The representations of two-parameter Kashiwara algebras

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.