On quantum toroidal algebra of type $A_1$

TL;DR

This paper introduces a new quantum algebra related to the 2-toroidal Lie algebra of type A1, establishing its algebraic structure and vertex operator realization.

Contribution

It presents a novel quantum toroidal algebra with a triangular decomposition, Hopf algebra structure, and vertex operator realization for type A1.

Findings

Established a triangular decomposition for the algebra

Proved the algebra has a Hopf algebra structure

Constructed a vertex operator realization

Abstract

In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.