Quantum symmetric Kac-Moody pairs

TL;DR

This paper develops a comprehensive theory of quantum symmetric pairs for symmetrizable Kac-Moody algebras, extending previous work from semisimple Lie algebras to a broader class with explicit structures and classifications.

Contribution

It generalizes the theory of quantum symmetric pairs to symmetrizable Kac-Moody algebras, including classification, explicit generators, and structural decompositions.

Findings

Constructed quantum symmetric pairs as right coideal subalgebras.

Derived triangular and Iwasawa decompositions for these pairs.

Identified centers and analyzed specializations.

Abstract

The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras. They give rise to triangular decompositions, including a quantum analog of the Iwasawa decomposition, and they can be written explicitly in terms of generators and relations. Moreover, their centers and their specializations are determined. The constructions follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras due to Kac and Wang. The resulting theory comprises various classes of examples which have previously appeared in the literature, such as q-Onsager algebras and the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba.