Some Hopf Algebras related to ${\mathfrak{sl}}_2$

Abstract

We define a series of Artin-Schelter Gorenstein Hopf algebras $H_{\beta }$ with injective dimensions 3. Radford's Hopf algebra and Gelaki's Hopf algebra are homomorphic images of $H_{\beta }$. We determine its Grothendieck ring $G_0(H_{\beta })$. Meanwhile we can obtain Grothendieck rings of Gelaki's Hopf algebras and Radford's Hopf algebras $U_{(N,\nu ,\omega )}$ in \cite{R}, and non-isomorphic Hopf algebras with isomorphic Grothendieck rings.