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Sugawara Operators for Classical Lie Algebras Mathematical Surveys and Monographs Volume 229 Sugawara Operators for Classical Lie Algebras Alexander Molev Sugawara Operators for Classical Lie Algebras Mathematical Surveys and Monographs Volume 229 Sugawara Operators for Classical Lie Algebras Alexander Molev EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 17B35, 17B63, 17B67, 17B69, 16S30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-229 Library of Congress Cataloging-in-Publication Data Names: Molev, Alexander, 1961- author. Title: Sugawara operators for classical Lie algebras / Alexander Molev. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Mathe- matical surveys and monographs ; volume 229 | Includes bibliographical references and index. Identifiers: LCCN 2017041529 | ISBN 9781470436599 (alk. paper) Subjects: LCSH: Lie algebras. | Affine algebraic groups. | Kac-Moody algebras. | AMS: Nonas- sociative rings and algebras – Lie algebras and Lie superalgebras – Universal enveloping (su- per)algebras. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Poisson algebras. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras. msc | Nonas- sociative rings and algebras – Lie algebras and Lie superalgebras – Vertex operators; vertex operator algebras and related structures. msc | Associative rings and algebras – Rings and algebras arising under various constructions – Universal enveloping algebras of Lie algebras. msc Classification: LCC QA252.3 .M6495 2018 | DDC 512/.482–dc23 LC record available at https://lccn.loc.gov/2017041529 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 select pages 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. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2018 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/ 10987654321 232221201918 Oruenoscu Kaxke Contents Preface xi Chapter 1. Idempotents and traces 1 1.1. Primitive idempotents for the symmetric group 1 1.2. Primitive idempotents for the Brauer algebra 6 1.3. Traces on the Brauer algebra 14 1.4. Tensor notation 17 1.5. Action of the symmetric group and the Brauer algebra 19 1.6. Bibliographical notes 21 Chapter 2. Invariants of symmetric algebras 23 2.1. Invariants in type A 23 2.2. Invariants in types B,C and D 28 2.3. Symmetrizer and extremal projector 39 2.4. Bibliographical notes 41 Chapter 3. Manin matrices 43 3.1. Definition and basic properties 43 3.2. Identities and invertibility 45 3.3. Bibliographical notes 51 Chapter 4. Casimir elements for glN 53 4.1. Matrix presentations of simple Lie algebras 53 4.2. Harish-Chandra isomorphism 55 4.3. Factorial Schur polynomials 58 4.4. Schur–Weyl duality 60 4.5. A general construction of central elements 61 4.6. Capelli determinant 63 4.7. Permanent-type elements 65 4.8. Gelfand invariants 66 4.9. Quantum immanants 67 4.10. Bibliographical notes 69 Chapter 5. Casimir elements for oN and spN 71 5.1. Harish-Chandra isomorphism 71 5.2. Brauer–Schur–Weyl duality 74 5.3. A general construction of central elements 76 5.4. Symmetrizer and anti-symmetrizer for oN 78 5.5. Symmetrizer and anti-symmetrizer for spN 83 5.6. Manin matrices in types B, C and D 89 5.7. Bibliographical notes 90 vii viii CONTENTS Chapter 6. Feigin–Frenkel center 91 6.1. Center of a vertex algebra 91 6.2. Affine vertex algebras 93 6.3. Feigin–Frenkel theorem 96 6.4. Affine symmetric functions 101 6.5. From Segal–Sugawara vectors to Casimir elements 103 6.6. Center of the completed universal enveloping algebra 104 6.7. Bibliographical notes 106 Chapter 7. Generators in type A 107 7.1. Segal–Sugawara vectors 107 7.2. Sugawara operators in type A 114 7.3. Bibliographical notes 117 Chapter 8. Generators in types B, C and D 119 8.1. Segal–Sugawara vectors in types B and D 119 8.2. Low degree invariants in trace form 128 8.3. Segal–Sugawara vectors in type C 134 8.4. Low degree invariants in trace form 142 8.5. Sugawara operators in types B, C and D 145 8.6. Bibliographical notes 147 Chapter 9. Commutative subalgebras of U(g) 149 9.1. Mishchenko–Fomenko subalgebras 149 9.2. Vinberg’s