Surface Algebras

Path algebras. A quiver is an oriented graph, consisting of a collection of vertices and arrows connecting the former. Concatenation of arrows leads to paths in the quiver. Viewing the paths as generators, with multiplication given by concatenation, the quiver defines an algebra, the path algebra of the quiver. Imposing relations on the paths induces relations between elements of the path algebra. By a result of Gabriel, this construction is very far- reaching, as it allows to describe any algebra (up to Morita equivalence). Surfaces with marked points on the boundary. To a given Riemann surface S with boundary components and a set of marked points on the boundary we can associate various algebras: The surface S can be triangulated with arcs connecting marked points in a way that no two arcs intersect. This determines a quiver whose vertices are the arcs in the triangulation and the boundary segments. The arrows of the quiver arise from rotations of arcs of the triangulation and of boundary segments. Taking the path algebra of this quiver thus defines an algebra associated to S. Quivers with potential. There is a canonical way to associate a potential W to a quiver Q: The summands of W are the oriented cycles in Q, clockwise oriented cycles being taken with positive sign, anti-clockwise oriented cycles with negative sign. The derivatives of W provide a set of relations on the quiver, (Q,W) hence give rise to a path algebra. Quivers with potentials have attracted a lot of interest in recent years as they lead to interesting examples of cluster algebras and cluster categories (Derksen-Weyman-Zelevinsky, Labardini-Fragoso, Amiot, etc.). This project is connected to the above areas and is inspired by recent developments in them. We aim to develop a general notion of surface algebras. We associate to a surface a class of algebras, by defining a quiver on the boundary segments, with arrows (and their reverses) joining adjacent boundary segments as well as arrows joining different boundary components. Furthermore, we impose certain relations on these arrows. Examples of such algebras have arisen in recent work by Jensen-King-Su and by Baur-King-Marsh. We call these algebras boundary algebras. A particular feature of our project is that we want to further the study of algebras associated to surfaces. We plan to encode properties of the surfaces in the relations directly, without considering a triangulation and its potential to start with. The boundary algebras will stand out among the algebras we study as a base point for our research. One of our goals in the project is to allow punctured surfaces, as they arise in the description of cluster algebras and cluster categories of Dynkin type D.

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