The representations of two-parameter Kashiwara algebras

TL;DR

This paper constructs and studies two-parameter quantum groups, Kashiwara algebras, and quantized Weyl algebras, providing a unified framework and analyzing their representations and module categories.

Contribution

It offers a unified construction of two-parameter quantum structures and analyzes their module categories, including semi-simplicity and classification of simple modules.

Findings

Unified construction of two-parameter quantum groups and related algebras.

Proved semi-simplicity of the category of certain modules.

Classified simple modules in the category.

Abstract

In this paper, we give a unified construction of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$, two-parameter Kashiwara algebras $B_{r,s}(\mathfrak{g})$, two-parameter quantized Weyl algebras $W_{r,s}(\mathfrak{g})$ and the action of $U_{r,s}(\mathfrak{g})$ on $B_{r,s}(\mathfrak{g})$ and $W_{r,s}(\mathfrak{g})$. Applying their properties to the category $\mathcal{O}(B_{r,s}(\mathfrak{g}))$, whose objects are "upper bounded" $B_{r,s}(\mathfrak{g})$-modules, We can give a conceptual proof of its semi-simplicity and the classification of simple modules in it.