Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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REPRESENTATIONS OVER GREEN ALGEBRAS OF WEAK HOPF ALGEBRAS BASED ON TAFT ALGEBRAS

  • Liufeng Cao (Department of Mathematical Sciences Yangzhou University)
  • Received : 2022.12.13
  • Accepted : 2023.04.21
  • Published : 2023.11.30
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Abstract

In this paper, we study the Green ring r(𝔴0n) of the weak Hopf algebra 𝔴0n based on Taft Hopf algebra Hn(q). Let R(𝔴0n) := r(𝔴0n) ⊗ℤ ℂ be the Green algebra corresponding to the Green ring r(𝔴0n). We first determine all finite dimensional simple modules of the Green algebra R(𝔴0n), which is based on the observations of the roots of the generating relations associated with the Green ring r(𝔴0n). Then we show that the nilpotent elements in r(𝔴0n) can be written as a sum of finite dimensional indecomposable projective 𝔴0n-modules. The Jacobson radical J(r(𝔴0n)) of r(𝔴0n) is a principal ideal, and its rank equals n - 1. Furthermore, we classify all finite dimensional non-simple indecomposable R(𝔴0n)-modules. It turns out that R(𝔴0n) has n2 - n + 2 simple modules of dimension 1, and n non-simple indecomposable modules of dimension 2.

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Acknowledgement

This work was financially supported by National Natural Science Foundation of China 12371041 and Scientific Research and Innovation Project of Graduate Students in Jiangsu Province (Grant No. KYCX22-3448).

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