Cluster categories and their relation to cluster algebras and semi-invariants

"Cluster categories were introduced in the paper ""Tilting theory and cluster combinatorics"" in order to better understand the combinatorics of cluster algebras, by giving new, module theoretic and categorical meanings to the combinatorics of the well known Cluster algebras of Fomin and Zelevinsky. Subsequently, we gave a very precise correspondence between the notions in these two areas. This proved to be quite useful and productive approach with even further connections to semi-invariants of quivers. However, in order to get this connection, we define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem (determinantal), the Saturation theorem and the Canonical Decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between the supports of the semi-invariants and the Tilting triangulation of the (n-1) sphere."