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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
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