2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64376We 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 ${\cal O}(B_q(\ge))$, whose objects are ''upper bounded '' $B_q(\ge)$-modules, we obtain its semi-simplicity and the classification of simple modules.LaTeX, 12 pagesQuantum AlgebraRepresentation Theorytext