Braid group actions on coideal subalgebras of quantized enveloping algebras

TL;DR

This paper constructs explicit braid group actions on coideal subalgebras of quantized enveloping algebras related to quantum symmetric pairs, confirming a conjecture and suggesting broader applicability.

Contribution

It introduces explicit braid group actions on coideal subalgebras associated with quantum symmetric pairs, confirming a conjecture for a specific case and proposing generalization.

Findings

Constructed braid group actions on coideal subalgebras.

Verified algebraic relations using computer algebra software.

Confirmed a conjecture by Molev and Ragoucy.

Abstract

We construct braid group actions on coideal subalgebras of quantized enveloping algebras which appear in the theory of quantum symmetric pairs. In particular, we construct an action of the semidirect product of Z^n and the classical braid group in n strands on the coideal subalgebra corresponding to the symmetric pair (sl_{2n}(C), sp_{2n}(C)). This proves a conjecture by Molev and Ragoucy. We expect similar actions to exist for all symmetric Lie algebras. The given actions are inspired by Lusztig's braid group action on quantized enveloping algebras and are defined explicitly on generators. Braid group and algebra relations are verified with the help of the package Quagroup within the computer algebra program GAP.