Cover

Title page

Contents

Preface

Chapter 1. Idempotents and traces

1.1. Primitive idempotents for the symmetric group

1.2. Primitive idempotents for the Brauer algebra

1.3. Traces on the Brauer algebra

1.4. Tensor notation

1.5. Action of the symmetric group and the Brauer algebra

1.6. Bibliographical notes

Chapter 2. Invariants of symmetric algebras

2.1. Invariants in type 𝐴

2.2. Invariants in types 𝐵,𝐶 and 𝐷

2.3. Symmetrizer and extremal projector

2.4. Bibliographical notes

Chapter 3. Manin matrices

3.1. Definition and basic properties

3.2. Identities and invertibility

3.3. Bibliographical notes

Chapter 4. Casimir elements for 𝔤𝔩_{𝔑}

4.1. Matrix presentations of simple Lie algebras

4.2. Harish-Chandra isomorphism

4.3. Factorial Schur polynomials

4.4. Schur–Weyl duality

4.5. A general construction of central elements

4.6. Capelli determinant

4.7. Permanent-type elements

4.8. Gelfand invariants

4.9. Quantum immanants

4.10. Bibliographical notes

Chapter 5. Casimir elements for 𝔬_{𝔑} and 𝔰𝔭_{𝔑}

5.1. Harish-Chandra isomorphism

5.2. Brauer–Schur–Weyl duality

5.3. A general construction of central elements

5.4. Symmetrizer and anti-symmetrizer for 𝔬_{𝔑}

5.5. Symmetrizer and anti-symmetrizer for 𝔰𝔭_{𝔑}

5.6. Manin matrices in types 𝐵, 𝐶 and 𝐷

5.7. Bibliographical notes

Chapter 6. Feigin–Frenkel center

6.1. Center of a vertex algebra

6.2. Affine vertex algebras

6.3. Feigin–Frenkel theorem

6.4. Affine symmetric functions

6.5. From Segal–Sugawara vectors to Casimir elements

6.6. Center of the completed universal enveloping algebra

6.7. Bibliographical notes

Chapter 7. Generators in type 𝐴

7.1. Segal–Sugawara vectors

7.2. Sugawara operators in type 𝐴

7.3. Bibliographical notes

Chapter 8. Generators in types 𝐵, 𝐶 and 𝐷

8.1. Segal–Sugawara vectors in types 𝐵 and 𝐷

8.2. Low degree invariants in trace form

8.3. Segal–Sugawara vectors in type 𝐶

8.4. Low degree invariants in trace form

8.5. Sugawara operators in types 𝐵, 𝐶 and 𝐷

8.6. Bibliographical notes

Chapter 9. Commutative subalgebras of 𝑈(𝔤)

9.1. Mishchenko–Fomenko subalgebras

9.2. Vinberg’s quantization problem

9.3. Generators of commutative subalgebras of 𝑈(𝔤𝔩_{𝔑})

9.4. Generators of commutative subalgebras of 𝑈(𝔬_{𝔑}) and 𝔘(𝔰𝔭_{𝔑})

9.5. Bibliographical notes

Chapter 10. Yangian characters in type 𝐴

10.1. Yangian for 𝔤𝔩_{𝔑}

10.2. Dual Yangian for 𝔤𝔩_{𝔑}

10.3. Double Yangian for 𝔤𝔩_{𝔑}

10.4. Invariants of the vacuum module over the double Yangian

10.5. From Yangian invariants to Segal–Sugawara vectors

10.6. Screening operators

10.7. Bibliographical notes

Chapter 11. Yangian characters in types 𝐵, 𝐶 and 𝐷

11.1. Yangian for 𝔤_{𝔑}

11.2. Dual Yangian for 𝔤_{𝔑}

11.3. Screening operators

11.4. Bibliographical notes

Chapter 12. Classical 𝒲-algebras

12.1. Poisson vertex algebras

12.2. Generators of 𝒲(𝔤)

12.3. Chevalley projection

12.4. Screening operators

12.5. Bibliographical notes

Chapter 13. Affine Harish-Chandra isomorphism

13.1. Feigin–Frenkel centers and classical 𝒲-algebras

13.2. Yangian characters and classical 𝒲-algebras

13.3. Harish-Chandra images of Sugawara operators

13.4. Harish-Chandra images of Casimir elements

13.5. Bibliographical notes

Chapter 14. Higher Hamiltonians in the Gaudin model

14.1. Bethe ansatz equations

14.2. Gaudin Hamiltonians and eigenvalues

14.3. Bibliographical notes

Chapter 15. Wakimoto modules

15.1. Free field realization of 𝔤𝔩_{𝔑}

15.2. Free field realization of 𝔬_{𝔑}

15.3. Free field realization of 𝔰𝔭_{2𝔫}

15.4. Wakimoto modules in type 𝐴

15.5. Wakimoto modules in types 𝐵 and 𝐷

15.6. Wakimoto modules in type 𝐶

15.7. Bibliographical notes

Bibliography

Index

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