Mutations for quivers with potentials and their representations

Andrei Zelevinsky (Northeastern University) March 23, 2007

This lecture is based on joint work in progress with H.Derksen and J.Weyman. We obtain a far-reaching generalization of classical Bernstein-Gelfand-Ponomarev reflection functors playing a fundamental role in the theory of quiver representations. These functors are defined only at a source or a sink of the quiver in question. We introduce a class of quivers with relations of a special kind given by non-commutative analogs of Jacobian ideals in the path algebra. We then define the mutations at arbitrary vertices for these quivers and their representations. If the vertex in question is a source or a sink, our mutations specialize to reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras. We will keep the exposition elementary, with all necessary background explained from scratch.