29 research outputs found
Categorification and the quantum Grassmannian
In \cite{JKS} we gave an (additive) categorification of Grassmannian cluster
algebras, using the category \CM(A) of Cohen-Macaulay modules for a certain
Gorenstein order . In this paper, using a cluster tilting object in the same
category \CM(A), we construct a compatible pair , which is the data
needed to define a quantum cluster algebra. We show that when is
defined from a cluster tilting object with rank 1 summands, this quantum
cluster algebra is (generically) isomorphic to the corresponding quantum
Grassmannian
Degenerations for derived categories
We propose a theory of degenerations for derived module categories, analogous
to degenerations in module varieties for module categories. In particular we
define two types of degenerations, one algebraic and the other geometric. We
show that these are equivalent, analogously to the Riemann-Zwara theorem for
module varieties. Applications to tilting complexes are given, in particular
that any two-term tilting complex is determined by its graded module structure