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

TL;DR

This paper introduces a new five-generator Hopf algebra that generalizes classical groups and quantum groups, unifies different deformations of $GL(2)$, and constructs associated noncommutative planes and coloured extensions.

Contribution

A novel Hopf algebra with five generators is constructed, unifying and extending known quantum groups and their coloured variants, with explicit realizations and geometric structures.

Findings

The ${ ilde G}_{r,s}$ Hopf algebra generalizes classical and quantum groups.

Two-parameter deformed $GL_{p,q}(2)$ can be realized within ${ ilde G}_{r,s}$.

Invariant noncommutative planes and coloured extensions are explicitly constructed.

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 $GL_q(2)$ quantum group (with $ q=r^{-1} $) as a Hopf subalgebra. However, we interestingly observe that the two parameter deformed $GL_{p,q}(2)$ 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 $GL_{p,q}(2)$ quantum group can easily be reproduced through such construction. Finally we consider the `coloured' extension of $GL_{p,q}(2)$ quantum group as well as corresponding Manin planes and explore their intimate connection with the `coloured' extension of ${\tilde G}_{r,s}$ Hopf structure.