The quantum double of the restricted quantum group $\mathbf{\overline{u}}_q(\mathfrak{sl_2})$

Hua Sun; Yijie Shen; Xiaoming Li; Huixiang Chen

The quantum double of the restricted quantum group $\mathbf{\overline{u}}_q(\mathfrak{sl_2})$

Quantum Algebra 2026-07-12 v1 Representation Theory

Authors: Hua Sun , Yijie Shen , Xiaoming Li , Huixiang Chen

Abstract

In this paper, we construct the quantum double $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ of the restricted quantum group $\mathbf{\overline{u}}_q(\mathfrak{sl_2})$ . We describe the algebraic structure of $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ by generators and relations. Moreover, we give the comultiplication $\Delta$ , the counit $\varepsilon$ and the antipode $S$ , respectively. Finally, we classify all irreducible representations of $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ when $p=2$ .

Keywords

quantum double restricted quantum group Hopf algebra generators and relations irreducible representations

Cite

@article{arxiv.2607.10602,
  title  = {The quantum double of the restricted quantum group $\mathbf{\overline{u}}_q(\mathfrak{sl_2})$},
  author = {Hua Sun and Yijie Shen and Xiaoming Li and Huixiang Chen},
  journal= {arXiv preprint arXiv:2607.10602},
  year   = {2026}
}