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This article is cited in 1 scientific paper (total in 1 paper)
Dual perfect bases and dual perfect graphs
Byeong Hoon Kahnga, Seok-Jin Kangab, Masaki Kashiwaraac, Uni Rinn Suhb a Department of Mathematical Sciences, Seoul National University, 599 Gwanak-Ro, Seoul 151-747, Korea
b Research Institute of Mathematics, Seoul National University, 599 Gwanak-Ro, Seoul 151-747, Korea
c Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
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 $U_q(\mathfrak g)$ 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.
Key words and phrases:
perfect basis, dual perfect basis, upper global basis, lower global basis.
Received: May 9, 2014
Citation:
Byeong Hoon Kahng, Seok-Jin Kang, Masaki Kashiwara, Uni Rinn Suh, “Dual perfect bases and dual perfect graphs”, Mosc. Math. J., 15:2 (2015), 319–335
Linking options:
https://www.mathnet.ru/eng/mmj561 https://www.mathnet.ru/eng/mmj/v15/i2/p319
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