ^c couldd be expressed with respect to £' by the formula l^^fydJ),^,.--. k'V , V* the same as earlier. Let А/^Лр Suppose ^ , i у 1,2; J^i,...,^ be 2^, independent odd elements, for simplicity, denote ^^ ^.-tuples ^ = ^5^ ...,?•/• ie^ us multiply elements of characteristic ee't X r^i.) ЬУ tbose of X tie) • According to the tbeom I , tt.14) .here Q, € Qo (Д/), *- ^^ » *' V o One easyly deduces from the Campbell-Hausdorff foroula that if- vf (|^ ^^ o.^ ^^ „в independent odd elements. The multiple 9, in the right hand «ide of (2.14) depends on Q± 4^ and on 0 -tuples J4 ? , i.e. We are able now to construct a supergroup proceeding from the Lie euperalgebra И , first of all, let us construct л Lie group Go proceeding from the Lie algebra ff .Next, let us consider an algebra Ofp (Go) consisting of ele- •e&te of the form (.2.1), £* being some independent elements of Qjp . Denote homomorphisms /'• 0[^ ( with respect to (2.13) and (2.14): and •£(<}'j')0J<: considered in the sence of Grassmann analytical oontinuation. Note, that the homomorphism 0 is isomorphism. (It follows from the possibility to express £'J by ^ ). Comoutativity of diagrams (2.7)-(2.6) follows easyly from (2.13), (2.14). Mote, that the first step in construction of a Lie super- group, i.e. the recovering of a Lie group Go with respect to its Lie algebra И , is not simple in a way, because Lie group Go Is defined only up to the local isomorphism. the next step has Go as the initial point and is unique. Let iv be the representation of Lie superalgebra Q ~ ^&®9 in the graded space E = Ц, + fc i ^ "(Yc] be a homoseneous base In 18 generates the representation of QfA/J in the envelope £ (M) otoE , If Xs"" Let T# be the corresponding repreeeatation of The representation T» being restricted on the characteris- tic set X (.?)•••; 1 у ее obtain the representation of Lie- supergroup G- . la. fact, the aatrix «f the operatorVin the homogeneous base has the form [%, fcj *be t **" ^* being put in f instead of A, О, С, <) ве obtain a honoaor- i л» /л i f \ n / f \\ ^ (t ff* | F * Ч*£.П\П \.{r *- i'^J * Ur ^ (*l /y >J ( vJO/" = It is evident that PC=u»*>FY Vr , «here Oy V; &*, 4* are the structural hamomorphisms of G-lfibjh/ and Gr respectively. (Cf. the example at the end of n° I). Xhe construction of recovering of the Lie algebra by its Lie group known from the Lie group theory ie generalised almost lit/ferally on supergroup case ( Г5] i. Thus, £u each of three corresponding to each other rep- resentation: that of the Lie superalgebra, that of the Lie su- pergroup and that of the group Q.[A/) for -Л/^9 the other two are recovered uniquely, for reasons that could be clear from the next a0, the just described representations of built with the use of representations of Lie ettperalgeoree are called Oraesmann analytical (G.a.r.). 3. Algebras JX.^ ( Ц/ and D/y (C?' . Let being odd elements of 0fv. Let us assigne to each element and each qc-C fj/) the бледен Denote Лд/(£/ a set of elements (2.17). It is evident tht\t ^jj (Qr) is an algebri and that the homomorphism J*' ^Jtf^oJ—* &• Kt (&У defined by (2.l5> is the isomorp- hism of algebras when л/ ^ ^ . It is natural to call the al- gebra Ау(сг) the algebra of Grassmann analytical (G.a.) functions on G-(A/J . Suppose Л/^У, and let At J , -•-, be characteristic set. The restriction of f f$.j e. С S ••-,§/ defines the inverse map ft ; ^^ Xhe ressemblance with analytical functions being continued, note that characteristic set serves as set of uniqueness,i.e. for a then Рл (%)= $а.(£) H 3 fe С /^,) . functions S"*4, ...,et'«- form the base of Лц/[(r). The 1 1 b e grading £ (в- " ... e 0 *» (n^odZ-) lc called natural.. 