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

RNA被认为是生命的基石之一。在其他功能中，RNA是连接DNA和蛋白质的信使，就像蛋白质一样，它可以催化反应。正是这一发现，使RNA兼具蛋白质和DNA的特性，从而产生了生命起源的“RNA世界”假说。这一假说规定，在从早期的“RNA世界”到现在的“RNA世界”的进化过程中，DNA和蛋白质取代了RNA的功能。核糖核酸是一种由核苷酸A(核苷)、C(胞嘧啶)、G(鸟氨酸)和U(Racil)链组成的分子。这些核苷酸以Watson-Crick(A-U，G-C)和(G-U)碱基对的形式相互形成氢键，形成RNA分子的二级结构。RNA二级结构通常用图形表示，其中核苷酸的线性序列用直线写成，氢键用从一个核苷酸到另一个核苷酸的直线上方的弧线表示。将RNA二级结构表示为这样的图已经启动了一个组合分析程序，导致了许多结构结果。我们建议在这一方法的基础上，通过为RNA二级结构下面的图配备更复杂的代数结构，我们预计这将产生关于RNA突变的结果。图是由一组顶点和连接这些顶点的一组边组成的数学对象。图可以是无向的，也可以是有向的，在后一种情况下，边被箭头替换。在代数的数学领域中，有向图被称为箭图。箭图是代数的基本构件之一。在这个项目中，我们考虑了簇倾斜代数和Brauer图代数。布劳尔图代数的显著之处在于，它们既可以用无向图表示，也可以用箭图和这个箭图上的关系来表示。在簇代数和簇范畴的新领域中，我们有类似的现象，代数可以用一个无向图来表示--例如，这可以是一个内接对角线形成三角形、四边形等的多边形-或者用箭图和关系来表示。在项目的第一阶段，我们将比较定义在同一图上的簇倾斜代数和Brauer图代数，我们将用Brauer图代数来解释簇情况下突变的含义，并比较这两个代数以及它们的基础图和关联曲面上覆盖的拓扑概念。此外，我们还将确定诸如Yoneda代数和Hochschild上同调的Brauer图代数的上同调结构和上同调不变量。然后我们将把这些结果联系起来并用于RNA二级结构的研究。在项目的第二阶段，我们建议从代数的角度通过将Brauer图代数与底层图相关联来研究RNA二级结构。然后，我们将把以前计算的同调不变量与二级结构的性质联系起来。在早期与R.Marsh的工作的基础上，我们已经在RNA二级结构和簇倾斜代数之间建立了组合联系，我们将展示簇突变和RNA点突变是如何相关的，以及一个人对另一个人的看法。我们将使用我们早期关于簇倾斜图代数和Brauer图代数的比较结果来确定我们的代数结果对RNA二级结构和RNA点突变研究的最重要影响。

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.