Representation theory of finite W algebras

TL;DR

This paper explores the structure, quantization, and representation theory of finite W algebras, revealing their connections to Lie algebra embeddings, integrable systems, and providing explicit constructions and realizations.

Contribution

It introduces a coordinate-free formula for finite W algebras, proves their embedding into Kirillov Poisson algebras, and develops a unified BRST quantization method.

Findings

Finite W algebras can be embedded into Kirillov Poisson algebras (generalized Miura map).

Finite Toda systems are reductions of free particle systems with finite W symmetry.

Explicit Fock space realizations of finite W algebras are constructed.

Abstract

In this paper we study the finitely generated algebras underlying $W$ algebras. These so called 'finite $W$ algebras' are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings of $sl_2$ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finite $W$ algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finite $W$ symmetry. In the second part we BRST quantize the finite $W$ algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finite $W$ algebras in one stroke. Explicit results for $sl_3$ and $sl_4$ are given. In the last part of the paper we study the representation theory of finite $W$ algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finite $W$ algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finite $W$ algebras.