Mathematics – Quantum Algebra
Scientific paper
2008-02-27
Adv. Math. 221 (2009), no. 1, 22-53
Mathematics
Quantum Algebra
28 pages; v2: some clarifications and corrections (mostly in Section 4); v3: minor typos corrected
Scientific paper
10.1016/j.aim.2008.11.016
Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commutor endows the category of crystals of a symmetrizable Kac-Moody algebra with the structure of a coboundary category, answering in the affirmative a question of Kamnitzer and Tingley.
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