On a nonstandard two-parametric quantum algebra and its connections with $U_{p,q}(gl(2))$ and $U_{p,q}(gl(1|1))$

TL;DR

This paper introduces a new two-parameter quantum algebra linked to a nonstandard R-matrix, deriving its universal R-matrix, exploring its representations, and connecting it with superalgebras and deformed super-Heisenberg algebra.

Contribution

It constructs a nonstandard two-parametric quantum algebra, derives its universal R-matrix, and explores its representations and superalgebra connections, extending the framework of quantum groups.

Findings

Derived the universal R-matrix for the nonstandard quantum algebra.

Constructed explicit (p,q)-dependent nonstandard R-matrix.

Connected the algebra to superalgebras and deformed super-Heisenberg algebra.

Abstract

A quantum algebra $U_{p,q}(\zeta ,H,X_\pm )$ associated with a nonstandard $R$-matrix with two deformation parameters$(p,q)$ is studied and, in particular, its universal ${\cal R}$-matrix is derived using Reshetikhin's method. Explicit construction of the $(p,q)$-dependent nonstandard $R$-matrix is obtained through a coloured generalized boson realization of the universal ${\cal R}$-matrix of the standard $U_{p,q}(gl(2))$ corresponding to a nongeneric case. General finite dimensional coloured representation of the universal ${\cal R}$-matrix of $U_{p,q}(gl(2))$ is also derived. This representation, in nongeneric cases, becomes a source for various $(p,q)$-dependent nonstandard $R$-matrices. Superization of $U_{p,q}(\zeta , H,X_\pm )$ leads to the super-Hopf algebra $U_{p,q}(gl(1|1))$. A contraction procedure then yields a $(p,q)$-deformed super-Heisenberg algebra $U_{p,q}(sh(1))$ and its universal ${\cal R}$-matrix.