Defining relations for quantum symmetric pair coideals of Kac-Moody type

TL;DR

This paper provides a comprehensive set of generators and relations for quantum symmetric pair coideals of Kac-Moody type, extending the classical symmetric pair theory into the quantum realm with broad applicability.

Contribution

It introduces a complete presentation by generators and relations for quantum symmetric pair coideals of Kac-Moody type, including inhomogeneous q-Serre relations valid for all generalized Cartan matrices.

Findings

Derived inhomogeneous q-Serre type relations for quantum symmetric pairs.

Established a presentation valid without restrictions on the Cartan matrix.

Connected quantum symmetric pairs to generalized q-Onsager algebras in the split case.

Abstract

Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra $U_q(\mathfrak{g})$. We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous $q$-Serre type and are valid without restrictions on the generalized Cartan matrix. We draw special attention to the split case, where the quantum symmetric pair coideal subalgebras are generalized $q$-Onsager algebras.