Crystals from categorified quantum groups

Abstract

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.

56 pages, 6 eps files. v2 corrects typos

References in corpus (13)

• A diagrammatic approach to categorification of quantum groups I
• 2-Kac-Moody algebras
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• A diagrammatic approach to categorification of quantum groups II
• Graded decomposition numbers for cyclotomic Hecke algebras
• A diagrammatic approach to categorification of quantum groups III
• A categorification of quantum sl(2)
• Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras
• Graded Specht modules
• Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
• Canonical bases and Khovanov-Lauda algebras
• Graded $q$-Schur algebras
• Nilpotency in type A cyclotomic quotients

Cited by in corpus (38)

• Knot invariants and higher representation theory
• Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras
• Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
• PBW bases and KLR algebras
• Simplicity of heads and socles of tensor products
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras
• On uniqueness of tensor products of irreducible categorifications
• Mirkovic-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras
• Homological properties of finite type Khovanov-Lauda-Rouquier algebras
• The sl_3 web algebra
• Trace as an alternative decategorification functor
• A comparison of Newton-Okounkov polytopes of Schubert varieties
• Combinatorial descriptions of the crystal structure on certain PBW bases
• Specht modules for quiver Hecke algebras of type $C$
• The Feigin Tetrahedron
• On the structure of cyclotomic nilHecke algebras
• Rigged configurations and the $\ast$-involution
• Homogeneous representations of Type A KLR-algebras and Dyck paths
• Combinatorial descriptions of the crystal structure on certain PBW bases (extended abstract)
• Biadjointness in cyclic Khovanov-Lauda-Rouquier Algebras
• Folding KLR algebras
• Representations of Quiver Hecke Algebras via Lyndon Bases
• Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types
• Young tableaux, multi-segments, and PBW bases
• Representations of orientifold Khovanov-Lauda-Rouquier algebras and the Enomoto-Kashiwara algebra
• Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$
• Representations of Khovanov-Lauda-Rouquier Algebras and Combinatorics of Lyndon Words
• Affinizations, R-matrices and reflection functors
• Crystal of affine ${\widehat{\mathfrak{sl}}}_{\ell }$ and Hecke algebras at a primitive $2\ell$th root of unity
• On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products
• Notes on parameters of quiver Hecke algebras
• Categorical crystals for quantum affine algebras
• Morita equivalences between cyclotomic KLR algebras in types ${\mathtt{C}}_{\infty }$ and ${\mathtt{A}}_{\infty }$
• Schubert Class and cyclotomic nilHecke algebras
• Representation type of higher level cyclotomic quiver Hecke algebras in affine type C
• A new description of the bicrystal $B(\infty )$ and the extended crystal
• Modified affine Hecke algebras and quiver Hecke algebras of type $A$
• Cellularity of KLR and weighted KLRW algebras via crystals