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Let be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra based on , then we investigate the structure of the representation ring of . Finally, we prove that the automorphism group of is just isomorphic to , where is the dihedral group with order 12.
Let be a finite union of Dynkin quivers, a finite abelian group, the generalized McKay quiver of and the Auslander-Reiten quiver of . Then acts functorially on the quiver . We show that the Auslander-Reiten quiver of coincides with the generalized McKay quiver of .
We first describe the Sekine quantum groups (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of and describe their representation rings . Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of .
We investigate the representation theory of the positively based algebra , which is a generalization of the noncommutative Green algebra of weak Hopf algebra corresponding to the generalized Taft algebra. It turns out that is of finite representative type if , of tame type if , and of wild type if In the case when , all indecomposable representations of are constructed. Furthermore, their right cell representations as well as left cell representations of are described.
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