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Simple modules of small quantum groups at dihedral groups

Gastón Andrés García, Cristian Vay

Abstract

Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over the dihedral groups $\mathbb{D}_{4t}$ with $t\geq 3$. To this end, we develop new techniques that can be applied to Nichols algebras over any Hopf algebra. Namely, we explain how to construct recursively irreducible representations when the Nichols algebra is generated by a decomposable module, and show that the highest-weight of minimum degree in a Verma module determines its socle. We also prove that tensoring a simple module by a rigid simple module gives a semisimple module.

Simple modules of small quantum groups at dihedral groups

Abstract

Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of bosonizations of finite-dimensional Nichols algebras over the dihedral groups with . To this end, we develop new techniques that can be applied to Nichols algebras over any Hopf algebra. Namely, we explain how to construct recursively irreducible representations when the Nichols algebra is generated by a decomposable module, and show that the highest-weight of minimum degree in a Verma module determines its socle. We also prove that tensoring a simple module by a rigid simple module gives a semisimple module.

Paper Structure

This paper contains 29 sections, 20 theorems, 72 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions
  4. The general framework
  5. Drinfeld doubles of bosonization of Nichols algebras
  6. Nichols algebras
  7. A generalized quantum group
  8. Simple D(V,H)-modules
  9. The character of the simple modules
  10. The action on the Verma modules
  11. The simple D(V,H)-modules as socles of the Verma modules
  12. Example
  13. Tensoring by rigids
  14. A recursive strategy for V decomposable
  15. A recursive example
  16. ...and 14 more sections

Key Result

Theorem 3.1

∎

Figures (8)

  • Figure 1: The Verma module $\mathsf{M}_H^V(\lambda)$. The (big) dots represent their (highest-)weights. The shadow regions indicate submodules generated by highest-weights. In particular, the region on the bottom is the socle $\mathsf{S}^{V}_H(\lambda)$ which is generated by the highest-weight of minimum degree. The white region on the top depicts its unique simple quotient $\mathsf{L}^{V}_H(\lambda)$.
  • Figure 2: The big dots represent the weights of ${\mathfrak B}(V)$ and $\mathsf{M}_H^V(\lambda)$. Their degrees are indicated on the right. Those in the shadow region form the socle $\mathsf{S}^{V}_H(\lambda)$ when $\deg\mu=-1$.
  • Figure 3: The dots represent the simple $\mathcal{D}(U,H)$-summands of ${\mathfrak B}(W)$ and $\mathsf{M}^{W}_{{\mathfrak B}(U)\#H}(\mathsf{L}^{U}_H(\lambda))$. Their degrees are indicated on the right. Those in the shadow region form its socle in the case that $\deg\mathsf{L}_H^U(\mu)=-1$.
  • Figure 4: The simple module $M_{0,s,t}$ associated with the conjugacy class of $x$.
  • Figure 5: The simple module $M_{1,s,t}$ associated with the conjugacy class of $xy$.
  • ...and 3 more figures

Theorems & Definitions (49)

  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • proof
  • Proposition 3.7
  • proof
  • ...and 39 more