Representations of shifted Yangians and finite W-algebras / Jonathan Brundan, Alexander Kleshchev.

Format:

Book

Author/Creator:

Brundan, Jonathan, 1970- author.

Kleshchëv, A. S. (Aleksandr Sergeevich), author.

Series:

Memoirs of the American Mathematical Society ; no. 918.

Memoirs of the American Mathematical Society, 0065-9266 ; number 918

Language:

English

Subjects (All):

Representations of quantum groups.

Lie superalgebras.

Physical Description:

1 online resource (122 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2008]

Language Note:

English

Summary:

The authors study highest weight representations of shifted Yangians over an algebraically closed field of characteristic $0$. In particular, they classify the finite dimensional irreducible representations and explain how to compute their Gelfand-Tsetlin characters in terms of known characters of standard modules and certain Kazhdan-Lusztig polynomials. The authors' approach exploits the relationship between shifted Yangians and the finite W-algebras associated to nilpotent orbits in general linear Lie algebras.

Contents:

""Contents""; ""Chapter 1. Introduction""; ""Acknowledgements""; ""Chapter 2. Shifted Yangians""; ""2.1. Generators and relations""; ""2.2. PBW theorem""; ""2.3. Some automorphisms""; ""2.4. Parabolic generators""; ""2.5. Hopf algebra structure""; ""2.6. The center of Y[sub(n)](Ï?)""; ""Chapter 3. Finite W-algebras""; ""3.1. Pyramids""; ""3.2. Finite W-algebras""; ""3.3. Invariants""; ""3.4. Finite W-algebras are quotients of shifted Yangians""; ""3.5. More automorphisms""; ""3.6. Miura transform""; ""3.7. Vanishing of higher T[sup(r)][sub(i,j)]'s""; ""3.8. Harish-Chandra homomorphisms""

""Chapter 4. Dual canonical bases""""4.1. Tableaux""; ""4.2. Dual canonical bases""; ""4.3. Crystals""; ""4.4. Consequences of the Kazhdan-Lusztig conjecture""; ""Chapter 5. Highest weight theory""; ""5.1. Admissible modules""; ""5.2. Gelfand-Tsetlin characters""; ""5.3. Highest weight modules""; ""5.4. Classification of admissible irreducible representations""; ""5.5. Composition multiplicities""; ""Chapter 6. Verma modules""; ""6.1. Parametrization of highest weights""; ""6.2. Characters of Verma modules""; ""6.3. The linkage principle""; ""6.4. The center of W(Ï€)""

""6.5. Proof of Theorem 6.2""""Chapter 7. Standard modules""; ""7.1. Two rows""; ""7.2. Classification of finite dimensional irreducible representations""; ""7.3. Tensor products""; ""7.4. Characters of standard modules""; ""7.5. Grothendieck groups""; ""Chapter 8. Character formulae""; ""8.1. Skryabin's theorem""; ""8.2. Tensor identities""; ""8.3. Translation functors""; ""8.4. Translation commutes with duality""; ""8.5. Whittaker functor""; ""Notation""; ""Bibliography""

Notes:

"November 2008, volume 196, number 918 (end of volume)."

Includes bibliographical references (pages 105-107).

Description based on print version record.

ISBN:

1-4704-0524-5