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 $A$. In this paper, using a cluster tilting object in the same category \CM(A), we construct a compatible pair $(B, L)$, which is the data needed to define a quantum cluster algebra. We show that when $(B, L)$ 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