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Published online by Cambridge University Press: 05 November 2009
Abstract
This paper suveys some recent algebraic developments in two parameter Quantum deformations and their Nonstandard (or Jordanian) counterparts. In particular, we discuss the contraction procedure and the quantum group homomorphisms associated to these deformations. The scheme is then set in the wider context of the coloured extensions of these deformations, namely, the so-called Coloured Quantum Groups.
Introduction
Recent years have witnessed considerable development in the study of multiparameter quantum deformations from both, the algebraic as well the differential geometric point of view. These have also found profound applications in many diverse areas of Mathematical Physics. Despite of the intensive and successful development of the mathematical theory of multiparameter quantum deformations or quantum groups, various important aspects still need thorough investigation. Besides, all quantum groups seem to have a natural coloured extension thereby defining corresponding coloured quantum groups. It is the aim of this paper to address some of the key issues involved.
Two parameter deformations provide an obvious step in constructing generalisations of single parameter deformations. Besides being mathematically interesting in their own right, two parameter quantum groups serve as very good examples in generalising physical theories based on the quantum group symmetry. GLp,q(2) and are well known examples of two parameter Quantum and Jordanian deformations of the space of 2 × 2 matrices.
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