Braid group actions for quantum symmetric pairs of type AIII/AIV

TL;DR

This paper constructs braid group actions on quantum symmetric pair coideal subalgebras of type AIII/AIV, confirming a conjecture for the case of rak{sl}_n and providing explicit calculations of the actions.

Contribution

It introduces explicit braid group actions on quantum symmetric pair coideal subalgebras of type AIII/AIV, completing the proof of a conjecture for rak{sl}_n.

Findings

Braid group actions are explicitly constructed on generators.

Verification of defining relations and braid relations through calculations.

Confirms conjecture by Kolb and Pellegrini for rak{sl}_n.

Abstract

In the present paper we construct braid group actions on quantum symmetric pair coideal subalgebras of type AIII/AIV. This completes the proof of a conjecture by Kolb and Pellegrini in the case where the underlying Lie algebra is $\mathfrak{sl}_n$. The braid group actions are defined on the generators of the coideal subalgebras and the defining relations and braid relations are verified by explicit calculations.