REALISATIONS OF QUANTUM GLp,q(2) AND JORDANIAN GLh,h′(2) DEEPAK PARASHAR and ROGER J. McDERMOTT

Non Standard (or Jordanian) deformations of Lie groups and Lie algebras has been a subject of considerable interest in the mathematical physics community. Jordanian deformations for GL(2) were introduced in [1,2], its two parametric generalisation given in [3] and extended to the supersymmetric case in [4]. Non Standard deformations of sl(2) (i.e. at the algebra level) were first proposed in [5], the universal R-matrix presented in [6-8] and irreducible representations studied in [9,10]. A peculiar feature of this deformation (also known as h-deformation) is that the corresponding R-matrix is triangular. It was shown in [11] that up to isomorphism, GLq(2) and GLh(2) are the only possible distinct deformations (with central determinant) of the group GL(2). In [12], an interesting observation was made that the h-deformation could be obtained by a singular limit of a similarity transformation from the q-deformations of the group GL(2). Given this contraction procedure, it would be useful to look for Jordanian deformations of other q-groups. In the present paper, we focus our attention on a particular two parameter quantum group, denoted Gr,s, which provides a realisation of the well known GLp,q(2). We investigate the contraction procedure on Gr,s, in order to obtain its non standard counterpart. The generators of the contracted structure are employed to realise the two parameter non standard GLh,h′(2). This is similar to what happens in the q-deformed case.