Serre-Lusztig relations for $\imath$quantum groups III

TL;DR

This paper establishes Serre-Lusztig relations for a class of $ extit{i}$-quantum groups related to Kac-Moody types, extending previous work to new root cases and proposing a related braid group symmetry conjecture.

Contribution

It introduces Serre-Lusztig relations for $ extit{i}$-quantum groups associated with roots not fixed by the diagram involution, complementing earlier results and formulating a new symmetry conjecture.

Findings

Serre-Lusztig relations established for roots i with i ≠ τi.

Complementary to earlier relations for roots i = τi.

Conjecture on braid group symmetries proposed.

Abstract

let $\widetilde{\bf U}^\imath$ be a quasi-split universal $\imath$quantum group associated to a quantum symmetric pair $(\widetilde{\bf U}, \widetilde{\bf U}^\imath)$ of Kac-Moody type with a diagram involution $\tau$. We establish the Serre-Lusztig relations for $\widetilde{\bf U}^\imath$ associated to a simple root $i$ such that $i \neq \tau i$, complementary to the Serre-Lusztig relations associated to $i=\tau i$ which we obtained earlier. A conjecture on braid group symmetries on $\widetilde{\bf U}^\imath$ associated to $i$ disjoint from $\tau i$ is formulated.