Wu–Liu–Ding algebras are a class of affine prime regular Hopf algebras of GK-dimension one, denoted by $D(m,d,\xi )$. In this paper, we consider their quotient algebras $D^{\prime }(m,d,\xi ),$ which are a new class of non-pointed semisimple Hopf algebras. We describe the Grothendieck rings of $D^{\prime }(m,d,\xi )$ when $d$ is odd. It turns out that the Grothendieck rings are commutative rings generated by three elements subject to some relations. Then we compute the Casimir numbers of the Grothendieck rings for $m=1$ and $m=3$.

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Wu-Liu-Ding 代数的格洛腾迪克环及其卡西米尔数（一）

Wu-Liu-Ding 代数是一类 GK 维的仿射素正则 Hopf 代数，记为$D(米,d,\xi )$. 在本文中，我们考虑他们的商代数$D^{\text{'}}(米,d,\xi ),$这是一类新的非指向半单 Hopf 代数。我们描述格洛腾迪克环$D^{\text{'}}(米,d,\xi )$什么时候$d$很奇怪。事实证明，格洛腾迪克环是由受一定关系约束的三个元素生成的交换环。然后我们计算格洛腾迪克环的卡西米尔数$米=1$和$米=3$.