**Mathematical Surveys and Monographs**

Volume: 229;
2018;
304 pp;
Hardcover

MSC: Primary 17; 16;

Print ISBN: 978-1-4704-3659-9

Product Code: SURV/229

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-4391-7
Product Code: SURV/229.E**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

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#### Supplemental Materials

# Sugawara Operators for Classical Lie Algebras

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*Alexander Molev*

The celebrated Schur-Weyl duality gives rise
to effective ways of constructing invariant polynomials on the
classical Lie algebras. The emergence of the theory of quantum groups
in the 1980s brought up special matrix techniques which allowed one to
extend these constructions beyond polynomial invariants and produce
new families of Casimir elements for finite-dimensional Lie
algebras. Sugawara operators are analogs of Casimir elements for the
affine Kac-Moody algebras.

The goal of this book is to describe algebraic structures
associated with the affine Lie algebras, including affine vertex
algebras, Yangians, and classical \(\mathcal{W}\)-algebras, which have numerous ties
with many areas of mathematics and mathematical physics, including
modular forms, conformal field theory, and soliton equations. An
affine version of the matrix technique is developed and used to
explain the elegant constructions of Sugawara operators, which
appeared in the last decade. An affine analogue of the Harish-Chandra
isomorphism connects the Sugawara operators with the classical
\(\mathcal{W}\)-algebras, which play the role of the Weyl group
invariants in the finite-dimensional theory.

#### Readership

Graduate students and researchers interested in algebraic aspects of representation theory and applications to mathematical physics.

#### Table of Contents

# Table of Contents

## Sugawara Operators for Classical Lie Algebras

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xi12
- Chapter 1. Idempotents and traces 116
- Chapter 2. Invariants of symmetric algebras 2338
- Chapter 3. Manin matrices 4358
- Chapter 4. Casimir elements for 𝔤𝔩_{𝔑} 5368
- 4.1. Matrix presentations of simple Lie algebras 5368
- 4.2. Harish-Chandra isomorphism 5570
- 4.3. Factorial Schur polynomials 5873
- 4.4. Schur–Weyl duality 6075
- 4.5. A general construction of central elements 6176
- 4.6. Capelli determinant 6378
- 4.7. Permanent-type elements 6580
- 4.8. Gelfand invariants 6681
- 4.9. Quantum immanants 6782
- 4.10. Bibliographical notes 6984

- Chapter 5. Casimir elements for 𝔬_{𝔑} and 𝔰𝔭_{𝔑} 7186
- 5.1. Harish-Chandra isomorphism 7186
- 5.2. Brauer–Schur–Weyl duality 7489
- 5.3. A general construction of central elements 7691
- 5.4. Symmetrizer and anti-symmetrizer for 𝔬_{𝔑} 7893
- 5.5. Symmetrizer and anti-symmetrizer for 𝔰𝔭_{𝔑} 8398
- 5.6. Manin matrices in types 𝐵, 𝐶 and 𝐷 89104
- 5.7. Bibliographical notes 90105

- Chapter 6. Feigin–Frenkel center 91106
- 6.1. Center of a vertex algebra 91106
- 6.2. Affine vertex algebras 93108
- 6.3. Feigin–Frenkel theorem 96111
- 6.4. Affine symmetric functions 101116
- 6.5. From Segal–Sugawara vectors to Casimir elements 103118
- 6.6. Center of the completed universal enveloping algebra 104119
- 6.7. Bibliographical notes 106121

- Chapter 7. Generators in type 𝐴 107122
- Chapter 8. Generators in types 𝐵, 𝐶 and 𝐷 119134
- Chapter 9. Commutative subalgebras of 𝑈(𝔤) 149164
- Chapter 10. Yangian characters in type 𝐴 169184
- 10.1. Yangian for 𝔤𝔩_{𝔑} 169184
- 10.2. Dual Yangian for 𝔤𝔩_{𝔑} 177192
- 10.3. Double Yangian for 𝔤𝔩_{𝔑} 180195
- 10.4. Invariants of the vacuum module over the double Yangian 183198
- 10.5. From Yangian invariants to Segal–Sugawara vectors 185200
- 10.6. Screening operators 186201
- 10.7. Bibliographical notes 190205

- Chapter 11. Yangian characters in types 𝐵, 𝐶 and 𝐷 191206
- Chapter 12. Classical 𝒲-algebras 213228
- Chapter 13. Affine Harish-Chandra isomorphism 243258
- Chapter 14. Higher Hamiltonians in the Gaudin model 269284
- Chapter 15. Wakimoto modules 277292
- Bibliography 295310
- Index 303318
- Back Cover Back Cover1321