Canonical bases and KLR-algebras

Abstract

We prove a recent conjecture of Khovanov-Lauda concerning the categorification of one-half of the quantum group associated with a simply laced Cartan datum.

Transitive $2$-representations of finitary $2$-categories

“…This means that L consists of direct summands of powers of the negative generators of U . Then, from [We,Theorem 3.17] in combination with [Ro1,Theorem 5.7] and [VV,Theorem 4.4], it follows that mapping an indecomposable 1-morphism F ∈ L to F L induces a bijection between L and the set of isomorphism classes of indecomposable objects in For any M ∈ B-proj we have ½ λ l M = 0 and therefore F M = 0 for any F ∈ L.…”

Canonical basis, KLR algebras and parity sheaves

“…To avoid this we must modify a bit the complex L V,k . We get the following theorem, see [25,Thm. 3.6].…”

“…To prove this we must check that the relations from Definition 2.1 hold for the elements [25,Thm. 3.6], this holds for A = C. We want to deduce from this that the relations hold for any A.…”

“…The goal of Section 2 is to give a geometric construction of KLR algebras in arbitrary characteristic. The zero characteristic case was done in [22] and [25]. The KLR algebras in positive characteristic were studied in [12].…”

Canonical bases in tensor products revisited

“…Note that for U of infinite type it is still an open question whether the projective indecomposable modules in Webster's category provide a categorification of the canonical basis of ω L(λ) ⊗ L(λ ′ ), for λ, λ ′ ∈ X + , even though the latter has been constructed in [Lu93]. Nevertheless, in the case of tensor products of highest weight integrable modules, combining [W12, Propositions 7.6, 7.7, Theorem 8.8] (where the hard work was done based on earlier works of Vasserot, Varagnolo and Rouquier [VV11,R12]) with our Theorem 2.9 provides the following theorem (which was known [W12] in finite ADE type). Note that we get the strongest result out of combining the categorification and algebraic approaches in (3) below.…”