Crystals from categorified quantum groups

By Aaron D. Lauda, Monica Vazirani. 2009

140 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b0d9d932e8adc9058969ee8096daabff106a2d70

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words (2009, unknown): “R T ] 1 0 Se p”
Quiver Hecke Algebras and 2-Lie Algebras (2011, background): “¤ Note that Lauda and Vazirani had shown earlier that B(λ) gives rise to the crystal graph of L(λ), in the symmetrizable case [24].”
Simplicity of heads and socles of tensor products (2014, background): “If L(i) is the one-dimensional simple R(αi)-module, then L(i) is real and M ⋄ L(i) corresponds to the crystal operator f̃iM and L(i) ⋄ M to the dual crystal operator f̃i ∨ M in [12].”
Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras (2011, background): “That is, in [13], Lauda and Vazirani investigated the crystal structure on the set of isomorphism classes of irreducible graded modules over R and R, and showed that these crystals are isomorphic to the crystals B(∞) and B(Λ), respectively.”