Dual Perfect Bases and dual perfect graphs

TL;DR

This paper introduces dual perfect bases and graphs in the context of quantum Kac-Moody algebras, establishing their existence in modules and connections to crystal bases, with applications to algebra representations.

Contribution

It defines dual perfect bases and graphs, proves their existence in modules over quantum Kac-Moody algebras, and links them to crystal bases and module categories.

Findings

Dual perfect bases exist in integrable highest weight modules.

Dual perfect graphs are isomorphic to known crystal bases.

Applications to module categories over Khovanov-Lauda-Rouquier algebras.

Abstract

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 $U_{q}(\mathcal{g})$ has a dual perfect basis and its dual perfect graph is isomorphic to the crystal $B(\lambda)$. We also show that the negative half $U_{q}^{-}(\mathcal{g})$ 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.