Crystal Bases for Quantum Generalized Kac-Moody Algebras

In this paper, we develop the crystal basis theory for quantum generalized Kac-Moody algebras. For a quantum generalized Kac-Moody algebra $U_q(\mathfrak g)$,wefirstintroducethecategory$\mathcal O_{int}$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 $U_q^{-}(\mathfrak{g})$.