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On classification of Hopf superalgebras of low dimension

Taiki Shibata, Ryota Wakao

Abstract

We examine the inverse procedure of the Radford-Majid bosonization for Hopf superalgebras and give a handy method for enumerating Hopf superalgebras whose bosonization is isomorphic to a given Hopf algebra. As an application, we classify Hopf superalgebras of dimension up to 5 and give examples of higher dimensions.

On classification of Hopf superalgebras of low dimension

Abstract

We examine the inverse procedure of the Radford-Majid bosonization for Hopf superalgebras and give a handy method for enumerating Hopf superalgebras whose bosonization is isomorphic to a given Hopf algebra. As an application, we classify Hopf superalgebras of dimension up to 5 and give examples of higher dimensions.
Paper Structure (27 sections, 42 theorems, 121 equations)

This paper contains 27 sections, 42 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. The Radford-Majid bosonization
  3. Coinvariant subspaces
  4. Yetter-Drinfeld categories
  5. The Radford-Majid bosonization
  6. Duals of coinvariant subalgebras
  7. Hopf superalgebras
  8. Superspaces
  9. Hopf superalgebras
  10. A construction of Hopf superalgebras
  11. Duals of Hopf superalgebras
  12. Bosonizations and super-forms
  13. Bosonization of Hopf superalgebras
  14. Admissible data for Hopf algebras
  15. Super-data for Hopf algebras
  16. ...and 12 more sections

Key Result

Theorem 1.1

Up to isomorphism, $\bigwedge(\Bbbk)$ is the only Hopf superalgebra of dimension $2$ whose $\bar{1}$-part is non-zero. If $p$ is an odd prime number, then a Hopf superalgebra of dimension $p$ is purely even, that is, its $\bar{1}$-part is zero (thus such a Hopf superalgebra is isomorphic to $\Bbbk\m

Theorems & Definitions (68)

  • Theorem 1.1: = Theorems \ref{['prp:2-dim']} and \ref{['thm:main1']}
  • Theorem 1.2: = Theorems \ref{['thm:ShiShiWak2']}, \ref{['thm:4ss']} and \ref{['thm:A_4-self-dual']}
  • Theorem 1.3: = Theorem \ref{['thm:H_6^5']}
  • Theorem 1.4: = Theorem \ref{['thm:K_8']}
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Lemma 3.1
  • Lemma 3.2
  • ...and 58 more