The Projective Class Rings of a family of pointed Hopf algebras of Rank two
The Projective Class Rings of a family of pointed Hopf algebras of Rank two
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DOI：
10.36045/bbms/1483671621
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发表时间：
2016-12
期刊：
影响因子：
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通讯作者：
Hui-xiang Chen;Hassan Suleman Esmael Mohammed;Weijun Lin;Hua Sun
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文献类型：
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作者：
Hui-xiang Chen;Hassan Suleman Esmael Mohammed;Weijun Lin;Hua Sun
本文计算了塔夫脱代数$An（q）$和$An（q^{-1}）$的张量积$\mathcal{H} n（q）= An（q）\otimes An（q ^{-1}）$及其上循环变形$H n（0，q）$和$H n（1，q）$的投射类环，其中$n>2$是正整数，$q$是本原$n$次单位根.证明了投射类环$r_p（\mathcal{H}_n（q））$，$r_p（H_n（0，q））$和$r_p（H_n（1，q））$分别是由三个元素，三个元素和两个元素满足某种关系生成的交换环.证明了即使$\mathcal{H}_n（q）$，$H_n（0，q）$和$H_n（1，q）$是相互等价的余圈扭，它们也分别具有不同的表示类型：wild，wild和驯服。
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.