On a nonstandard two-parametric quantum algebra and its connections withU _{p, q} (gl(2)) andU _{p, q}(gl(1/1))

Abstract

A quantum algebraU _{p, q} (ζ,H,X _±) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal ℛ-matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal ℛ-matrix of the standardU _{p, q}(gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal ℛ-matrix ofU _{p, q}(gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandardR-matrices. Superization ofU _{p, q}(ζ,H,X _±) leads to the super-Hopf algebraU _{p, q}(gl(1/1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebraU _{p, q}(sh(1)) and its universal ℛ-matrix.