Weighted Khovanov-Lauda-Rouquier algebras

TL;DR

This paper introduces weighted Khovanov-Lauda-Rouquier algebras, generalizing the original algebras with new structures, quotients, and geometric interpretations, impacting categorification and crystal basis theory.

Contribution

The paper defines weighted KLR algebras, explores their structures, quotients, and geometric interpretations, extending the categorification framework and connecting to Hall algebras.

Findings

Weighted KLR algebras retain key structures of original KLR algebras.

Natural quotients carry categorical Lie algebra actions.

Geometric interpretation as convolution algebras generalizes previous results.

Abstract

In this paper, we define a generalization of Khovanov-Lauda-Rouquier algebras which we call weighted Khovanov-Lauda-Rouquier algebras. We show that these algebras carry many of the same structures as the original Khovanov-Lauda-Rouquier algebras, including induction and restriction functors which induce a twisted biaglebra structure on their Grothendieck groups. We also define natural quotients of these algebras, which in an important special case carry a categorical action of an associated Lie algebra. Special cases of these include the algebras categorifying tensor products and Fock spaces defined by the author and Stroppel in past work. For symmetric Cartan matrices, weighted KLR algebras also have a natural gometric interpretation as convolution algebras, generalizing that for the original KLR algebras by Varagnolo and Vasserot; this result has positivity consequences important in the theory of crystal bases. In this case, we can also relate the Grothendieck group and its bialgebra structure to the Hall algebra of the associated quiver.