A diagrammatic approach to categorification of quantum groups I

Abstract

Abstract. To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the Kac-Moody Lie algebra associated with the graph.

Degenerate affine Hecke–Clifford algebras and type Q Lie superalgebras

“…In particular, we hope that the functor introduced here will play a role in showing that the 2-category for b ∞ , introduced by Khovanov-Lauda and independently by Rouquier, acts on O(q(n)), see [16][17][18]36]. Additionally, in [45], Wang and Zhao initiated a study of super analogues of W -algebras.…”

Implicit structure in 2-representations of quantum groups

“…Cyclotomic quotients. For Λ = i∈I λ i Λ i ∈ X + consider the two-sided ideal J Λ of R = R Q generated by elements y y • The cyclotomic quotient conjecture from [14,16] states that the category Proj(R Λ ) of finitely generated graded projective R Λ -modules categorifies the irreducible highest weight representation of U q (g) with dominant integral weight Λ. This implies there is an isomorphism of U q (g)-modules…”

Nilpotency in type A cyclotomic quotients

“…For more details see [7,9]. The results in this note do not depend on the ground field k; they remain valid when considering the ring R defined as above with Z-linear combinations of diagrams.…”