Realisations of $G{L}_{p,q}(2)$ quantum group and its coloured extension through a novel Hopf algebra with five generators

Abstract

A novel Hopf algebra $({\tilde{G}}_{r,s})$, depending on two deformation parameters and five generators, has been constructed. This ${\tilde{G}}_{r,s}$ Hopf algebra might be considered as some quantisation of classical $GL(2)\otimes GL(1)$ group, which contains the standard $G{L}_q(2)$ quantum group (with $q=r^{-1}$) as a Hopf subalgebra. However, we interestingly observe that the two parameter deformed $G{L}_{p,q}(2)$ quantum group can also be realised through the generators of this ${\tilde{G}}_{r,s}$ algebra, provided the sets of deformation parameters $p,q$ and $r,s$ are related to each other in a particular fashion. Subsequently we construct the invariant noncommutative planes associated with ${\tilde{G}}_{r,s}$ algebra and show how the two well known Manin planes corresponding to $G{L}_{p,q}(2)$ quantum group can easily be reproduced through such construction. Finally we consider the `coloured' extension of $G{L}_{p,q}(2)$ quantum group as well as corresponding Manin planes and explore their intimate connection with the `coloured' extension of ${\tilde{G}}_{r,s}$ Hopf structure.

arXiv comment24 pages