We investigate some essential algebraic as well as geometric structures
underlying biparametric and coloured quantum groups. GL(2) is known to
admit two distinct quantisations: the standard GL_q(2) and the Jordanian
GL_h(2). The same is true for their respective biparametric versions
GL_p,q(2) and GL_h,h'(2). Though distinct, both these deformations
are related to each other by means of a contraction procedure. There
exists a particularly interesting biparametric quantum deformation of
GL(2) x GL(1) which provides a realization of GL_p,q(2), and also relates
several other examples of quantum groups in a coherent way. We contract
this deformation to obtain its Jordanian analogue which also provides a
realization of GL_h,h'(2) in a manner similar to the q-deformed case. The
scheme is then set in the wider context of `coloured extensions' of
quantum groups. Focussing on the most intuitive example of GL_q(2), I
will present basic results from the theory of coloured quantum groups such
as establishing the picture of duality between the quantised algebra of
functions on a group and the quantised universal enveloping algebra,
constructing differential calculi on these structures within the framework
of the R-matrix approach. The quantum plane covariant under the action of
coloured GL_q(2) will also be exhibited.
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