Deformed Virasoro algebras from elliptic quantum algebras

TL;DR

This paper revisits the construction of deformed Virasoro algebras from elliptic quantum algebras, generalizing previous methods and identifying critical central charge values with extended centers and a Liouville formula.

Contribution

It introduces a generalized bilinear trace procedure connecting elliptic quantum algebras with vertex operator techniques, revealing new algebraic structures and critical points.

Findings

Identification of two critical central charge values with extended centers

Connection established between elliptic quantum algebras and vertex operator methods

Derivation of a Liouville formula in the $gl(2)$ case)

Abstract

We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 90's. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period. As a by-product, the method pinpoints two critical values of the central charge for which the center of the algebra is extended, as well as (in the $gl(2)$ case) a Liouville formula.