Weighted KhovanovLaudaRouquier algebras
Abstract
In this paper, we define a generalization of KhovanovLaudaRouquier algebras which we call weighted KhovanovLaudaRouquier algebras. We show that these algebras carry many of the same structures as the original KhovanovLaudaRouquier 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2012
 arXiv:
 arXiv:1209.2463
 Bibcode:
 2012arXiv1209.2463W
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Algebraic Geometry
 EPrint:
 37 pages