Algebra Seminar: Seminar Day with Seven Seminars

Welcome to join on site or via Zoom link (Meeting ID: 645 5572 6999, please contact the organizer for passcode)

Talk 1.

Time: 08.30 - 09.25 CEST

Speaker: Zbigniew Wojciechowski (Dresden)

Title: Free monoidal and module categories from a commutativity perspective

Abstract: Module categories are to monoidal categories what k-algebras are to commutative rings over k. I will make this analogy precise, discuss non-standard definitions of monoidal and module categories, and apply these ideas to construct free monoidal and module categories generalizing constructions from ring theory. This is based on joint work with Anna Rodriguez Rasmussen, Mateusz Stroinski, and Tony Zorman.

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09.25-09.50 Tea and coffee break

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Talk 2.

Time: 09.50 - 10.45 CEST

Speaker: Catherine Meusburger (Erlangen)

Title: 3d Dijkgraaf-Witten TQFT with defects

Abstract: We give a gauge theoretical construction of 3d (untwisted) Dijkgraaf-Witten TQFT with defects a la Freed-Quinn.

This gives a symmetric monoidal functor from a category of stratified surfaces and stratified 3d cobordisms into the category of finite-dimensional vector spaces. It does not require choices of triangulations, allows an easy computation of examples and can be applied to defects in Kitaev's quantum double model.

This is joint work with Joao Faria Martins, arXiv: 2410.18049

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Talk 3.

Time: 10.50 - 11.45 CEST

Speaker: Christoph Schweigert (Hamburg)

Title: Covariance of graphical calculus for pivotal

bicategories and RCFT correlators

Abstract: We develop a graphical calculus for pivotal bicategories and analyze how it transforms under appropriate classes of functors.

This leads to relations between skein-theoretic constructions based on different pivotal bicategories.

We demonstrate how such a relations encodes important information about correlators in two-dimensional conformal field theories.

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11.45-13.00 Lunch break

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Talk 4.

Time: 13.00 - 13.55 CEST

Speaker: Juergen Fuchs (Karlstad)

Title: Grothendieck-Verdier module categories and relative Serre functors

Abstract: A Grothendieck-Verdier category is a monoidal category having a duality structure more general than rigidity. It comes with two monoidal structures, which are related by mixed associators that are generically not invertible. I will discuss aspects of a natural class of module categories over Grothendieck-Verdier categories. These have two distinguished subcategories on which certain lax or oplax module functors are actually strong. The module category can be realized as a category of (co)modules over a (co)algebra if and only if the distinguished subcategories contain a (co)generator. Moreover, the subcategories can be characterized via a comparison between internal Hom and internal coHom functors. This allows one to construct a partial relative Serre functor for the module category, which provides an equivalence between the two subcategories. The internal Hom of a fixed point of the relative Serre functor carries a natural structure of a Grothendieck-Verdier Frobenius algebra.

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Talk 5.

Time: 14.00 - 14.55 CEST

Speaker: Joost Vercruysse (Brussels)

Title: Generalizations of Yetter-Drinfeld modules and the center construction of bicategories

Abstract: A Yetter-Drinfeld module over a bialgebra $H$, is at the same time a module and a comodule over $H$ satisfying a particular compatibility condition. It is well-known that the category of Yetter-Drinfeld modules (say, over a finite dimensional Hopf algebra $H$) is equivalent to the center of the monoidal category of $H$-(co)modules as well as to the category of modules over the Drinfel'd double of $H$. Caenepeel, Militaru and Zhu introduced a generalized version of Yetter-Drinfeld modules. More precisely, they consider two bialgebras $H$, $K$, together with an bimodule coalgebra $C$ and a bicomodule algebra $A$ over them. A generalized Yetter-Drinfel'd module in their sense, is an $A$-module that is at the same time a $C$-comodule satisfying a certain compatibility condition. Under finiteness conditions, they showed that these modules are exactly modules of a suitably constructed smash product build out of $A$ and $C$. The aim of this talk is to show how the category of these generalized Yetter-Drinfel'd can be obtained as a "relative center" of the category of $A$-modules, viewed as a bi-actegory (or a bimodule-category) over the categories of $H$-modules and $K$-modules. Some other variations of Yetter-Drinfel'd modules, such as anti-Yetter-Drinfel'd modules, arise as a particular case. Furthermore, we give a bicategorical interpretation of our results, showing how generalized Yetter Drinfeld modules can be structured in a bicategory that is graded over the groupoid of Galois (co)objects, analogous to Turaev's group-graded monoidal categories.

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14.55-15.30 Tea and Coffee break

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Talk 6.

Time: 15.30 - 16.25 CEST

Speaker: Marco Mackaay (Algarve)

Title: Induction for extended affine type A Soergel bimodules from a maximal parabolic

Abstract: Parabolic induction plays an important role in the finite-dimensional representation theory of extended affine type A Hecke algebras. In a forthcoming paper with Vanessa Miemietz and Pedro Vaz, we introduce and study a categorical analog of induction for maximal parabolics using Soergel bimodules and Rouquier complexes. In my talk, I will sketch some of our results.

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Talk 7.

Time: 16.30 - 17.25 CEST

Speaker: Mateusz Stroinski (Uppsala)

Title: A coboundary Temperley-Lieb category for sl2-crystals

Abstract: This talk is based on joint work with Moaaz Alqady (University of Oregon). In it, I will present a diagrammatic category TL0, which is monoidally equivalent to the category of Kashiwara crystals for sl2. It can be viewed as a renormalization of the Temperley-Lieb category, specialized at q=0.

I will give concise, closed formulas for the Jones-Wenzl projectors, explain how the resulting bases for Hom-spaces have much nicer structure constants than for non-zero q, and how the projectors can be obtained from Möbius inversion for finite inverse monoids. Additionally, we will discuss a diagrammatic realization of the coboundary structure on g-crystals defined by Henriques and Kamnitzer, in the case g=sl2. This amounts to describing an action of the n-fruit cactus group on the set of cup diagrams on n strands. Time permitting, I will also talk about the spaces of fiber functors for TL0 and their parametrization via degenerate bilinear forms, which proves to be more complex than that for non-zero q, where the bilinear forms are non-degenerate.

While these results require a lot of categorical and representation-theoretic language to be formulated, they can be illustrated using elementary combinatorics of binary strings and cup diagrams, and I intend to focus on this perspective.