Article

Representation theory of rectangular finite $W$-algebras

Journal of Algebra (Impact Factor: 0.6). 03/2010; 340(1). DOI: 10.1016/j.jalgebra.2011.05.014
Source: arXiv

ABSTRACT

We classify the finite dimensional irreducible representations of rectangular finite $W$-algebras, i.e., the finite $W$-algebras $U(\mathfrak{g}, e)$ where $\mathfrak{g}$ is a symplectic or orthogonal Lie algebra and $e \in \mathfrak{g}$ is a nilpotent element with Jordan blocks all the same size. Comment: 34 pages

Full-text preview

Available from: ArXiv
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A finite W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. In this survey we review recent developments in the representation theory of W-algebras. We emphasize various interactions between W-algebras and universal enveloping algebras. Comment: The text of a sectional talk for ICM 2010. 21 pages
    Full-text · Article · Mar 2010 · Physics Letters B
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider finite W-algebras U(g, e) associated to even multiplicity nilpotent elements in classical Lie algebras. We give a classification of finite dimensional irreducible U(g, e)-modules with integral central character in terms of the highest weight theory from Brundan et al. (Int. Math. Res. Notices 15, art. ID rnn051, 2008). As a corollary, we obtain a parametrization of primitive ideals of U(g, e) with associated variety the closure of the adjoint orbit of e and integral central character.
    Full-text · Article · Sep 2010 · Mathematische Zeitschrift
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A highest weight theory for a finite W-algebra U(g,e) was developed in [BGK]. This leads to a strategy for classifying the irreducible finite dimensional U(g,e)-modules. The highest weight theory depends on the choice of a parabolic subalgebra of g leading to different parameterizations of the finite dimensional irreducible U(g,e)-modules. We explain how to construct an isomorphism preserving bijection between the parameterizing sets for different choices of parabolic subalgebra when g is of type A, or when g is of types C or D and e is an even multiplicity nilpotent element
    Full-text · Article · May 2011 · Journal of Algebraic Combinatorics
Show more