'let JTK/v(lr / be the algebra analogouse to ьЖд/С&у, fc Cenot e tne me that ls-iepmorphic to 0j K,, TG ) • fin °*i~4 oned isomorphism. It is nature to call elements of \А*.л/ [^ J G.a. functions in к variables: £ = f ($i,- •• i%±) Let us consider the isomorphism A ; J\,*i fG) "~* «Лу and homomorphiem v^: AJJ(Q) ~~Ъ n»u (G у defined by 1 Isomorphism O . ^[t,(Or0)—"» 9f [ &оУ and homomorp- hiem 20 So axioms (2.4)-(2.6) for о} on is a G.a.c. of function "i*t... 4Л,ТХ... i» (}) £ JJenote ву (Gr) the set of functioaa on Cr (M) »itb Talu- es in Olfi, of the forms for J.= £#fe), f.»e' being the same as in (2.17) u^ Q ' ~ ' (2.16/ It is natural to call elements of SA/ (Oy Craseaann analyti- cal functions with Graesmann coefficients. As the elements of Л"д/ (Q/ they are uniquely determined by their restricti- ons on characteristiacteristic sets. . Denote в0/А, (Cyiesp. в4 *, {Gy the subset 0^ &v (Q-J consisting of elements satisfying d"(fC4f 4... ^ f 'rii--- iJs k + ij • ** ie evident that i are algebras, &, h jQ) is a linear space, ® ^ 1 velope of j(v fQ.^ provided witn S /в *-. . . <У^)- ^И ри»^ It ie important that &o t/ [&) ,ttJj. i / ( Q^) and tnerefore &u (Gc) are Invariant with respect to left and right shifts. In fact, let . The element Qs л/ /v|| depends on Q, and Q. in a Grassmann analytical way. So the coefficient functions of £/£)5-£ . ё. в// iGj oouW be represented in the Г-ii.... -Cfcf f'^anu ф? ... ^(fli/ J\ are G.a. funotiona in tb** first variable. further on © ^ ~ ^ (?у S') ? ^ GT " J depends on 0. and H in a Grassmann analytical fashion and it is the odd element of Фу ( Ge у . So where «*^..^к , ^,.,-t, _ are G.a. functi- ons In «he first two variables, $ (li'f + 1, if ^ 6 вс,# (G-) . -t - 0,1 then /, (p*4(}f) In case Gro is a complex group we are needed besides JKu o () the invo v; n A/ [G'J l '- e the complex structure of Gr0 • Ihe algebra Ж# (Q-J is built w.r.t. 07^ fff^ (cf.n°I) fl as A/y (Gr) i built w.r.t. ОТ. ((то) , i-e. it consists of all elements of the form (20l5) where ^ (Q ^) e ^^(ОтЛ. The space в^д/ ("Й and algebras (B^ (0); (B^J (G; consist of elements of the form (2.l£), where /^ . , ^ ^i?) V are complex analytical functionctions of Q 0 eQ , parity being the same ar in & • § Finally, note that G.a. continuation in' of о defined in the previous n° have the following characteristic 22 properties 1) The representation To acta in е/л£/ whieb la th* Grassmann envelope of graded space £F* ^e* ^4. • 2) Matrix elements 4. Ые operators in algebra <8д/ (Qj. Let в (b ) = &№ *X. be a I-parameter subgroup of Or (№) generated by an ele- ment x feQf f/V) • Define as usual left and right Lie opera- tors corresponding to the element CC ', It would be convenient later on to discuss'left and right Lie operators simultaneously. Рот Щ£в -рпгровв. *a introduce tfao sign Ф for г or -t . Let ^ Ъе an odd element of Ojfv} t € Ojfv} ; It ia evident that the operator Lx has the fo*m /. «ruere A a;, is the uniquely determined operator in Suppose is a homogeneous base of /* Let L'~ are defined from the identity £ t «• oC A* > where is a odd element of • • л* ^.-^у. . Operators Zj and /I- generate a repree-- ectation of the Ыв superelgebra п • lo shorten, put we have ; X constants being the same in both cases. '£he proof of this proposition is simple, so it is. ommited. An arbitrary Lie operator in (8л/ (Or) could be expressed with respect to H: by the formula where Q. are homogeneous elements of satisfying S( 2* that acts on 0ft (Qo ) , i.e. W; «Й ^ J* Call operators П£ lie operators in Us supergroup (r . 