The bar involution for quantum symmetric pairs -- hidden in plain sight

TL;DR

This paper demonstrates that quantum symmetric pair coideal subalgebras of Kac-Moody type possess a bar involution for appropriate parameters, using a generalized quasi K-matrix approach without explicit generators and relations.

Contribution

It introduces a new method to establish bar involutions in quantum symmetric pairs via a generalized quasi K-matrix, avoiding explicit algebra presentations.

Findings

All quantum symmetric pair coideal subalgebras of Kac-Moody type have a bar involution with suitable parameters.

The proof employs a generalized quasi K-matrix, not requiring explicit algebra presentations.

The approach simplifies understanding of the structure of quantum symmetric pairs.

Abstract

We show that all quantum symmetric pair coideal subalgebras $B_\mathbf{c}$ of Kac-Moody type have a bar involution for a suitable choice of parameters $\mathbf{c}$. The proof relies on a generalized notion of quasi K-matrix. The proof does not involve an explicit presentation of $B_\mathbf{c}$ in terms of generators and relations.