Geometric representations of cluster categories, Brauer graph algebras and RNA secondary structures

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RNA is considered to be one of the building blocks of life. Amongst other functions, RNA is a messenger linking DNA and proteins and just like proteins it can catalyze reactions. It is this discovery that RNA has features both of proteins and DNA that led to the "RNA world" hypothesis for the origin of life. This hypothesis stipulates that DNA and proteins took over the functions of RNA in the evolution from the early "RNA world" to the present one. RNA is a molecule that consists of a chain of nucleotides A(denine), C(ytosine), G(uanine), and U(racil). These nucleotides form hydrogen bonds with each other in the form of Watson-Crick (A-U, G-C) and (G -U) base pairs and form the secondary structure of the RNA molecule. RNA secondary structures are often represented by a graph where the linear sequence of nucleotides is written in a straight line and the hydrogen bonds are represented by arcs above the line from one nucleotide to another. The representation of RNA secondary structure as such a graph has initiated a programme of combinatorial analysis leading to many structural results. We propose to build on this approach by equipping the graph underlying the RNA secondary structure with a more complex algebraic structure which we expect to yield results on RNA mutations. A graph is a mathematical object consisting of a set of vertices and a set of edges connecting these vertices. Graphs can be undirected or directed, and in the latter case the edges are replaced by arrows. In the mathematical field of algebra a directed graph is called a quiver. Quivers are one of the basic building blocks of algebras. In this project we consider cluster-tilted algebras and Brauer graph algebras. Brauer graph algebras are remarkable in the fact that they can either be represented by an undirected graph or in a different representation by a quiver and relations on this quiver. In the new field of cluster algebras and cluster categories, we have similar phenomena where algebras can either be represented by an undirected graph - this can for example be a polygon where the inscribed diagonals form triangles, quadrangles etc - or in a different representation by a quiver and relations.In a first phase of the project we will compare cluster-tilted algebras and Brauer graph algebras defined on the same graph, we will interpret the meaning of mutation in the cluster case in terms of Brauer graph algebras and compare the topological notion of coverings on both algebras as well as their underlying graphs and associated surfaces. Furthermore, we will determine the cohomological structure and homological invariants of Brauer graph algebras such as the Yoneda algebra and Hochschild cohomology. We will then connect and use these results in the study of RNA secondary structures.In the second phase of the project, we propose to study RNA secondary structures from an algebraic point of view by associating a Brauer graph algebra to the underlying graph. We will then relate the previously calculated homological invariants to properties of the secondary structure. Building on earlier work with R. Marsh, where we have established a combinatorial connection between RNA secondary structures and cluster-tilted algebras we will show how cluster mutations and RNA point mutations are related and what one says about the other. We will use our earlier results on the comparison of cluster-tilted and Brauer graph algebras to determine the most significant impact of our algebraic results on the study of RNA secondary structures and RNA point mutations.