quantization problem 155 9.3. Generators of commutative subalgebras of U(glN ) 157 9.4. Generators of commutative subalgebras of U(oN )andU(spN ) 165 9.5. Bibliographical notes 167 Chapter 10. Yangian characters in type A 169 10.1. Yangian for glN 169 10.2. Dual Yangian for glN 177 10.3. Double Yangian for glN 180 10.4. Invariants of the vacuum module over the double Yangian 183 10.5. From Yangian invariants to Segal–Sugawara vectors 185 10.6. Screening operators 186 10.7. Bibliographical notes 190 Chapter 11. Yangian characters in types B, C and D 191 11.1. Yangian for gN 191 11.2. Dual Yangian for gN 202 11.3. Screening operators 206 11.4. Bibliographical notes 211 Chapter 12. Classical W-algebras 213 12.1. Poisson vertex algebras 213 12.2. Generators of W(g) 216 12.3. Chevalley projection 226 12.4. Screening operators 228 12.5. Bibliographical notes 241 CONTENTS ix Chapter 13. Affine Harish-Chandra isomorphism 243 13.1. Feigin–Frenkel centers and classical W-algebras 243 13.2. Yangian characters and classical W-algebras 255 13.3. Harish-Chandra images of Sugawara operators 259 13.4. Harish-Chandra images of Casimir elements 263 13.5. Bibliographical notes 268 Chapter 14. Higher Hamiltonians in the Gaudin model 269 14.1. Bethe ansatz equations 269 14.2. Gaudin Hamiltonians and eigenvalues 271 14.3. Bibliographical notes 275 Chapter 15. Wakimoto modules 277 15.1. Free field realization of glN 277 15.2. Free field realization of oN 280 15.3. Free field realization of sp2n 284 15.4. Wakimoto modules in type A 287 15.5. Wakimoto modules in types B and D 289 15.6. Wakimoto modules in type C 292 15.7. Bibliographical notes 294 Bibliography 295 Index 303 Preface In representation theory of Lie algebras, Casimir operators are commonly un- derstood as certain expressions constructed from generators of a Lie algebra which commute with its action. Their spectra are useful for understanding the represen- tation. In particular, finite-dimensional irreducible representations of a simple Lie algebra g over the field of complex numbers are characterized by the eigenvalues of the Casimir operators. This fact is based on a theorem of Harish-Chandra describ- ing the center Z(g) of the associated universal enveloping algebra U(g). The center is isomorphic to an algebra of polynomials via the Harish-Chandra isomorphism ∼ (0.1) Z(g) = C L1,...,Ln . Here n is the rank of g and L1,...,Ln are polynomial functions in the highest weights of the representations, each Li is invariant under a certain action of the Weyl group of g. The isomorphism (0.1) relies on a theorem of Chevalley which can also be recovered as a ‘classical limit’ of (0.1). Namely, the symmetric algebra S(g) is isomorphic to the graded algebra gr U(g), and the subalgebra of g-invariants in S(g) is isomorphic to gr Z(g). Taking the symbols Mi of the polynomials Li,we get the Chevalley isomorphism g ∼ (0.2) S(g) = C M1,...,Mn . The respective degrees d1,...,dn of the Weyl group invariants M1,...,Mn coincide with the exponents of g increased by 1. A vast amount of literature both in mathematical physics and representation theory has been devoted to understanding the correspondence in (0.1) in terms of concrete generators on both sides, especially for the Lie algebras g of classical types A, B, C and D. Various families of generators of the center Z(g) were discovered together with their Harish-Chandra images. The simple Lie algebras g can be regarded as a part of the family of Kac–Moody algebras parameterized by generalized Cartan matrices. Of particular importance is the class of affine Kac–Moody algebras which admits a simple presentation. The (untwisted) affine Kac–Moody algebra g is the central extension g[t, t−1] ⊕ CK of the Lie algebra of Laurent polynomials with coefficients in g. Basic results of representation theory of these Lie algebras together with applications to conformal field theory, modular forms and soliton equations can be found in the book by V. Kac [86]. Motivated by the significance of the Lie algebras g,onecomesto wonder what the center of U(g) looks like.
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