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
Finite Quantum Groups and Pointed Hopf Algebras
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
We show that under certain conditions a finite dimensional graded pointed Hopf algebra is an image of an algebra twist of a quantized enveloping algebra Uq(b) when q is aro ot of unity. In addition we obtain a classification of Hopf algebras H such that G(H) ha s odd prime order p > 7 a nd grH is of Cartan type.
Throughout this paper K will denote an algebraically closed base field of characteristic zero. Recently there has been considerable interest in the structure of finite dimensional pointed Hopf algebras over K. For example if p is prime all Hopf algebras of dimension p are group algebras. Also any pointed Hopf algebra of dimension p2 is either a group algebra or a Taft algebra, while those of dimension p3 have been classified [AS1], [CD], [SvO], [Z]. In addition there are infinitely many isomorphism classes of pointed Hopf algebras of dimension p4, [AS1], [BDG], see also [G].
If H is pointed the coradical filtration {Hn} on H is a Hopf algebra filtration and the associated graded algebra is a Hopf algebra, see [M1], also [M2,Lemma 5.5.1]. In [AS2] pointed Hopf algebras H such that H ≅ grH are studied using methods from Lie theory and quantum groups.
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- Quantum Groups and Lie Theory , pp. 191 - 205Publisher: Cambridge University PressPrint publication year: 2002
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