Extremal projectors of two-parameter Kashiwara algebras

We define a two-parameter Kashiwara algebra $B_{r,s}(\mathfrak{g})$ and its extremal projector, and study their basic properties. 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 obtain its semi-simplicity and the classification of simple modules.