# Semi-invariants and cluster categories I and II

ARGOMENTI: Convegni

SEMINARI DI ALGEBRA
la professoressa GORDANA TODOROV della Northeastern University di Boston terra' due seminari dal titolo "Semi-invariants and cluster categories I and II"

-Date
Venerdi' 21 settembre in aula 1AD/30 alle ore 15.30
Lunedi' 24 settembre in aula 1BC/45 alle ore 16.15

-Abstract
We apply quiver representations and their semi- invariants to expose compatible combinatorial underpinnings for the tilting objects of cluster categories (and hence, clusters for cluster algebras), and for the homology of nilpotent groups. We focus on semi-invariants and tilting objects in cluster categories, by extending the classical semi-invariant results of Schofield, Derksen and Weyman, and interpreting the fundamental results about cluster categories to this setting. Modeling from K-theory, for an arbitrary quiver without oriented cycles, we consider semi-invariants in the derived category by extending the definition of representation spaces to virtual dimension vectors of virtual modules over the path algebra of the quiver. Such virtual dimension vectors have both positive and negative coordinates. Specifically, instead of working with representation spaces of the quiver acted upon by products of general linear groups, we work with generalized representation spaces, the spaces HomQ(P0, P1) for projective modules P0, P1.
The natural action of the group Aut(P0) × (Aut(P1))op replaces the action of the product of general linear groups. The semi-invariants for these actions are called the generalized semi-invariants. We construct the virtual
representation space and we prove the three basic theorems in the virtual setting: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem.
In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n - 1)-sphere.

Rif. int. S. Bazzoni