Contraction of the Gr,s quantum group to its nonstandard analog and corresponding colored quantum groups

Abstract

The quantum group G r,s provides a realisation of the two parameter quantum GL p,q (2) which is known to be related to the two parameter non standard GL h,h ′ (2) group via the contraction method. We apply the contraction procedure to G r,s and obtain a new Jordanian quantum group G m,k . Furthermore, we provide a realisation of GL h,h ′ (2) in terms of G m,k . The contraction procedure is then extended to the coloured quantum group GL λ,µ r (2) to yield a new Jordanian quantum group GL λ,µ m (2). Both G r,s and G m,k are then generalised to their coloured versions which inturn provide similar realisations of GL λ,µ r (2) and GL λ,µ m (2).

On the biparametric quantum deformation of GL(2)⊗GL(1)

“…The differential calculus obtained on A r,s enables us to investigate the associated gauge theory from a noncommutative perspective. It would be useful to repeat the analysis presented in this paper for the biparametric Jordanian deformation of GL(2) ⊗ GL(1) obtained in [3] and also to investigate corresponding hybrid (q, h)-deformations [11,12]. Furthermore, it would indeed be interesting to generalise the setting to the case of coloured quantum and Jordanian deformations.…”

“…It was shown in [3] that the A r,s deformation could be contracted (by means of singular limit of similarity transformations) to obtain a nonstandard or Jordanian analogue, say A m,k , with deformation parameters {m, k} and the associated R-matrix is triangular. In analogy with A r,s , A m,k can also be considered as the semidirect or cross-product…”

“…• Both the biparametric quantum and Jordanian deformations of GL(2) ⊗ GL(1) admit coloured extensions [3] which also commute with the contraction procedure.…”

“…• Another interesting feature of the A r,s deformation is that it can be contracted (by means of the contraction procedure [2] based on the concept of singular limit of a similarity transformation) to yield the corresponding biparametric Jordanian deformation of GL(2) ⊗ GL(1), which in turn provides a complete realisation of the biparamet- (2), i.e., GL h,h ′ (2) in a manner similar to that for the q-deformed case [3].…”

The coloured quantum plane

“…i.e. the coloured h-plane corresponding to the coloured Jordanian quantum group GL λ,µ h (2) [9,10]. Again, for vanishing colour parameters, this reduces to the wellknown h-plane [x, y] = hy 2 for the Jordanian quantum group GL h (2).…”