Dual perfect bases and dual perfect graphs

We introduce the notion of dual perfect bases and dual perfect graphs. We show that every integrable highest weight module $V_q(\lambda)$ over a quantum generalized Kac–Moody algebra $U_q(\mathfrak g)$ has a dual perfect basis and its dual perfect graph is isomorphic to the crystal $B(\lambda)$. We also show that the negative half $U_q^-(\mathfrak g)$ has a dual perfect basis whose dual perfect graph is isomorphic to the crystal $B(\infty)$. More generally, we prove that all the dual perfect graphs of a given dual perfect space are isomorphic as abstract crystals. Finally, we show that the isomorphism classes of finitely generated graded projective indecomposable modules over a Khovanov–Lauda–Rouquier algebra and its cyclotomic quotients form dual perfect bases for their Grothendieck groups.