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

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    A novel Hopf algebra (G~r,s) ( {\tilde G}_{r,s} ), depending on two deformation parameters and five generators, has been constructed. This G~r,s {\tilde G}_{r,s} Hopf algebra might be considered as some quantisation of classical GL(2)GL(1)GL(2) \otimes GL(1) group, which contains the standard GLq(2)GL_q(2) quantum group (with q=r1 q=r^{-1} ) as a Hopf subalgebra. However, we interestingly observe that the two parameter deformed GLp,q(2)GL_{p,q}(2) quantum group can also be realised through the generators of this G~r,s {\tilde G}_{r,s} algebra, provided the sets of deformation parameters p, qp,~q and r, sr,~s are related to each other in a particular fashion. Subsequently we construct the invariant noncommutative planes associated with G~r,s {\tilde G}_{r,s} algebra and show how the two well known Manin planes corresponding to GLp,q(2)GL_{p,q}(2) quantum group can easily be reproduced through such construction. Finally we consider the `coloured' extension of GLp,q(2)GL_{p,q}(2) quantum group as well as corresponding Manin planes and explore their intimate connection with the `coloured' extension of G~r,s{\tilde G}_{r,s} Hopf structure.Comment: 24 page
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