http://dx.doi.org/10.1090/surv/143 Yangians and Classical Lie Algebras Mathematical Surveys and Monographs Volume 143 Yangians and Classical Lie Algebras Alexander Molev American Mathematical Society ^VDED" EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 17B37, 81R50; Secondary 17B10, 17B20, 81R10, 05E10, 05E15. For additional information and updates on this book, visit www.ams.org/bookpages/SURV-143 Library of Congress Cataloging-in-Publication Data Molev, Alexander, 1961- Yangians and classical Lie algebras / Alexander Molev. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 143) Includes bibliographical references and index. ISBN 978-0-8218-4374-1 (alk. paper) 1. Lie algebras. 2. Quantum groups. 3. Matrices. 4. Representations of algebras. 5. Symmetry (Mathematics). I. Title. QA252.3.M65 2007 512/.482—dc22 2007060781 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 ITocBumaeTCH B.-JL Contents Preface xi Chapter 1. Yangian for giN 1 1.1. Defining relations 1 1.2. Matrix form of the defining relations 2 1.3. Automorphisms and anti-automorphisms 5 1.4. Poincare-Birkhoff-Witt theorem 7 1.5. Hopf algebra structure 9 1.6. Quantum determinant and quantum minors 12 1.7. Center of the algebra Y(giN) 16 1.8. Yangian for 5 liv 18 1.9. Quantum Liouville formula 20 1.10. Factorization of the quantum determinant 23 1.11. Gauss decomposition 27 1.12. Quantum Sylvester theorem 31 1.13. Gelfand-Tsetlin subalgebra 33 1.14. Bethe subalgebras 34 1.15. Examples 36 Bibliographical notes 42 Chapter 2. Twisted Yangians 45 2.1. Defining relations 45 2.2. Matrix form of the defining relations 47 2.3. Automorphisms and anti-automorphisms 48 2.4. Embedding into the Yangian 49 2.5. Sklyanin determinant 52 2.6. Sklyanin minors 56 2.7. Explicit formula for the Sklyanin determinant 59 2.8. The center of the twisted Yangian 64 2.9. The special twisted Yangian 66 2.10. Coideal property 67 2.11. Quantum Liouville formula 68 2.12. Factorization of the Sklyanin determinant 70 2.13. Extended twisted Yangian 72 2.14. Quantum Sylvester theorem 77 2.15. An equivalent presentation of Y(QN) 82 2.16. Examples 85 Bibliographical notes 91 Chapter 3. Irreducible representations of Y(Q[N) 93 viii CONTENTS 3.1. Drinfeld presentation of the Yangian 93 3.2. Highest weight representations 104 3.3. Representations of Y(g[2) HI 3.4. Representations of Y^^) 120 3.5. Examples 124 Bibliographical notes 128 Chapter 4. Irreducible representations of Y(QN) 131 4.1. Split presentation of the twisted Yangian 131 4.2. Highest weight representations 137 4.3. Representations of Y(sp2n) 145 4.4. Representations of Y(o2n) 154 4.5. Representations of Y(o2n+i) 168 4.6. Examples 182 Bibliographical notes 184 Chapter 5. Gelfand-Tsetlin bases for representations of Y(g[Ar) 185 5.1. Yangian of level p 185 5.2. Lowering and raising operators 189 5.3. Action of the Drinfeld generators 195 5.4. Gelfand-Tsetlin bases for representations of glN 197 5.5. Examples 200 Bibliographical notes 201 Chapter 6. Tensor products of evaluation modules for Y(g[iV) 203 6.1. Sufficient conditions 203 6.2. Necessary conditions 208 6.3. Irreducibility criterion 220 6.4. Fusion procedure for the symmetric group 223 6.5. Multiple tensor products 229 6.6. Examples 236 Bibliographical notes 237 Chapter 7. Casimir elements and Capelli identities 239 7.1. Newton's formulas 239 7.2. Noncommutative Cayley-Hamilton theorem 245 7.3. Graphical constructions of Casimir elements 246 7.4. Higher Capelli identities and quantum immanants 253 7.5. Noncommutative Pfaffians and Hafnians 262 7.6. Capelli identities for ON and sp2n 266 7.7. Examples 271 Bibliographical notes 275 Chapter 8. Centralizer construction 279 8.1. Olshanski algebra associated with QI^ 279 8.2. Virtual Casimir elements and highest weight modules for gl^ 281 8.3. Polynomial invariants for giN 286 8.4. Algebraic structure of A(glOQ) 291 8.5. Skew representations of Y(g[iV) 295 8.6. Olshanski algebra associated with g^ 300 CONTENTS ix 8.7. Virtual Casimir elements and highest weight modules for g^ 301 8.8. Polynomial invariants for $N 304 8.9. Algebraic structure of A(^oo) 311 8.10. Examples 316 Bibliographical notes 320 Chapter 9. Weight bases for representations of QN 323 9.1. The