Moscow Mathematical Journal

Volume 15, Issue 2, April–June 2015 pp. 319–335.

Dual Perfect Bases and Dual Perfect Graphs

Authors: Byeong Hoon Kahng (1), Seok-Jin Kang (2), Masaki Kashiwara (3), and Uhi Rinn Suh (1)
Author institution:(1) Department of Mathematical Sciences, Seoul National University, 599 Gwanak-Ro, Seoul 151-747, Korea
(2) Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, 599 Gwanak-Ro, Seoul 151-747, Korea
(3) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 6068502, Japan, and Department of Mathematical Sciences, Seoul National University, 599 Gwanak-Ro, Seoul 151-747, Korea

Summary:

We introduce the notion of dual perfect bases and dual perfect graphs. We show that every integrable highest weight module V_q(λ) 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(λ). 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(∞). 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.

2010 Math. Subj. Class. 20G42.

Keywords: Perfect basis, dual perfect basis, upper global basis, lower global basis.

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