The bar involution for quantum symmetric pairs

TL;DR

This paper constructs a bar involution for quantum symmetric pair coideal subalgebras associated with involutive automorphisms of Kac-Moody algebras, extending previous algebraic presentations and identifying parameter conditions.

Contribution

It introduces a unified presentation of these algebras and specifies parameter conditions for the existence of an intrinsic bar involution.

Findings

Unified algebraic presentations for quantum symmetric pairs.

Explicit parameter conditions for bar involution existence.

Extension of previous algebraic results by Letzter and others.

Abstract

We construct a bar involution for quantum symmetric pair coideal subalgebras $B_{\mathbf{c},\mathbf{s}}$ corresponding to involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras. To this end we give unified presentations of these algebras in terms of generators and relations extending previous results by G. Letzter and the second named author. We specify precisely the set of parameters $\mathbf{c}$ for which such an intrinsic bar involution exists.