Weighted Khovanov-Lauda-Rouquier Algebras

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 bialgebra structure on their Grothendieck groups. We also define natural steadied quotients of these algebras, which in an important special cases give categorical actions of an associated Lie algebra. These include the algebras categorifying tensor products and Fock spaces defined by the author and C. Stroppel [B. Webster, Mem. Am. Math. Soc. 1191, iii-vi, 146 p. (2017; Zbl 07000045), p. 141, and C. Stroppel and B. Webster, “Quiver Schur algebras and $q$-Fock space”, Preprint, arxiv:1110.1115]. For symmetric Cartan matrices, weighted KLR algebras also have a natural geometric interpretation as convolution algebras, generalizing that for the original KLR algebras by M. Varagnolo and E. Vasserot [J. Reine Angew. Math. 659, 67–100 (2011; Zbl 1229.17019)]; 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.