A matrix realization of the quantum group g_{p, q}

TL;DR

This paper constructs a matrix realization of the quantum group g_{p, q} by developing primitive idempotents, a basis, and analyzing their action on modules, leading to insights into symmetric linear functions.

Contribution

It provides a novel matrix realization of g_{p, q} and characterizes symmetric linear functions in terms of this realization, advancing understanding of its algebraic structure.

Findings

Constructed all primitive idempotents of g_{p, q}

Developed a basis and matrix representation of g_{p, q}

Expressed symmetric linear functions via this basis

Abstract

In this paper we will find a matrix realizations of the quantum group g_{p, q}. For this purpose, we construct all primitive idempotents and a basis of g_{p, q}. We determine the action of elements of the basis on the indecomposable projective modules, which give rise to a matrix realization of g_{p, q}. By using this result, we obtain a basis of the space of symmetric linear functions on g_{p, q}} and express the symmetric linear functions obtained by the left integral, the balancing element and the center of g_{p, q} in term of this basis.