Representation Theory of W-Algebras

TL;DR

This paper investigates the representation theory of W-algebras associated with simple Lie algebras, establishing the exactness of the reduction functor and linking characters of irreducible modules to those of affine Lie algebras.

Contribution

It proves the exactness of the reduction functor for W-algebras and relates their irreducible module characters to affine Lie algebra representations.

Findings

The reduction functor is exact and maps irreducible modules to either zero or irreducible modules.

Characters of irreducible W-algebra modules are determined by affine Lie algebra characters.

Provides extended results on the representation theory of W-algebras.

Abstract

This paper is the detailed version of math.QA/0403477 (T. Arakawa, Quantized Reductions and Irreducible Representations of W-Algebras) with extended results; We study the representation theory of the W-algebra $W_k(g)$ associated with a simple Lie algebra $g$ (and its principle nilpotent element) at level k. We show that the "-" reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k. Moreover, we show that the character of each irreducible highest weight representation of $W_k(g)$ is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of $g$.