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Pointed Hopf algebras of odd dimension and Nichols algebras over solvable groups

N. Andruskiewitsch, I. Heckenberger, L. Vendramin

Abstract

We classify finite-dimensional Nichols algebras of Yetter-Drinfeld modules with indecomposable support over finite solvable groups in characteristic 0, using a variety of methods including reduction to positive characteristic. As a consequence, all Nichols algebras over groups of odd order are of diagonal type, which allows us to describe all pointed Hopf algebras of odd dimension.

Pointed Hopf algebras of odd dimension and Nichols algebras over solvable groups

Abstract

We classify finite-dimensional Nichols algebras of Yetter-Drinfeld modules with indecomposable support over finite solvable groups in characteristic 0, using a variety of methods including reduction to positive characteristic. As a consequence, all Nichols algebras over groups of odd order are of diagonal type, which allows us to describe all pointed Hopf algebras of odd dimension.

Paper Structure

This paper contains 30 sections, 47 theorems, 121 equations, 1 table.

Table of Contents

  1. Introduction
  2. Pointed Hopf algebras of odd dimension
  3. Nichols algebras over solvable groups
  4. The roadmap of the proof of Theorem \ref{['mainthm:nichols-solvable']}
  5. Affine simple racks
  6. Notations
  7. Yetter-Drinfeld modules and racks
  8. Yetter-Drinfeld modules
  9. Yetter-Drinfeld modules over groups
  10. Change of group
  11. Racks
  12. Affine racks
  13. Braided vector spaces of rack type
  14. Main examples
  15. Graded and filtered Hopf algebras
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $\Bbbk$ be an algebraically closed field of characteristic 0. Let $H$ be a pointed Hopf algebra such that $d = \dim H$ is odd. Then:

Theorems & Definitions (102)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5: MR2426855
  • ...and 92 more