General properties of classical W algebras

TL;DR

This paper reviews the construction, classification, and new properties of classical W algebras, including secondary reductions and the role of Kac-Moody subalgebras, highlighting their structures and realizations.

Contribution

It introduces the concept of secondary reduction of W algebras and analyzes the impact of Kac-Moody subalgebras on their structure and realizations.

Findings

Chains of W algebras can be generated by constraints (secondary reduction).

Factorizing out Kac-Moody subalgebras leads to rational or polynomial W algebra structures.

New realizations of W algebras as finitely generated or W_infinity types are identified.

Abstract

After some definitions, we review in the first part of this talk the construction and classification of classical $W$ (super)algebras symmetries of Toda theories. The second part deals with more recently obtained properties. At first, we show that chains of $W$ algebras can be obtained by imposing constraints on some $W$ generators: we call secondary reduction such a gauge procedure on $W$ algebras. Then we emphasize the role of the Kac-Moody part, when it exists, in a $W$ (super) algebra. Factorizing out this spin 1 subalgebra gives rise to a new $W$ structure which we interpret either as a rational finitely generated $W$ algebra, or as a polynomial non linear $W_\infty$ realization. (Plenary talk presented by P. SORBA at the $XXII^{th}$ International Conference on Differential Geometric Methods in Theoretical Physics. Ixtapa Mexico, September 1993.)