Weight modules for quantum symmetric pair subalgebras

TL;DR

This paper develops a weight theory for quantum symmetric pair subalgebras, classifies Verma modules, constructs weight bases, and explores applications like rational representations and the Harish-Chandra isomorphism.

Contribution

It introduces a new weight theory for quantum symmetric pairs, classifies Verma modules, and constructs weight bases, extending classical representation theory to the quantum setting.

Findings

Classified weight Verma modules for the quantum symmetric pair

Constructed explicit highest weight vectors in tensor products

Established foundational tools for quantum symmetric pair representation theory

Abstract

We develop a theory of weights for a quantum analogue of the symmetric pair (gl4,gl2 x gl2) realised as a quantum symmetric pair subalgebra. Based on Letzter's triangular decomposition we define Verma modules. Using magical operators that are compatible with weight spaces, we classify weight Verma modules and characterise their irreducible finite dimensional quotients. We then prove the existence of weight bases in tensor products by explicitly constructing some highest weight vectors. These constructions allow us to mimic the important aspects of the classical finite dimensional representation theory. Applications include a definition of rational representations, the BGG resolution, a Clebsch--Gordan formula, the Harish-Chandra isomorphism and central characters, as well as a classification and description of all irreducible polynomial representations.