5. Various coordinates in Ые supergroup. Let *ic vr© be coordinate neighbourhood of flome element and ОЦ»|»Ч/Ь# the algebra' analogous to 07 j, (Go) oonsieting of elements of the form {2.1), 0 fc 1С. A set of p even elements "4 and 9, odd elantnta tl9 la celled a system of generators in Utp «, I *v ^ вяоЬ element e C6p * (V) could be represented in the form where 1± ^K (<4j are infinetely differentlable functions. Call local.coordinates an arbitrary system of generators of Let S^ be local coordinates 61S.J-0 «hen С * p.? §* ISCV d , when ^ > P .It follows froifi the definition of ope- rators Ht* (see sec, 4) that in local coordinates these ope- rators are differential operators of the first order ( -~^ stands for left .derisatira fat к >• P X*. Ibe. local К coordinates being changed, the coefficients Q • transfer by tensor law; (the derivative is supposed to be a left one, if fc > p). ?here eoald be given to operators L x acting in Drf(Q) a sore usual air of differential operates of order I. Уог this purpuee let us take an agreement. Let ILcR be an open set, 01^^(11) be the Qraeemann algebra with Vv\ generators, its coeffici- ents are infinitly differentiable functions A- • ('X.) de- fined on ll « (iSlements ffx-j^) €. Oih, ъ (It) have the same form, as elements t2.I} ). Let l fc 1 obtained from tf fx }... X J .,, J ; by substitution of some' even elements -U"1- e. O(SK ТП I ^") instead of J; and и some odd elements *i& в (Xh h, /'Z'V instead of X •<> independency is supposed. Define the derivatives; as the resuxt of substitution 4 t ^ instead of JC, 4 in Thie agreeaient in mind, we can write an arbitrary opera- 1 tor N^ acting in Юд/ 1Ч / in the ечте form as operator "t" acting in \JL> a ( G"o) • '^hus, an-arbitrary operator ^--^—^.C^M- obtainee the form of a 1-order differential operator, if consi- dtired w.r.t. coordinates. Let us give examples of the most important local coordi- nate systems. EKample I. bet £ be fixed odd generators. Let ue ta- ke the local coordinates on li as even generators. Let us fix a system of independent odd elements of the Greseroann algebra Щ-» generated by £ * and suppo- se dCf)= ^pC^^f^^i ) • Generators 3C*, |* me caanf^ with the decomposition |= £o (x) &(Т) . We call these generators Cartал generators, because of exceptional role that the decomposition Q=QO& plays in Cartan's theory of symmetric spaces. The other coordinate systems are introduced by compari- son with the Cartan coordinate system. £xample_2. bet Ц be a neighbourhood of the unity of the group GrV\ that admitts the canonical coordinates, й fa^ 9(^ j having the same meaning as earlier. According to the third assirtion of the Theorem 1 ^4; + г : :, „ч i.i £•« It is easy to check that 4 ; 4 (X. £Л S = ^ 0 Q \ are generators of the algebra 0£ф в (lA.) • ^e call them the Grassmann canonical coordinates. We met tham earlier. ijxaiD£le 3. -Let CJ. be a reductive Lie group, H be its Cartan subgroup. Let us fix a regular element ^^' no^ " and denote (/. the neighbourhood of h0 in G with the *' Lie group is called reductive if its Lie algebra is reductive. Kecall a definition. Let Z-L c(7 be a subgroup of the к о О . . The element h,o is oalled regu- lar if Hi - H "0 property I) И/In consists of regular elements. 2). fiacn element Qo 6 M could be represent ей la the form }4V|eZ4"V^^Zr , ITIT being neighbourhood of the unity of Q . the coordinates In (A are introduced as follows. Denote С и the manifold with the property that each element 0 б И la uniquely presented in tne form Q.o~Q.hq~\ h e Ч()№> Я&Н (Such a manifold у exicts if the neighbourhood и is suf- ficiently small, see below}. Let %\..., J^ *~c&m ^ be coordinates in ^ ? -£;... , i ; 6= dim И be coordinates in fi • therefore V X are the coordinates in Coordinates $>^ are called the polar coordinates, those . цs fu1 ....