**Mathematical Surveys and Monographs**

Volume: 143;
2007;
400 pp;
Hardcover

MSC: Primary 17; 81;
Secondary 05

Print ISBN: 978-0-8218-4374-1

Product Code: SURV/143

List Price: $108.00

Individual Member Price: $86.40

**Electronic ISBN: 978-1-4704-1370-5
Product Code: SURV/143.E**

List Price: $108.00

Individual Member Price: $86.40

#### Supplemental Materials

# Yangians and Classical Lie Algebras

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*Alexander Molev*

The Yangians and twisted Yangians are remarkable associative algebras taking their origins from the work of St. Petersburg's school of mathematical physics in the 1980s. The general definitions were given in subsequent work of Drinfeld and Olshansky, and these algebras have since found numerous applications in and connections with mathematical physics, geometry, representation theory, and combinatorics.

The book is an introduction to the theory of Yangians and twisted Yangians, with a particular emphasis on the relationship with the classical matrix Lie algebras. A special algebraic technique, the \(R\)-matrix formalism, is developed and used as the main instrument for describing the structure of Yangians. A detailed exposition of the highest weight theory and the classification theorems for finite-dimensional irreducible representations of these algebras is given.

The Yangian perspective provides a unifying picture of several families of Casimir elements for the classical Lie algebras and relations between these families. The Yangian symmetries play a key role in explicit constructions of all finite-dimensional irreducible representations of the orthogonal and symplectic Lie algebras via weight bases of Gelfand-Tsetlin type.

#### Table of Contents

# Table of Contents

## Yangians and Classical Lie Algebras

- Contents vii8 free
- Preface xi12 free
- Chapter 1. Yangian for gl[sub(N)] 120 free
- 1.1. Defining relations 120
- 1.2. Matrix form of the defining relations 221
- 1.3. Automorphisms and anti-automorphisms 524
- 1.4. Poincaré–Birkhoff–Witt theorem 726
- 1.5. Hopf algebra structure 928
- 1.6. Quantum determinant and quantum minors 1231
- 1.7. Center of the algebra Y(gl[sub(N)]) 1635
- 1.8. Yangian for sl[sub(N)] 1837
- 1.9. Quantum Liouville formula 2039
- 1.10. Factorization of the quantum determinant 2342
- 1.11. Gauss decomposition 2746
- 1.12. Quantum Sylvester theorem 3150
- 1.13. Gelfand–Tsetlin subalgebra 3352
- 1.14. Bethe subalgebras 3453
- 1.15. Examples 3655
- Bibliographical notes 4261

- Chapter 2. Twisted Yangians 4564
- 2.1. Defining relations 4564
- 2.2. Matrix form of the defining relations 4766
- 2.3. Automorphisms and anti-automorphisms 4867
- 2.4. Embedding into the Yangian 4968
- 2.5. Sklyanin determinant 5271
- 2.6. Sklyanin minors 5675
- 2.7. Explicit formula for the Sklyanin determinant 5978
- 2.8. The center of the twisted Yangian 6483
- 2.9. The special twisted Yangian 6685
- 2.10. Coideal property 6786
- 2.11. Quantum Liouville formula 6887
- 2.12. Factorization of the Sklyanin determinant 7089
- 2.13. Extended twisted Yangian 7291
- 2.14. Quantum Sylvester theorem 7796
- 2.15. An equivalent presentation of Y(g[sub(N)]) 82101
- 2.16. Examples 85104
- Bibliographical notes 91110

- Chapter 3. Irreducible representations of Y(gl[sub(N)]) 93112
- Chapter 4. Irreducible representations of Y(g[sub(N)]) 131150
- Chapter 5. Gelfand–Tsetlin bases for representations of Y(gl[sub(N)] 185204
- Chapter 6. Tensor products of evaluation modules for Y(gl[sub(N)] 203222
- Chapter 7. Casimir elements and Capelli identities 239258
- 7.1. Newton's formulas 239258
- 7.2. Noncommutative Cayley–Hamilton theorem 245264
- 7.3. Graphical constructions of Casimir elements 246265
- 7.4. Higher Capelli identities and quantum immanants 253272
- 7.5. Noncommutative Pfaffians and Hafnians 262281
- 7.6. Capelli identities for o[sub(N)] and sp[sub(2n)] 266285
- 7.7. Examples 271290
- Bibliographical notes 275294

- Chapter 8. Centralizer construction 279298
- 8.1. Olshanski algebra associated with gl[sub(∞)] 279298
- 8.2. Virtual Casimir elements and highest weight modules for gl[sub(∞)] 281300
- 8.3. Polynomial invariants for gl[sub(N)] 286305
- 8.4. Algebraic structure of A(gl[sub(∞)]) 291310
- 8.5. Skew representations of Y(gl[sub(N)]) 295314
- 8.6. Olshanski algebra associated with g[sub(∞)] 300319
- 8.7. Virtual Casimir elements and highest weight modules for g[sub(∞)] 301320
- 8.8. Polynomial invariants for g[sub(N)] 304323
- 8.9. Algebraic structure of A(g[sub(∞)]) 311330
- 8.10. Examples 316335
- Bibliographical notes 320339

- Chapter 9. Weight bases for representations of g[sub(N)] 323342
- 9.1. The Mickelsson algebra theory 323342
- 9.2. Mickelsson–Zhelobenko algebra Z(g[sub(N)],g[sub(N–2)]) 326345
- 9.3. Twisted Yangian and Mickelsson–Zhelobenko algebra 331350
- 9.4. Yangian action on the multiplicity space 341360
- 9.5. Basis of the multiplicity space 352371
- 9.6. Basis of V(λ) 356375
- 9.7. Examples 375394
- Bibliographical notes 379398

- Bibliography 381400
- Index 399418

#### Readership

Graduate students and research mathematicians interested in representation theory and quantum groups.

#### Reviews

This book is well written and will be indispensable for anyone working on Yangians. It will also be of interest for the new point of view it brings to branching rules. Some versions of the theorems proved hold for other algebras, in particular quantized enveloping algebras of finite and affine type. Each chapter concludes with examples of these versions, and bibliographic notes linking the chapter to the extensive bibliography at the end of the book.

-- Mathematical Reviews