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

TL;DR

This paper computes the projective class rings of certain pointed Hopf algebras of rank two, revealing their algebraic structure and differing representation types despite cocycle twist equivalences.

Contribution

It provides explicit descriptions of the projective class rings for a family of rank two pointed Hopf algebras and analyzes their representation types.

Findings

The projective class rings are generated by three or two elements with specific relations.

The algebras are of different representation types: wild and tame.

Cocycle twist-equivalent algebras can have different representation types.

Abstract

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.