Finite $W$ symmetry in finite dimensional integrable systems

TL;DR

This paper introduces finite versions of $W$ algebras via Poisson reduction of Lie algebra structures, explores their properties, representations, and connections to finite Toda systems, expanding the understanding of $W$ symmetries in finite dimensions.

Contribution

It constructs finite $W$ algebras from Lie algebra Poisson structures, providing explicit formulas, examples, and representation theory insights, thus extending $W$ algebra theory to finite-dimensional systems.

Findings

Finite $W$ algebras can be obtained by Poisson reduction of Lie algebra structures.

Explicit coordinate-free formulas for the reduced Poisson structures are provided.

Finite Toda systems exhibit the non-linear finite $W$ symmetry discussed.

Abstract

By generalizing the Drinfeld-Sokolov reduction a large class of $W$ algebras can be constructed. We introduce 'finite' versions of these algebras by Poisson reducing Kirillov Poisson structures on simple Lie algebras. A closed and coordinate free formula for the reduced Poisson structure is given. These finitely generated algebras play the same role in the theory of $W$ algebras as the simple Lie algebras in the theory of Kac-Moody algebras and will therefore presumably play an important role in the representation theory of $W$ algebras. We give an example leading to a quadratic $sl_2$ algebra. The finite dimensional unitary representations of this algebra are discussed and it is shown that they have Fock realizations. It is also shown that finite dimensional generalized Toda theories are reductions of a system describing a free particle on a group manifold. These finite Toda systems have the non-linear finite $W$ symmetry discussed above. Talk given at the `workshop on low dimensional topology and physics', Cambridge, September 1992.