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

## TL;DR

This paper introduces a new family of Artin-Schelter Gorenstein Hopf algebras, explores their relationships with known Hopf algebras, and computes their Grothendieck rings, revealing non-isomorphic algebras with identical Grothendieck rings.

## Contribution

It defines a new series of Hopf algebras $H_eta$, analyzes their properties, and determines their Grothendieck rings, connecting them to existing Hopf algebras.

## Key findings

- Radford's and Gelaki's Hopf algebras are homomorphic images of $H_eta$.
- The Grothendieck ring $G_0(H_eta)$ is explicitly determined.
- Non-isomorphic Hopf algebras with isomorphic Grothendieck rings are identified.

## 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.