$W$-algebras via Lax type operators

Abstract

$W$-algebras are certain algebraic structures associated to a finite dimensional Lie algebra $\mathfrak{g}$ and a nilpotent element $f$ via Hamiltonian reduction. In this note we give a review of a recent approach to the study of (classical affine and quantum finite) $W$-algebras based on the notion of Lax type operators. For a finite dimensional representation of $\mathfrak{g}$ a Lax type operator for $W$-algebras is constructed using the theory of generalized quasideterminants. This operator carries several pieces of information about the structure and properties of the $W$-algebras and shows the deep connection of the theory of $W$-algebras with Yangians and integrable Hamiltonian hierarchies of Lax type equations.

15 pages. This is a (short) review and summary of the theory of Lax type operators for $W$-algebras for the proceedings of the XIth International Symposium Quantum Theory and Symmetry in Montreal