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, (A) over a quantum generalized Kac-Moody algebra U-q (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(-) (g) has a dual perfect basis whose dual perfect graph is isomorphic to the crystal B(infinity). 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.