Finite W-algebras

TL;DR

This paper introduces finite W-algebras as symplectic reductions of simple Lie algebras, constructs a finite analogue of W^{(2)}_3, and explores its representations.

Contribution

It presents the first detailed study of finite W-algebras, including the construction and analysis of their highest weight representations.

Findings

Finite W-algebras are realized via symplectic reductions.

A finite analogue of W^{(2)}_3 is constructed and analyzed.

Unitary and non-unitary representations are explicitly built.

Abstract

Finite versions of W-algebras are introduced by considering (symplectic) reductions of finite dimensional simple Lie algebras. In particular a finite analogue of $W^{(2)}_3$ is introduced and studied in detail. Its unitary and non-unitary, reducible and irreducible highest weight representations are constructed.