Remarks on Finite W Algebras

TL;DR

This paper discusses properties of finite W algebras, their role in constructing realizations of Lie algebras, and explores their applications in representation theory and affine extensions, including a brief overview of alternative constructions.

Contribution

It introduces new insights into finite W algebras' properties, their use in Lie algebra realizations, and extends these ideas to affine cases with an alternative Wakimoto construction.

Findings

Finite W algebras can serve as commutants to realize Lie algebras.

Unitary representations of conformal and Poincare algebras are constructed via W algebras.

An alternative Wakimoto construction for sl(2) level k is presented.

Abstract

The property of some finite W algebras to be the commutant of a particular subalgebra of a simple Lie algebra G is used to construct realizations of G. When G=so(4,2), unitary representations of the conformal and Poincare algebras are recognized in this approach, which can be compared to the usual induced representation technique. When G=sp(2,R) or sp(4,R), the anyonic parameter can be seen as the eigenvalue of a W generator in such W representations of G. The generalization of such properties to the affine case is also discussed in the conclusion, where an alternative of the Wakimoto construction for sl(2) level k is briefly presented. This mini review is based on invited talks presented by P. Sorba at the ``Vth International Colloquium on Quantum Groups and Integrable Systems'', Prague (Czech Republic), June 1996; ``Extended and Quantum Algebras and their Applications to Physics'', Tianjin (China), August 1996; ``Selected Topics of Theoretical and Modern Mathematical Physics'', Tbilisi (Georgia), September 1996; to be published in the Proceedings.