Abstract

Wu-Liu-Ding algebras, that is $D(m,d,\xi ),$ are a class of non-pointed affine prime regular Hopf algebras of GK-dimension one. In this paper, we mainly study a class of quotient algebras of $D(m,d,\xi ),$ denoted by $D\prime (m,d,\xi ),$ which are 2$m^{2}d$-dimensional non-pointed semisimple Hopf algebras. For a better understanding of the structure of the Hopf algebra $D\prime (m,d,\xi ),$ we have considered the Grothendieck rings of $D\prime (m,d,\xi )$ and their Casimir numbers when d is odd in our previous paper. In this paper we continue dealing with the more complex case when d is even. It turns out that the Grothendieck rings of $D\prime (m,d,\xi )$ are generated by four elements subject to some relations. Then we give the Casimir numbers of the Grothendieck rings of $D\prime (1,d,\xi )$ and $D\prime (2,d,\xi ).$

摘要

五六丁代数 $d（米，d，\xi ），$是一类GK维度的无尖仿射素数正规Hopf代数。在本文中，我们主要研究一类商代数$d（米，d，\xi ），$ 表示为 $d\prime （米，d，\xi ），$ 这是2${米}^{2}d$维无尖半简单Hopf代数。为了更好地了解Hopf代数的结构$d\prime （米，d，\xi ），$ 我们已经考虑了格罗腾迪克环 $d\prime （米，d，\xi ）$和我们在前一篇论文中当d为奇数时的卡西米尔数。在本文中，我们将继续处理d为偶数时的更复杂情况。原来，格罗腾迪克环$d\prime （米，d，\xi ）$是由受某些关系约束的四个元素生成的。然后我们给出格罗腾迪克环的卡西米尔数$d\prime （\mathrm{1个}，d，\xi ）$ 和 $d\prime （2，d，\xi ）。$