Weyl algebras for quantum homogeneous spaces

TL;DR

This paper introduces a new family of quantum Weyl algebras associated with homogeneous spaces of matrices, utilizing twisted tensor products and quantum symmetric pairs, with applications to quantum bounded symmetric domains.

Contribution

It constructs novel quantum Weyl algebras for matrix-related homogeneous spaces using twisted tensor products and invariance properties, expanding the algebraic framework for quantum symmetric spaces.

Findings

Quantum Weyl algebras are constructed for various matrix spaces.

These algebras have compatible $U_q(rak{gl}_N)$-module structures.

Relations are expressed via matrices and relate to quantum bounded symmetric domains.

Abstract

We present a new family of quantum Weyl algebras where the polynomial part is the quantum analog of functions on homogeneous spaces corresponding to symmetric matrices, skew symmetric matrices, and the entire space of matrices of a given size. The construction uses twisted tensor products and their deformations combined with invariance properties derived from quantum symmetric pairs. These quantum Weyl algebras admit $U_q(\mathfrak{gl}_N)$-module algebra structures compatible with standard ones on the polynomial part, have relations that are expressed nicely via matrices, and are closely related to an algebra arising in the theory of quantum bounded symmetric domains.