ч"") are called angular coordinates and "4 =(£•)—> J ar« OEtlled radial ones. Let U5 consider now the Lie supergroup Q- with the Lie algebra £j. of the type X, Qo being its basic group. Let H c be toe Carten subgroup of £o , И (Л/) G- (A/) be the c«atraii«er of H in G-(^) "*^, Denote ^ c^ a Lie sub- •' in other words n[N) consists of all those elements 0 eCi^/tfaat satisfy a V» — ^* ^ tor ааУ U П . It is easy to Ш9Ч that if vr is a Lie supergroup of the type I then ele- •tnts of W(tJj have the form Q = exp (H. Х^УгУ , where •i^iil is a base of the Lie algebra 7€ corresponding to H * and ЗС^аге arbitrary even elements of *• algebra of Ц corresponding to the Lie group И • Lot м eon- eider in the Lie algebra > a linear transformation ft* (h) '• X н» Г к xl} Ъ & 3£ .The Lie superalgebra 4 ie deooa- posed as a linear space into the direct BUB of two oubapacea invariant with respect to act- (L) . One of these subepaoea ie "P? itself, denote б^0 being the invariant complement to Ж in v Let /.£ be a homogeneous base in the Lie superalgebra 4 euch that \УЛ for 1S-C * ^ *o™ a base in JC i *or /»-i « i-t & P they £orm. a base in И, . Element в € Cr0 ie called eu- perregular if g = exp эс^ X € ^( , X is superregular. Lemma 2.1. Let n, € tfC fee a_ Buperregular_elamen$. ) k K/VJ agjd ^ & ^ (2Л7) 1) The^i^meat n^ is uniquely determined and could b represented in the form «here К /И/)/) ) «Mi. • 2> Therecould be^ghoeea the element «hgre \ Ij j 1д tfae_dgaS£^lied above^baeg ^n theLie^superglgefara C\ «• The e1ement K^ with this, property is uniquely determi- ned. The ease lemma is valid also without the presumptftan on the type of the Lie supergroup С • but in this paper such a geaeral form is not needed. The proof o» leiuma S.I is in the following section. 'She lemma 2.1 allows «e to introduce in a lie supergroup Q po- lar coordinates. Suppose (lc(J is a domain consisting of dndj ° superregular elements 'admitting polar coordinates, let be independent odd elements of Recondition | 'с М implysthat j' where к (b) G W} о fy} € ^ С ?Г . Hence, Zj ) , \*я1 (%»&)(%»&)*** independenpt ^ele- ж ments. According to the lemma 2}I k. (*) ®[\)' <л n-i К-л where k,. It, Ы>*Т (¥^(М) Yi \ \f U, KW)l) n[N), we find, that with respect to the theo- where (\, ("tj^Jare nilpotent even elements. Xhus, It is evident that the set of even and odd elements V,. • •,H л f л * ° S, ..•, & j \ f •••Л/ le tne eet of generators of the al- gebra O"Lp, 30 There is another possible way to Introduce polar coordi- nates. If the neighbourhood U ie sufficiently smell, then the manifold 'У could be chosen so that it consists of the elements of the type 9 , fy- £*p ( 2. # Y^ . The theorem 2Л being applied, aN)i([i:,i) ~ e.xP[Zа 'Ъ) an* *ne U elements & = ft [ 4 . •? }~Ь ) are nilpotent for С > ~t . Ihe Cajrpbell-Mauedorff formula impliesJ In turn, this implies with respect to (2.21) bj (2.19' The new polar coordinates (3CV,S ) differ from the old ones in the set of angular variables, i.e. ЭС instead of uv , >ii , £(*')= 0 for ^ + 4*1 - p and S(xl) • i for p + 1 ^ i & p + q. 6. frroof of the lemma 2.1. It follows from Theorem 2.1 that lemma 2.1 is the corollary of the analogous lemma in Lie algebra case that we formulate in the ваше form that shall be needed in ch.3. Lemma 2.2. Let к £ Ч„ be superregular element , 31 and__hag^the_form 2) Let ^ Vt| be_^^baee_^onnec$ad_juth J\ in the ваше fas- hionas the eiql^ar .base In-jtwgna jj j. e^ could__fee_choosen_JLn a tbe foxa Ri ^ RY V^; • Under_Jthesa_condition3 й1 is and ^ where (2.20) yields that &а* A ~ A £ 3d Ш) . het us expand &} LL in the power series in generators of Од/ and deno- * 1" i te fc« j r\£. b.1 « the sum of elements homogeneous w.r.t. = degree. Substituting &=.Z£Sjli, 2bS) 1^= 2^.J)S in (2.20) we obtain the recurrent equations for п§} ^-iys Ц ^° (2.2JJ where Suppose that \^.