## The projective class rings of a family of pointed Hopf algebras of rank two.(English)Zbl 1411.16029

Summary: In this paper, we compute the projective class rings of the tensor product $$\mathcal{H}_n(q)=A_n(q)\otimes A_n(q^{-1})$$ 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 $$r_p(\mathcal{H}_n(q))$$, $$r_p(H_n(0,q))$$ 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 $$\mathcal{H}_n(q)$$, $$H_n(0,q)$$ and $$H_n(1,q)$$ are cocycle twist-equivalent to each other, they are of different representation types: wild, wild and tame, respectively.

### MSC:

 16T05 Hopf algebras and their applications 16G60 Representation type (finite, tame, wild, etc.) of associative algebras

### Keywords:

Green ring; indecomposable module; Taft algebra
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