Crystals from categorified quantum groups

Details Crystals from categorified quantum groups Crystals from categorified quantum groups

2009-09-10

arxiv.org/abs/0909.1810v2

Advances in Mathematics, Volume 228, Issue 2, 1 October 2011, Pages 803-861

Mathematics

Representation Theory

56 pages, 6 eps files. v2 corrects typos

Scientific paper

10.1016/j.aim.2011.06.009

We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. We identify this crystal with Kashiwara's crystal for the corresponding negative half of the quantum Kac-Moody algebra. As a consequence, we show the simple graded modules for certain cyclotomic quotients carry the structure of highest weight crystals, and hence compute the rank of the corresponding Grothendieck group.

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