^ are defined for S 38 (2.23) Suppose ЛЬ»п (2.23) is uniquely solvable. In fact, the equation has the following form componentwise j where II A ,' II is the matrix of 0Ло (k) . Thie matrix is the solution of (2.23) Bed with the use of (2.24). у-ти / The last assertion. Suppose в / i where ft. ^ yy dependp s polynomiallp y on n , we find 1л (li)y h C^-t-t » where ^и + -< depends po- lynomially on K. . It follows from (2.25) that gv л+J has the form analogouse to that of fe^ к ^ 1П ^ Ц>^ + ^_ . bet Д Y^ I is the ваше base as earlier, U s ZZ ЦУ J^ ~ iTopoeitioa 2.1. Por d. where С £•£ ^e__g^uotural_jcoaetecte^of у- injthebase Proof. U = £хь -fOjuLfac^Js vthe aext lB evident. 7, Ide operators in в canonical and polar coordinates. Canonical coordinates, het ti С Q be a neighbourhood of ttae unity that, admltte the canonical coordinates, Ц,\N) С QIN) be a domi2i sonsietlng of elements oS the form U-0o tnere are canonical coordiaatee in W My also. bet \ YX'J be a homogeneous base in j- , o(Yi) = O . , when '/;> 1 when p + ^ i S f>+b j ОС = Ы% "Z. ^°Уо be elements ot fy {/I/) . fixpaad -6t ft e into the power series witb respect,to the real parameter £ ; iie operator of left translation c.x corresponding to the el-eat * has the for» (£^)ф. iL j? ^ + 6t) j^ Oar goal is to find the relation between Z, X. and M . In «bat follows it is convenient to present the elements of the Lie algebra Q. (A/) by Lie operators of left translati- ons in the spaoe $дг (Q-J and to present the elements of the group Q [M) as left shifts. Xhls interpretation in miljfci. we obtain the usual integral equation for U (tf £J , i.e. Proa this equation Я у It is easyly proved in the usual way that if belongs to the sufficiently small neighbourhood of zero of the l £ie algebra Qo then there exiete the operator 9*r. vJ. and therefore Ъ could be expressed with respect to According to (2.26), the Lie operator l~x in the cano- nical coordinates has the form (2.28) It follows from (2.27) that Z * ЭС when U~0. 4. Sac «u •* In the similar'way the element в 6 •= € * being considered, we find the operator L^. \ (2.29) the relation between the element 1r - £. v". 7-t, V and ОС. being given by Яепсе, If ~ 31 when 4=0. Polar^coordinates. We shallfneed\notythe exact expression of Lie operators buu the first terms of decomposition of these Lie operators wif;b respect to angular coordinates, only. Let T Y< (D e the base in ^ needed for the construction of polar coordinates ; Yi , . • } У I is a base in a Carton sub- algebra Ж, V^i ... , Yp+£ ls a base in $. . Let 4Z * Y* % ? * $+1 * ' * J роев U is superregular and let us introduce the polar coordi- nates in the neighbourhood of the elements, h [t. ) = •& let us interpret© elements of the lie algebra <-d /И/^) аз Lie operators of left translation in S/i/(G~J and let us realize the elements of Q {A/j as the left ehifta in вд/ (Qj. Let us present the element в $- 'Н»^/ ^ £ being the real pa- rameter, in the form *Ь" l^ ^) 6 4C^"J, let us sxpand the both sides in the pcvrer series with respect to £ and write out the terms of the degree О and 1 ( ) ( ) t*t эв Where tritr(oiy)i , О'-О-ф,^) . Pat f« 05+^» x where o"£ , . vTc are constants, OTj. , 0"4 are linear tune- tionnes in 4 . Jrwn (2.31) we obtain Suppose J«. . As 1r € *J (2.33) where for V»U we have Q, - •* when <• - ^ and u - «r - "v. •*- depend on ОС , • » • , X only, when-С >f In case X * 5" Ti'= and (2.3$) implies that <Г„ = v, <ГО - 0, <*»J It follows from (2.34) that tor -Х.