On Certain Quantum Deformations of $gl(N,R)$

TL;DR

This paper explores specific quantum deformations of the general linear group, providing explicit representations using generalized creation and annihilation operators, and introduces a natural concept of q-direct sums of q-algebras.

Contribution

It characterizes all deformations of $gl(N,R)$ with smooth Lie group limits and defines q-direct sums of q-algebras, expanding the understanding of quantum group structures.

Findings

Classified all deformations of $gl(N,R)$ with smooth limits.

Constructed representations using generalized creation and annihilation operators.

Introduced the concept of q-direct sums of q-algebras.

Abstract

In this paper all deformations of the general linear group, subject to certain restrictions which in particular ensure a smooth passage to the Lie group limit, are obtained. Representations are given in terms of certains sets of creation and annihilation operators. These creation and annihilation operators may belong to a generalisation of the $q$-quark type or $q$-hadronic type, of $q$-boson or $q$-fermion type. We are also led to a natural definition of $q$-direct sums of q-algebras.