Key words: triangulations/tilings, frieze patterns, Grassmannian cluster structures, root systems, gentle algebras/string algebras
Abstract: Surface combinatorics have been instrumental in describing algebraic structures such as cluster algebras, cluster categories, gentle algebras, etc. In this course, I will show how triangulations and tilings of surfaces give rise to cluster categories and to module categories for gentle algebras.
In the case of a disk, triangulations and certain diagrams yield a combinatorial approach to cluster structures on the (coordinate ring of the) Grassmannians. These diagrams provide cluster-tilting objects corresponding to Pluecker coordinates. In general, the categories are of infinite type. The objects corresponding to Pluecker coordinates may be viewed as building blocks for arbitrary objects. We explain constructions of higher rank objects and will discuss a link to root systems associated to the Grassmannian cluster structure.
Key words: Hall algebras, Preprojective algebras, Kac polynomials, Yangians, Quiver varieties, Geometric representation theory, Kac-Moody Lie algebras, Borcherds algebras
Abstract: Cohomological Hall algebras (CoHAs) are certain associative algebras, for which the underlying vector spaces are given by the equivariant cohomology of moduli spaces of objects in an Abelian category, and for which the product is induced via the structure of extensions in the same category. The CoHAs I will focus on in these talks are built in this way from the category of representations of preprojective algebras. We will cover general structure theorems regarding CoHAs, including a Poincaré-Birkhoff-Witt theorem that enables us to encode the graded dimensions of these CoHAs using the coefficients of Kac polynomials. The Kac polynomials are defined by counting representations of quivers (up to isomorphism) over finite fields, and so are quite classical. Secondly, we will see how parts of these CoHAs arise as universal enveloping algebras of Kac-Moody Lie algebras, and their generalisation due to Borcherds. Finally, we will study the representation theory of these CoHAs, which is approached via Nakajima's quiver varieties and geometric representation theory, and use this as a bridge to compare these CoHAs with generalised Yangians (a class of quantum groups) introduced by Maulik and Okounkov.