Crystals and Nakajima monomials for quantum generalized Kac–Moody algebras

We introduce the notion of Nakajima monomials for quantum generalized Kac–Moody algebras and construct the crystals B(∞) and B(λ) in terms of Nakajima monomials. We also give an explicit description of the Nakajima monomials in the crystals B(∞) and B(λ) for the rank 2 quantum generalized Kac–Moody algebras and for the quantum Monster algebra.