Abstract
Let K be a field. Let H be a finite-dimensional semisimple and cosemisimple K-Hopf algebra. In this paper, we introduce a notion of β-character algebra C β (H) for each group-like element β in H ∗. We prove that Radford’s action of the Drinfel’d double D(H) on H β (see Radford, J. Algebra, 270:670–695, 2003) and the right hit action of the β-character algebra C β (H) on H β form a commuting pair. This generalizes an earlier result of Zhu (Proc. Amer. Math. Soc., 125(10):2847–2851, 1997). A K-basis of C β (H) is given when H is split semisimple. Finally, as an example, we explicitly construct all the simple modules for the Drinfel’d double of the unique 8-dimensional non-commutative and non-cocommutative semisimple Hopf algebra.
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Hu, J., Zhang, Y. β-character Algebra and a Commuting Pair in Hopf Algebras. Algebr Represent Theor 10, 497–516 (2007). https://doi.org/10.1007/s10468-007-9055-4
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DOI: https://doi.org/10.1007/s10468-007-9055-4