Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras

Abstract

In this paper, we prove Khovanov-Lauda's cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let U q (g) be the quantum group associated with a symmetrizable Cartan datum and let V (Λ) be the irreducible highest weight U q (g)-module with a dominant integral highest weight Λ. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra R Λ gives a categorification of V (Λ).2000 Mathematics Subject Classification. 05E10, 16G99, 81R10.This paper is organized as follows. In Section 2 and Section 3, we recall basic properties of quantum groups, integrable highest weight modules and the Khovanov-Lauda-Rouquier algebra R. In Section 4, we investigate the structure of cyclotomic Khovanov-Lauda-Rouquier algebra R Λ and prove the exact sequence (1.2), and then show that the functors E Λ i , F Λ i (i ∈ I) are exact and send projectives to projectives. Section 5 is devoted to the sl 2 -categorification theory. In Section 6, we finish the proof of Khovanov-Lauda's cyclotomic categorification conjecture.

Implicit structure in 2-representations of quantum groups

“…Recently Kang and Kashiwara [12] prove this conjecture by showing that i-induction and i-restriction functors induce a functorial action of U q (g) on the categories Proj(R Λ ). They also define natural transformations between these functors giving an action of the KLR algebra [12,Section 6].…”

Folding KLR algebras

“…It is a bicrystal by Lemma 10.7. Proposition 10.5 shows that it satisfies conditions (1)- (6) in the statement of Theorem 10.2. Applying Theorem 10.2 completes the proof.…”

“…For condition (6), first apply condition (5) to deduce * i (f i M ) = * i (M ). This implies that f i M also satisfies condition (5).…”

“…Remark 12.10. When there is no diagram automorphism, a categorified version of Corollary 12.9 is proved in [6,22].…”

Symmetric quiver Hecke algebras andR-matrices of quantum affine algebras III: Table 1.

“…The quiver Hecke algebra R, introduced independently by Khovanov-Lauda [18] and Rouquier [23], provides a categorification of the negative half U − q (g) of a quantum group U q (g). Moreover, its cyclotomic quotients R Λ , depending on dominant integral weights Λ, also provide a categorification of the integrable highest weight modules V q (Λ) over U q (g) [11]. Recall that the cyclotomic quotients of an affine Hecke algebra give a categorification of integrable highest weight U (A (1) N −1 )-modules.…”