AIMS Mathematics (Jan 2022)

Automorphism groups of representation rings of the weak Sweedler Hopf algebras

  • Dong Su ,
  • Shilin Yang

DOI
https://doi.org/10.3934/math.2022131
Journal volume & issue
Vol. 7, no. 2
pp. 2318 – 2330

Abstract

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Let $ \mathfrak{w}^{s}_{2, 2}(s = 0, 1) $ be two classes of weak Hopf algebras corresponding to the Sweedler Hopf algebra, and $ r(\mathfrak{w}^{s}_{2, 2}) $ be the representation rings of $ \mathfrak{w}^{s}_{2, 2} $. In this paper, we investigate the automorphism groups $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $ of $ r(\mathfrak{w}^{s}_{2, 2}) $, and discuss some properties of $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $. We obtain that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{0}_{2, 2})) $ is isomorphic to $ K_4 $, where $ K_4 $ is the Klein four-group. It is shown that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is a non-commutative infinite solvable group, but it is not nilpotent. In addition, $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is isomorphic to $ (\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2} $, and its centre is isomorphic to $ \mathbb{Z}_{2} $.

Keywords