Mickelsson algebra theory 323 9.2. Mickelsson-Zhelobenko algebra Z(g^v, 97v-2) 326 9.3. Twisted Yangian and Mickelsson-Zhelobenko algebra 331 9.4. Yangian action on the multiplicity space 341 9.5. Basis of the multiplicity space 352 9.6. Basis of V(X) 356 9.7. Examples 375 Bibliographical notes 379 Bibliography 381 Index 399 Preface The theory of classical matrix Lie algebras can be viewed from at least two related but different perspectives. On the one hand, the special linear, orthogonal and symplectic Lie algebras form four infinite series, An, Bn, Cn, Dn, which to­ gether with five exceptional Lie algebras, E6, £7, E$, F4, £?2, comprise a complete list of the simple Lie algebras over the field of complex numbers. The structure of these Lie algebras is uniformly described in terms of certain finite sets of vectors in a Euclidean space called the root systems. The symmetries of the root systems play a key role in the representation theory of all simple Lie algebras providing the dimension and character formulas for the representations. On the other hand, the matrix realizations of the classical Lie algebras allow some specific tools to be used for their study which are not always available for the exceptional Lie algebras. The theory of Yangians and twisted Yangians which we develop in this book is one of such tools bringing in new symmetries and shedding new light on this classical subject. The Yangians and twisted Yangians are associative algebras whose defining re­ lations are written in a specific matrix form. We describe the structure of these algebras and classify their finite-dimensional irreducible representations. The re­ sults exhibit many analogies with the representation theory of the classical Lie algebras themselves, including the triangular decompositions of the (twisted) Yan­ gians and the parametrization of the representations by their highest weights. In the simplest cases explicit constructions of the irreducible representations are also given. Then we apply the Yangian symmetries to the classical Lie algebras. The ap­ plications include constructions of several families of Casimir elements, derivations of the characteristic identities and Capelli identities, and explicit constructions of all finite-dimensional irreducible representations of the classical Lie algebras via weight bases of Gelfand-Tsetlin type. Let us discuss the relationship between the classical Lie algebras and the (twisted) Yangians in more detail. The term Yangian was introduced by V. G. Drin- feld (in honor of C. N. Yang) in his fundamental paper (1985). In that paper, Drinfeld also defined the quantized Kac-Moody algebras, which together with the work of M. Jimbo (1985), who introduced these algebras independently, marked the beginning of the era of quantum groups. The Yangians form a remarkable family of quantum groups related to rational solutions of the classical Yang-Baxter equa­ tion. For each simple finite-dimensional Lie algebra a over the field C of complex numbers, the corresponding Yangian is defined as a canonical deformation of the universal enveloping algebra U(a[z]) for the polynomial current Lie algebra a[z\. Importantly, the deformation is considered in the class of Hopf algebras, which guarantees its uniqueness under some natural homogeneity conditions. Another presentation of the Yangian for a was given later by Drinfeld (1988). xi xii PREFACE A few years earlier, the algebra, which is now called the Yangian for the gen­ eral linear Lie algebra #lN and denoted by Y^t^), was considered in the work of L. D. Faddeev and the St. Petersburg (Leningrad) school. The defining relations of the Yangian Y(g(iV) can be written in the form of a single ternary (or RTT) relation on the matrix of generators. This relation has a rich and extensive back­ ground. It originates from the quantum inverse scattering method', see for instance L. A. Takhtajan and Faddeev (1979), P. P. Kulish and E. K. Sklyanin (1982), and Faddeev (1984). The Yangians were primarily regarded as a vehicle for producing rational solutions of the Yang-Baxter equation; cf. Drinfeld (1985). Conversely, the ternary relation is a powerful tool for studying quantum groups themselves; see e.g.