  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.
