# Cartan Subalgebras for Quantum Symmetric Pair Coideals

## TL;DR

This paper introduces quantum Cartan subalgebras for quantum symmetric pair coideals, providing a foundational step towards understanding their finite-dimensional modules and representation theory.

## Contribution

It constructs quantum Cartan subalgebras for all such coideals, extending classical theory and analyzing their action on finite-dimensional modules.

## Key findings

- Quantum Cartan subalgebras exist for all quantum symmetric pair coideals.
- These subalgebras act semisimply on finite-dimensional unitary modules.
- Explicit generators are identified for specific examples.

## Abstract

There is renewed interest in the coideal subalgebras used to form quantum symmetric pairs because of recent discoveries showing that they play a fundamental role in the representation theory of quantized enveloping algebras. However, there is still no general theory of finite-dimensional modules for these coideals. In this paper, we establish an important step in this direction: we show that every quantum symmetric pair coideal subalgebra admits a quantum Cartan subalgebra which is a polynomial ring that specializes to its classical counterpart. The construction builds on Kostant and Sugiura's classification of Cartan subalgebras for real semisimple Lie algebras via strongly orthogonal systems of positive roots. We show that these quantum Cartan subalgebras act semisimply on finite-dimensional unitary modules and identify particularly nice generators of the quantum Cartan subalgebra for a family of examples.