Seminar

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Seminar Archive

13:15, Room V5-148
Markus Schmidmeier (Boca Raton): Hammocks to visualize the support of finitely presented functors
Abstract: Many properties of a module can be expressed in terms of the dimension of the vector space obtained by applying a finitely presented functor to that module. For example, the dimension of the kernel, image or cokernel of the multiplication map given by an algebra element; or the number of summands of a certain type when the module is considered as a module over a subalgebra.
When the indecomposable modules over the algebra are arranged in the Auslander-Reiten quiver, the support of the finitely presented functor typically has the shape of a hammock, spanned between sources and sinks. There may also be tangents which are meshes where the hammock function at the middle term exceeds the sum of the values at the start and end terms. We describe how sources, sinks and tangents of the hammock relate to the modules which define the projective resolution of the finitely presented functor.
Examples include quiver representations and invariant subspaces of nilpotent linear operators.

13:15, Zoom
Haruhisa Enomoto (Osaka): From the lattice of torsion classes to wide and ICE-closed subcategories
Abstract: For a module category, we can consider the posets of various subcategories. In this talk, I explain that we can compute the posets of wide and ICE-closed (Image, Cokernel, Extension-closed) subcategories from the lattice of torsion classes in a purely combinatorial way. Also I give a simple expression of the poset of wide subcategories from the lattice of torsion classes, and give an application to the combinatorics of Reading's shard intersection order on Coxeter groups.

13:15, Room T2-149
Vanessa Miemietz (Norwich): Uniqueness of bocses corresponding to a quasi-hereditary algebra
Abstract: Koenig-Kuelshammer-Ovsienko proved that an algebra is quasi-hereditary if and only if it is Morita equivalent to the right algebra of a normal directed bocs. I will review their construction and explain my joint work with Julian Kuelshammer, which proves that the basic bocs associated to a Morita equivalence class of quasi-hereditary algebras is unique up to isomorphism.

13:15, Room T2-149
Georges Neaime (Bielefeld): Towards the linearity of complex braid groups
Abstract: Complex braid groups are a generalisation of Artin–Tits groups. They are attached to complex reflection groups, which are themselves a generalisation of finite Coxeter groups. It is an ongoing challenge to extend the theory of Artin–Tits groups to all complex braid groups. A part of this theory extension was established by Broué, Malle, and Rouquier in their seminal work. An important feature of spherical Artin groups is that they are linear groups, i.e., they admit a faithful linear representation of finite dimension. For the usual braid group, this property was shown to hold independently by Bigelow and Krammer. We seek to extend the theory of linearity to the context of complex braid groups, with a focus on the infinite families. Indeed, we will describe a definition of BMW and Brauer algebras, from which we can construct suitable linear representations and conjecture their faithfulness. We present a number of theorems and conjectures related to the structure of the aforementioned algebras, as well as properties of the relevant representations. We will finally propose a research programme for the sequel.

13:15, Room T2-149
Estanislao Herscovich (Grenoble): Vertex algebras and 2-monoidal categories
Abstract: In an attempt to make the theory of vertex algebras more natural, R. Borcherds proposed a new foundation based on two tensor products. In this talk I will explain how these ideas sit well within the framework of 2-monoidal categories. More precisely, I will present a certain 2-monoidal categories of functors and show how vertex algebras can be regarded as commutative algebras with respect to one of the tensor products.
14:30, Room T2-149
Thorsten Heidersdorf (Bonn): Categorical quotient constructions and representations of the general linear supergroup GL(m|n)
Abstract: Finite dimensional representations of complex algebraic supergroups are nowadays reasonably well understood as abelian categories (in terms of blocks, extensions etc) but their monoidal structure is largely unknown. I will describe some quotient constructions to approximate this monoidal structure in the case of GL(m|n).

13:15, Room T2-149
Isaac Bird (Prague): Duality and definable categories in triangulated categories
Abstract: Over a ring, a submodule is pure if and only if its dual is a summand, while a module is pure-injective if and only if it is a direct summand of a dual module. I will discuss how these statements can be generalised to triangulated categories through the framework of duality triples, before turning attention to duality and definable categories. This is done through the use of duality pairs, which I will introduce. Several applications and examples will be given. This talk is based on joint work with Jordan Williamson.

13:15, Room T2-149
Achim Krause (Münster): Integral endotrivial modules
Abstract: Endotrivial modules are invertible objects in stable module categories, i.e. modules "invertible modulo projectives". These have been studied intensively over p-groups with coefficients in a finite field, and more recently also for more general finite groups. In joint work with Jesper Grodal, we generalize stable module categories to arbitrary coefficients, and study the resulting notion of endotrivial modules over arbitrary rings, with special focus on integer coefficients. The relationship between the integer coefficient case and the finite field coefficient case leads to surprising character-theoretic integrality conditions.

For information on earlier talks please check the complete seminar archive.