Factorisation of quasi K-matrices for quantum symmetric pairs

TL;DR

This paper provides an explicit factorisation formula for the quasi K-matrix in quantum symmetric pairs of type sl(n), extending recursive constructions and conjecturing broader applicability.

Contribution

It introduces a new explicit product formula for the quasi K-matrix in specific cases, linking it to the restricted Weyl group, and conjectures its general validity.

Findings

Explicit formula for quasi K-matrix as a product of rank-one quasi K-matrices

Dependence of factorisation on the restricted Weyl group

Conjecture on the general applicability of the formula

Abstract

The theory of quantum symmetric pairs provides a universal K-matrix which is an analogue of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far only been constructed recursively. In this paper we restrict to the cases where the underlying Lie algebra is sl(n) or the Satake diagram has no black dots. In these cases we give an explicit formula for the quasi K-matrix as a product of quasi K-matrices for Satake diagrams of rank one. This factorisation depends on the restricted Weyl group of the underlying symmetric Lie algebra in the same way as the factorisation of the quasi R-matrix depends on the Weyl group of the Lie algebra. We conjecture that our formula holds in general.