2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67592In this paper, we develop the crystal basis theory for quantum generalized Kac-Moody algebras. For a quantum generalized Kac-Moody algebra $U_q(\mathfrak g)$, we first introduce the category $\mathcal O_{int}$ of $U_q(\mathfrak g)$-modules and prove its semisimplicity. Next, we define the notion of crystal bases for $U_q(\mathfrak g)$-modules in the category $\mathcal O_{int}$ and for the subalgebra $U_q^-(\mathfrak g)$. We then prove the tensor product rule and the existence theorem for crystal bases. Finally, we construct the global bases for $U_q(\mathfrak g)$-modules in the category $\mathcal O_{int}$ and for the subalgebra $U_q^-(\mathfrak g)$.60 pages, no figuresQuantum Algebra17B37text