Algebras From Surfaces and Their Derived Equivalences

Date of Completion

January 2012






In 2002, Fomin and Zelevinsky introduced cluster algebras in the hopes of providing a new algebraic framework to study Lusztig's canonical basis. The original definition is elementary but the calculations can quickly become involved. To further study cluster algebras, cluster categories and cluster-tilted algebras were introduced. These categories and algebras allow us to study cluster algebras via the representation theory of quivers. The theory of cluster-tilted algebras closely follows the theory of tilted algebras developed in the 1980's. Not surprisingly, the cluster-tilted algebras are closely related to tilted algebras via the algebraic construction of the relation extension of an algebra and the combinatorial construction of admissible cuts in quivers. ^ We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. We show that surface algebras are endomorphism algebras of partial cluster-tilting objects in generalized cluster categories, we compute the invariant of Avella-Alaminos and Geiss for surface algebras and we provide a geometric model for the module category of surface algebras. Further, we determine some of the derived equivalences of these surface algebras. In particular, we fix a triangulation of a surface and determine when different cuts produce derived equivalent algebras. Additionally, we consider derived equivalences arising from reflections of gentle algebras. These reflections can be expressed in the surface triangulation in terms of cluster mutations and surface cuts. This allows us to see some of the derived equivalences between surface algebras coming from different triangulations of the same surface. ^