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