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Automorphism group of representation ring of the weak Hopf algebra H 8 ˜

Dong SuShilin Yang — 2018

Czechoslovak Mathematical Journal

Let H 8 be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra H 8 ˜ based on H 8 , then we investigate the structure of the representation ring of H 8 ˜ . Finally, we prove that the automorphism group of r ( H 8 ˜ ) is just isomorphic to D 6 , where D 6 is the dihedral group with order 12.

The duality of Auslander-Reiten quiver of path algebras

Bo HouShilin Yang — 2019

Czechoslovak Mathematical Journal

Let Q be a finite union of Dynkin quivers, G Aut ( 𝕜 Q ) a finite abelian group, Q ^ the generalized McKay quiver of ( Q , G ) and Γ Q the Auslander-Reiten quiver of 𝕜 Q . Then G acts functorially on the quiver Γ Q . We show that the Auslander-Reiten quiver of 𝕜 Q ^ coincides with the generalized McKay quiver of ( Γ Q , G ) .

Remarks on Sekine quantum groups

Jialei ChenShilin Yang — 2022

Czechoslovak Mathematical Journal

We first describe the Sekine quantum groups 𝒜 k (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 𝒜 k and describe their representation rings r ( 𝒜 k ) . Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of r ( 𝒜 k ) .

Representations of a class of positively based algebras

Shiyu LinShilin Yang — 2023

Czechoslovak Mathematical Journal

We investigate the representation theory of the positively based algebra A m , d , which is a generalization of the noncommutative Green algebra of weak Hopf algebra corresponding to the generalized Taft algebra. It turns out that A m , d is of finite representative type if d 4 , of tame type if d = 5 , and of wild type if d 6 . In the case when d 4 , all indecomposable representations of A m , d are constructed. Furthermore, their right cell representations as well as left cell representations of A m , d are described.

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