paper

The Projective Class Rings of a family of pointed Hopf algebras of Rank two

arXiv:1709.08782

Abstract

In this paper, we compute the projective class rings of the tensor product $H_{n} (q) = A_{n} (q) \otimes A_{n} (q^{- 1})$ of Taft algebras $A_{n} (q)$ and $A_{n} (q^{- 1})$ , and its cocycle deformations $H_{n} (0, q)$ and $H_{n} (1, q)$ , where $n > 2$ is a positive integer and $q$ is a primitive $n$ -th root of unity. It is shown that the projective class rings $r_{p} (H_{n} (q))$ , $r_{p} (H_{n} (0, q))$ and $r_{p} (H_{n} (1, q))$ are commutative rings generated by three elements, three elements and two elements subject to some relations, respectively. It turns out that even $H_{n} (q)$ , $H_{n} (0, q)$ and $H_{n} (1, q)$ are cocycle twist-equivalent to each other, they are of different representation types: wild, wild and tame, respectively.

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