A Serre presentation for the $\imath$quantum covering groups

TL;DR

This paper extends Serre presentations to $ extit{i}$-quantum covering groups, incorporating a parameter $ extit{ extpi}$ that interpolates between Lusztig quantum groups and quantum supergroups, with new $ extit{ extpi}$-Serre relations.

Contribution

It generalizes Serre presentations to quantum covering algebras with an additional parameter, unifying quantum groups and supergroups within a single framework.

Findings

Established Serre presentation for $ extit{i}$-quantum covering groups.

Introduced $ extit{ extpi}$-Serre relations and $ extit{ extpi}$-divided powers.

Unified quantum groups and supergroups via a parameter $ extit{ extpi}$.

Abstract

Let $(\mathbf{U}, \mathbf{U}^\imath)$ be a quasi-split quantum symmetric pair of Kac-Moody type. The $\imath$quantum group $\mathbf{U}^\imath$ admits a Serre presentation featuring the $\imath$-Serre relations in terms of $\imath$-divided powers. Generalizing this result, we give a Serre presentation $ \mathbf{U}^\imath_\pi $ of quantum symmetric pairs $ (\mathbf{U}_\pi, \mathbf{U}^\imath_\pi) $ for quantum covering algebras $\mathbf{U}_\pi$, which have an additional parameter $ \pi $ that specializes to the Lusztig quantum group when $ \pi = 1 $ and quantum supergroups of anisotropic type when $ \pi = -1 $. We give a Serre presentation for $ \mathbf{U}^\imath_\pi $, introducing the $\imath^\pi$-Serre relations and $\imath^\pi$-divided powers.