Yangians and Classical Lie Algebras
About this Title
Alexander Molev, University of Sydney, Sydney, Australia
Publication: Mathematical Surveys and Monographs
Publication Year 2007: Volume 143
ISBNs: 978-0-8218-4374-1 (print); 978-1-4704-1370-5 (online)
MathSciNet review: MR2355506
MSC (2000): Primary 17B37; Secondary 16W35, 82B23
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.
Graduate students and research mathematicians interested in representation theory and quantum groups.
Table of Contents
- 1. Yangian for
- 2. Twisted Yangians
- 3. Irreducible representations of
- 4. Irreducible representations of
- 5. Gelfand-Tsetlin bases for representations of
- 6. Tensor products of evaluation modules for
- 7. Casimir elements and Capelli identities
- 8. Centralizer construction
- 9. Weight bases for representations of