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Weight modules for quantum symmetric pair subalgebras

Catharina Stroppel, Liao Wang

TL;DR

This work develops a weight-theoretic framework for the quantum symmetric pair coideal subalgebra $U_q'$ associated with the classical pair $( ext{gl}_4, ext{gl}_2 imes ext{gl}_2)$. By constructing a triangular decomposition and a quantized Cartan subalgebra, the authors define Verma modules and classify their finite-dimensional irreducible quotients via explicit weight-parameter conditions, including good vs exceptional Verma modules and associated quotients. They further build weight bases in tensor powers using magical operators, establish a rational representation theory with a Clebsch–Gordan rule, and realize a Schur–Weyl duality with the type B Hecke algebra, enabling a complete decomposition of tensor powers and a Bernstein–Gelfand–Gelfand-like resolution framework. The center is explicitly described and linked to a Harish-Chandra-type homomorphism, enabling central characters that separate irreducibles; a Weyl-group action governs equivalence of central characters, mirroring the classical theory. Together, these results extend highest-weight techniques to quantum symmetric pairs, yielding a robust toolkit for rational and polynomial representations and central-character theory in this non-Hopf setting, with concrete combinatorial labelling via bipartitions and parity-standard tableaux.

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.

Weight modules for quantum symmetric pair subalgebras

TL;DR

This work develops a weight-theoretic framework for the quantum symmetric pair coideal subalgebra associated with the classical pair . By constructing a triangular decomposition and a quantized Cartan subalgebra, the authors define Verma modules and classify their finite-dimensional irreducible quotients via explicit weight-parameter conditions, including good vs exceptional Verma modules and associated quotients. They further build weight bases in tensor powers using magical operators, establish a rational representation theory with a Clebsch–Gordan rule, and realize a Schur–Weyl duality with the type B Hecke algebra, enabling a complete decomposition of tensor powers and a Bernstein–Gelfand–Gelfand-like resolution framework. The center is explicitly described and linked to a Harish-Chandra-type homomorphism, enabling central characters that separate irreducibles; a Weyl-group action governs equivalence of central characters, mirroring the classical theory. Together, these results extend highest-weight techniques to quantum symmetric pairs, yielding a robust toolkit for rational and polynomial representations and central-character theory in this non-Hopf setting, with concrete combinatorial labelling via bipartitions and parity-standard tableaux.

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.
Paper Structure (29 sections, 56 theorems, 146 equations)

This paper contains 29 sections, 56 theorems, 146 equations.

Table of Contents

  1. Preliminaries on quantum symmetric pairs
  2. Quantum groups and quantum symmetric pairs
  3. Schur--Weyl duality with the type $\operatorname{B}$ Hecke algebra
  4. The main player
  5. Classical symmetric pair $(\mathfrak{gl}_4,\mathfrak{gl}_2\times\mathfrak{gl}_2)$
  6. Classical tensor power decomposition
  7. Finite dimensional representations for $\mathfrak{gl}_4^\theta$.
  8. Quantized Cartan subalgebra and PBW theorem
  9. Quantized Cartan subalgebra
  10. A PBW type basis
  11. Verma modules and classification of finite dimensional irreducible modules
  12. Definition and basics
  13. Magical operators
  14. Action on Verma modules
  15. Good Verma modules
  16. ...and 14 more sections

Key Result

Theorem A

Consider a good Verma module $\mathbb{M}=\mathbb{M}({\kappa_{\diamondsuit}},{\kappa_1},[\mu;0],\zeta)$ with highest weight vector $v$. Then it has a finite dimensional irreducible quotient $L$ if and only if In this case $L$ is unique and has a weight basis $\{F_+^a F_-^b v\mid 0\le a\leq i ,\, 0\le b\le \kappa-i\}.$

Theorems & Definitions (150)

  • Theorem A: \ref{['fd quot good verma']}
  • Theorem B
  • Definition 1.2
  • Definition 1.3
  • Lemma 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8: Ehrig--Stroppel, Bao--Wang
  • Corollary 1.9
  • ...and 140 more