Extremal Projectors of q-Boson Algebras

We define the extremal projector of the $q$-boson Kashiwara algebra $B_q(\ge )$ and study their basic properties. Applying their proerties to the representation theory of the category $\mathcal{O}(B_q(\ge ))$, whose objects are ''upper bounded ‘’ $B_q(\ge )$-modules, we obtain its semi-simplicity and the classification of simple modules.