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
An Introduction to Group Doublecross Products and Some Uses
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
INTRODUCTION Factorisations of groups have been sudied for a long time, and it is well known that Hopf algebras can be constructed from them [16, 11]. In this article I shall review this material in the finite group case, and then comment on some more recent developments on quantum doubles and duality [2, 5]. Then I shall discuss the relation between group factorisations and integrable models, including the Hamiltonian structure and some speculations on the quantum theory. This is based on the inverse scattering process [8, 14, 15], using a formalism emphasising the algebraic structure [3, 4]. Finally I shall mention some recent work connecting group doublecrossproducts and T-duality in sigma models in classical field theory [9, 10, 6].
I have worked in these areas jointly with S. Majid (on Hopf algebras and T-duality) and with P.R. Johnson (integrable models). I would like to thank the organisers of the symposium for inviting me to speak.
Group doublecross product
A group doublecross product is a group X which has two subgroups G and M so that every element x ∈ X can be factored uniquely as x = su for s ∈ M and u ∈ G, and also as x = vt for t ∈ M and v ∈ G. We use the notation to denote a doublecross product.
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- Quantum Groups and Lie Theory , pp. 23 - 32Publisher: Cambridge University PressPrint publication year: 2002