University of Glasgow, UK
For finite dimensional algebras, there is a bijection between the projective and simple modules. To pass from the projectives to the simples, one takes the quotient by the Jacobson radical. This process retains some information about the algebra; for example it produces an isomorphism on Grothendieck groups and can detect projective dimension. A construction due to Orlov lets one generalise this process to finite dimensional DGAs. These are chain complexes with a compatible multiplication which appear in geometry and representation theory. In this talk I’ll explain how we can quotient a fd DGA by its radical and what information about the DGA this retains.
Institut des Hautes Etudes Scientifiques, France
One of the most celebrated theorem in the theory of quiver representations is undoubtedly Gabriel’s theorem, which reveals a deep connection between quiver representations and root systems arising from Lie algebras. In particular, Gabriel’s theorem shows that the finite-type quivers are classified by the ADE Dynkin diagrams and the indecomposable representations are in bijection with the underlying positive roots. Following the works of Dlab—Ringel, the classification can be generalised to include all the other Dynkin diagrams (including BCFG) if one considers the more general notion of valued quiver representations instead.
The aim of this talk is to introduce a new notion of Coxeter quivers, where their representations are built in certain fusion categories. While valued quivers categorify root systems arising from semisimple Lie algebras, Coxeter quivers categorify root systems over fusion rings arising from Coxeter groups. The relevance to Gabriel’s theorem is as follows: a Coxeter quiver has finitely many indecomposable representations if and only if its underlying graph is a Coxeter-Dynkin diagram — including the non-crystallographic types H and I. Moreover, the indecomposable representations of a Coxeter quiver are in bijection with the positive roots of the underlying Coxeter group.
University of Glasgow, UK
Discrete cluster categories of Dynkin type A∞ were introduced initially by Holm-Jørgensen and more generally by Igusa-Todorov. They are well behaved triangulated categories that provide a practical setting in which to begin the study of cluster combinatorics in a category with infinitely many isomorphism classes of indecomposable objects.
In this talk we compute the Grothendieck groups of these categories, making use of work by Palu on the Grothendieck group of a 2-Calabi-Yau triangulated category with a cluster tilting subcategory. We follow this up by subsequently computing the Grothendieck groups of the completions of the discrete cluster categories of Dynkin type A∞ in the sense of Paquette-Yıldırım, a closely related category found as a Verdier localisation of a discrete cluster category of Dynkin type A∞.
Ghent University, Belgium
We propose a candidate for a continuous generalization of the type A associahedron. Our construction is motivated by the world sheet in string theory and by the continuous cluster category. We will show that that choosing an appropriate zigzag Z in a continuous version of the derived category D yields: (i) an abelian category that looks like finitely presented representations of a continuous quiver where Z plays the role of the projective slice, (ii) a truncated subcategory of D that contains a continuous version of the cluster combinatorics of a type A cluster algebra, and (iii) a candidate for the continuous generalization of a type A associahedron. For a particular choice of zigzag, there is a chain finite type A associahedra that embed into our continuous candidate. This is joint work with Maitreyee C. Kulkarni, Jacob P. Matherne, and Kaveh Mousavand (arXiv:2108.12927).
Charles University, Czech Republic
Both Happel and Ladkani proved that, for commutative rings, the quiver An is derived equivalent to the diagram generated by An where any composition of two consecutive arrows vanishes. We give a purely derivator-theoretic reformulation of this result, implying that it occurs uniformly across stable derivators and is then independent of coefficients. The equivalence we obtain provides a bridge between homotopy theory and representation theory; in fact we will see how our result is a derivator-theoretic version of the ∞-Dold-Kan correspondence for bounded cochain complexes.
University of Cologne, Germany
Based on joint work with Sibylle Schroll, Hipolito Treffinger and Nicholas J Williams. The $\tau$-cluster morphism category of an algebra (also known as the `cluster morphism category' or the `category of wide subcategories') encodes information about the $\tau$-tilting theory, cluster combinatorics and picture group of the algebra. In this talk, we discuss this category and show how it may be defined in terms of the wall-and-chamber structure of the algebra. This geometric perspective leads to a simplified proof that the category is well-defined.