W-Algebras via Lax Type Operators

Abstract

W -algebras are certain algebraic structures associated to a finitedimensional Lie algebra 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 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.

A slow review of the AGT correspondence

“…These developments are based on various underlying symmetry algebras that generalize W-algebras [122,746] and would deserve a review of their own by someone more qualified ( [458] looks like a good starting point): qW-algebra symmetries and quiver W-algebras [77,126,467,472,523,591,593,594,[747][748][749][750][751][752][753][754][755][756][757][758][759][760][761][762], the spherical Hecke central algebra [763][764][765], the Ding-Iohara-Miki algebra [34,212,213,458,461,464,654,670,697,706,707,736,737,741,[766][767][768]…”