Non-Polynomial Realizations of W-Algebras

TL;DR

This paper explores new realizations of W-algebras by relaxing constraints in Hamiltonian reduction, providing explicit quantum expressions for several important algebras in the context of non-exceptional simple Lie algebras.

Contribution

It introduces non-polynomial realizations of W-algebras through a modified Hamiltonian reduction process, extending the understanding of their structure and explicit quantum forms.

Findings

Derived explicit quantum expressions for Virasoro, W3, and Bershadsky algebras.

Provided general results for non-exceptional simple Lie algebras.

Connected the realizations to the commutant of nilpotent subalgebras in enveloping algebras.

Abstract

Relaxing first-class constraint conditions in the usual Drinfeld-Sokolov Hamiltonian reduction leads, after symmetry fixing, to realizations of W algebras expressed in terms of all the J-current components. General results are given for G a non exceptional simple (finite and affine) algebra. Such calculations directly provide the commutant, in the (closure of) G enveloping algebra, of the nilpotent subalgebra $G_-$, where the subscript refers to the chosen gradation in G. In the affine case, explicit expressions are presented for the Virasoro, $W_3$, and Bershadsky algebras at the quantum level.