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Examples of solvable and nilpotent finite quantum groups

Gerard Glowacki, Masamune Hattori, Masato Tanaka

Abstract

We prove the solvability and nilpotency of Kac--Paljutkin's finite quantum group and Sekine quantum groups and we classify the solvable series of Kac--Paljutkin's finite quantum group via Cohen--Westreich's Burnside theorem. Some semisimple quasitriangular Hopf algebras of dimensions $2pq$ are also studied. In Appendix A, we give a direct computation of the universal $R$-matrices for Kac--Paljutkin's $8$-dimensional finite quantum group.

Examples of solvable and nilpotent finite quantum groups

Abstract

We prove the solvability and nilpotency of Kac--Paljutkin's finite quantum group and Sekine quantum groups and we classify the solvable series of Kac--Paljutkin's finite quantum group via Cohen--Westreich's Burnside theorem. Some semisimple quasitriangular Hopf algebras of dimensions are also studied. In Appendix A, we give a direct computation of the universal -matrices for Kac--Paljutkin's -dimensional finite quantum group.
Paper Structure (10 sections, 17 theorems, 63 equations)

This paper contains 10 sections, 17 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. Finite quantum groups
  5. Coideals, idempotent states and group-like projections
  6. Universal $R$-matrices
  7. Nilpotency, solvability and conjugacy classes for finite dimensional Hopf algebras
  8. On nilpotency and solvability
  9. Hopf algebras of dimension $2pq$
  10. A direct computation of the universal $R$-matrices for Kac--Paljutkin's finite quantum group

Key Result

Proposition 2.6

Any finite dimensioanl $C^*$-algebra is of the form $\bigoplus_{k=1}^N\mathbb{M}_{n_k}$ for some positive integers $N$ and $n_1,\ldots,n_N$.

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Example
  • Definition 2.3
  • Example
  • Definition 2.4
  • Example
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • ...and 44 more