Short star products for quantum symmetric pairs and applications

TL;DR

This paper proves the star product for quantum symmetric pair coideal subalgebras is short and uses this to provide new, elementary proofs of fundamental properties and conjectures in the theory of quantum symmetric pairs.

Contribution

It introduces the short star product property for quantum symmetric pairs and applies it to give elementary proofs of key structural results and conjectures.

Findings

Established the existence of algebra anti-automorphism and bar involution without quasi K-matrix.

Provided a new proof of the fundamental lemma for quantum symmetric pairs.

Derived a formula relating the tensor quasi K-matrix to the quasi R-matrix and Letzter map.

Abstract

We prove that the star product for quantum symmetric pair coideal subalgebras is short. We apply this result to obtain new conceptual proofs, from first principles, of several fundamental facts about quantum symmetric pairs. In particular, we establish the existence of the algebra anti-automorphism $\sigma_\tau$ and of the bar involution, without making use of the quasi K-matrix. We give a new elementary proof of a conjecture by Balagovi\'c and Kolb, sometimes referred to as the fundamental lemma for quantum symmetric pairs. We obtain a conceptual formula expressing the tensor quasi K-matrix in terms of the much studied quasi R-matrix and the Letzter map. This also allows for a new independent proof of the intertwiner property of the quasi K-matrix.