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.

I.2. Summary for public relations work In this project, properties of triangulations of surfaces, associated algebras and combinatorial objects were studied. Among the main results of this research project is the characterization of periodic infinite frieze patterns and of their growth. In the 70s, Coxeter and Conways studied frieze patterns of numbers, arrays formed by finitely many rows of positive integers satisfiying the diamond rule: in any rhombus formed by with positive integers the numbers satisfy ad - bc = 1. An example (rhombus on left): 3 2 1 11 11 b 13 21 3 2 ad2 51 21 1 1 c 13 21 3 2 1 11 112 3 The first two rows determine the patterns completely. These patterns ex- hibit a horizontal symmetry as they are invariant under a glide reflection. Furthermore, the patterns correspond to triangulations of polygons. The first non-constant row is the given by the numbers of triangles meeting at the vertices of the polygon (hexagon on right). The discovery of links with cluster algebras revived the interest in frieze patterns. In this context, infi- nite such patterns were described in a paper by Baur and Marsh on friezes for cluster algebras. An example of an infinite frieze pattern: 1 11 11 2 23 22 3 2 3 55 35 3 7 78 77 8 .. .. .. 2 We were able to prove that every infinite (periodic) frieze pattern corre- sponds to a triangulation of an annulus ([3, 2]). We showed that the surface determines the growth of the pattern. Furthermore, we studied the growth of such frieze patterns, showing that they either have linear or exponential growth. The growth coecients are given by Chebychev polynomials, [1]. References 1. K. Baur, K. Fellner, M. J. Parsons, and M. Tschabold, Growth behaviour of periodic tame friezes, ArXiv e-prints (2016). 2. K. Baur, M. J. Parsons, and M. Tschabold, Infinite friezes, European J. Combin. 54 (2016), 220237. MR 3459066 3. M. Tschabold, Arithmetic infinite friezes from punctured discs, ArXiv e-prints (2015).