Two-parameter quantum vertex representations via finite groups and the McKay correspondence

TL;DR

This paper introduces two-parameter quantum toroidal algebras of simply laced types, using finite group representations via McKay correspondence, and constructs their vertex representations for quantum affine algebras of ADE types.

Contribution

It provides a novel group-theoretic realization of two-parameter quantum toroidal algebras and their vertex representations for ADE types.

Findings

Realization of two-parameter quantum toroidal algebras using finite groups.

Construction of vertex representations for quantum affine algebras of ADE types.

Connection established between McKay correspondence and quantum algebra representations.

Abstract

We introduce two-parameter quantum toroidal algebras of simply laced types and provide their group theoretic realization using finite subgroups of $SL_2(\mathbb C)$ via McKay correspondence. In particular our construction contains a realization of the vertex representation of the two-parameter quantum affine algebras of $ADE$ types.