Ь~Ж the coefficients CL" of the operator (2.33) being expanded into the power series with respect to u*0 have the form end the terms ft. [4,X.; involve V In the power no lesa than 2, If* feeing the coefficient of V corresponding to le*e Y*' <• > ^ > defined by (2.34). She analogous analysis may be held {or a Lie operators of right translation. a. The invariant integral. In this section we define the linear functional on Q^ (QJ invariant with respect to the shifts. This functional has values in the Grassmann algebra on €> •/({j-/and in numbemwhen restricted to A(M)C On/ As £ С ^((J-Jas uniquly determined by its values on the cha- racteristic set, it is natural to find this functional in the £orm J where J» J.ett), JJ.. € € G Ge,e y & Z is a measure on Cr0 and the integral with respect to л' is understood as in Гб J. In future, I (•£) is called the Integral on the Lie supergroup. The integral I(-fy is called left-invariant if it does not change when | (<£) is replaced by $»> ($)~ H^%) •?* definition of the right-invariant integral is analogous. The integral is called Invariant if it is both left- and right- invariant. Let %~%Л « Gr M) §oS G0) t € ) •.when ILv = J (2.36) where PM (Ct^) = a{$*fr),t) #&)• Puttie called density. Coordlaa-fces Xl and elements jT ^ aerve ae a syetes of generatora in CtypaiU). In what follows it ie iaportant to know the description of the integral j(jf ) in the fora analogous to (2.36) w^th respect to an arbitrary eyetex of i5iMerui;o:?Li. The density depends on a system of generators. To underline this circumstance we shall write sometimes j^, £ where'stands for a system of generators, -instead of mere Ру( Let '&U ?J' T*ft) C2.37) where 3=^."» Л 2e^'-' '^^ ?S Z* oeing generators in (hp* & ) • йю transformation of genera- tors (2.37) could be dependable on parameters, i.e. or1— = гЧПК^); 1*= 7'fail**)- 3й) are even and odd para- meters respectively. Exact meaning of parameter' in (2.37) is that the isomorpfaic injection с ^; - J Ь <*fci), Щ, г)) ?u> ft* г) (2.39) and 2Г stands for a eyatem д$ i^^Zstande for a sys- tem Г7]. There is a relation analogous to (2.39) Iheorem_2.3.I. Ther^are left- and right-invariant 2. Two sides lnvariantjtntegral on^ Lj^_aupergrottp exiats^iff s^* aJ(ct) ^o f« aU Ot tf ^ . ( atands for eupartra^ce, а ее the deflniUon in Appendix|^. Proof. Ibejuilgnjaees. let {А С Go be a eooxdinoto aetgbbourbood and ULcU be a eubdoaain,end -if be & faouzbood of tbe unity at Go «ice tbat g^ € U for fo 6 a g Hi . Ш* to Tbeoreo 2.1 tbe identity is Talid SOT у a ($l,- are generatore in ^p<^ Wl\ ~$6 being para»etW3. Sappoee , ^ U*^) veciabeeon tbe boundary of t^i ea£ eo во all its derivations. Let The lstwrloDtneM oondltlon ie \ Lj(* According to tbe formula of tbe «ranefoaaatioo of reexite a left band aide of (2.40) »i«j respect to tbe genera- tors и *} jj in ^^, Tbe variables of integration In (2.4OJ being redeaotect, i.e. * "* ?/ ^ -• 2 ^г f ind th0 ioveriantnega oondition . witb respect to the deaeity (Ям condition (2.4i) differs froe (2.39): there is f>M z in toe both sides of the equality (2.44), while there is P^r' ia the left bead side of (2.39)). Suppose, there is a left-invariant integral with the