Book contents
- Frontmatter
- Contents
- Introduction
- Lectures on Cyclotomic Hecke Algebras
- An Introduction to Group Doublecross Products and Some Uses
- Canonical Bases and Piecewise-linear Combinatorics
- Integrable and Weyl Modules for Quantum Affine sl2
- Notes on Balanced Categories and Hopf Algebras
- Lectures on the dynamical Yang-Baxter Equations
- Quantized Primitive Ideal Spaces as Quotients of Affine Algebraic Varietie
- Representations of Semisimple Lie Algebras in Positive Characteristic and Quantum Groups at Roots of Unity
- The Yang-Baxter Equation for Operators on Function Fields
- Noncommutative Differential Geometry and Twisting of Quantum Groups
- Finite Quantum Groups and Pointed Hopf Algebras
- On Some Two Parameter Quantum and Jordanian Deformations, and their Coloured Extensions
- Tensor Categories and Braid Representations
On Some Two Parameter Quantum and Jordanian Deformations, and their Coloured Extensions
Published online by Cambridge University Press: 05 November 2009
- Frontmatter
- Contents
- Introduction
- Lectures on Cyclotomic Hecke Algebras
- An Introduction to Group Doublecross Products and Some Uses
- Canonical Bases and Piecewise-linear Combinatorics
- Integrable and Weyl Modules for Quantum Affine sl2
- Notes on Balanced Categories and Hopf Algebras
- Lectures on the dynamical Yang-Baxter Equations
- Quantized Primitive Ideal Spaces as Quotients of Affine Algebraic Varietie
- Representations of Semisimple Lie Algebras in Positive Characteristic and Quantum Groups at Roots of Unity
- The Yang-Baxter Equation for Operators on Function Fields
- Noncommutative Differential Geometry and Twisting of Quantum Groups
- Finite Quantum Groups and Pointed Hopf Algebras
- On Some Two Parameter Quantum and Jordanian Deformations, and their Coloured Extensions
- Tensor Categories and Braid Representations
Summary
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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- Chapter
- Information
- Quantum Groups and Lie Theory , pp. 206 - 215Publisher: Cambridge University PressPrint publication year: 2002