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

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 $\mathcal{R}$-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 $\mathcal{R}$-matrix of the standard $U_{p,q}(gl(2))$ corresponding to a nongeneric case. General finite dimensional coloured representation of the universal $\mathcal{R}$-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 $\mathcal{R}$-matrix.

17pages, LaTeX, Preprint No. imsc-94/43 Revised version: A note added at the end of the paper correcting and clarifying the bibliography

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