Rita Fioresi: Quantum principal bundles on quantum projective varieties

24th June, 2024

In non-commutative geometry, a quantum principal bundle over an affine base is recovered through a deformation of the algebra of its global sections: the property of being a principal bundle is encoded by the notion of Hopf-Galois extension, while the local triviality is expressed by the cleft property. We examine the case of a projective base X in the special case X = G/P, where G is a complex semisimple group and P is a parabolic subgroup. The quantization of G will then be interpreted as the quantum principal bundle on the quantum base space X, obtained via a quantum section.

Sondre Kvamme: Higher torsion classes and silting complexes

20th June, 2024

Higher Auslander-Reiten theory was introduced by Iyama in 2007 as a generalization of classical Auslander-Reiten theory. The main objects of study in the theory are d-cluster tilting subcategories of module categories. It turns out that many notions in algebra and representation theory have generalizations to higher Auslander-Reiten theory. In particular, in 2016 Jørgensen introduced a generalization of torsion classes, called higher torsion classes.

In this talk, I will recall the definition of higher torsion classes. I will then explain how functorially finite d-torsion classes give rise to (d+1)-term silting complexes, and hence to derived equivalences. The construction is analogous to the construction of 2-term silting complexes due to Adachi-Iyama-Reiten in 2014. I will illustrate the constructions and results on higher Nakayama algebras of type An.

Paul Laubie: Combinatorics of free pre-Lie algebras and algebras with several pre-Lie products sharing the Lie bracket

17th June, 2024

Using the theory of algebraic operads, we give a combinatorial description of free pre-Lie algebras (also known as left-symmetric algebras) with rooted trees. A numerical coincidence hints a similar description for algebras with several pre-Lie products sharing the Lie bracket using rooted Greg trees which are rooted trees with black and white vertices such that black vertices have at least two children. We then show that those Greg trees can be used to give a description of the free Lie algebras.

Marc Stephan: An equivariant BGG correspondence and applications to free A4-actions

13th June, 2024

A classical question in the theory of transformation groups asks which finite groups can act freely on a product of spheres. For instance, Oliver showed that the alternating group A4 can not act freely on any product of two equidimensional spheres.

I will report on joint projects with Henrik Rüping and Ergün Yalcin and explain that for 'most' dimensions m and n, there is no free A4-action on Sm × Sn and whenever there exists such a free action, then the corresponding cochain complex with mod 2 coefficients is rigid: its equivariant homotopy type only depends on m and n.

This involves an equivariant extension of Carlsson’s BGG correspondence in order to classify perfect complexes over 𝔽2[A4] with 4-dimensional total homology.

Erhard Neher: Corestriction

10th June, 2024

Corestriction is an important technique in the theory of central-simple associative algebras over a field. Given a finite étale extension K/F, e.g. a Galois extension, corestriction associates a central-simple associative F-algebra with every central-simple associative K-algebra. In this talk, I will give an introduction to corestriction over fields, applicable to non-associative algebras. Towards the end of my talk, I will indicate why it is of interest to generalize corestruction to schemes and sketch how this can be done.

Kent Vashaw: A Chinese remainder theorem and Carlson theorem for monoidal triangulated categories

6th June, 2024

Carlson's connectedness theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. In this talk, we will discuss a generalization, where it is proved that the Balmer support for an arbitrary monoidal triangulated category satisfies the analogous property. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.

Calum Hughes: An elementary theory of the 2-category of small categories

5th June, 2024

Lawvere’s Elementary Theory of the Category of Sets (ETCS) posits that the category Set is a well-pointed elementary topos with natural numbers object satisfying the axiom of choice. This provides a category theoretic foundation for mathematics which axiomatises the properties of function composition in contrast to Zermelo-Fraenkel set theory with the axiom of choice (ZFC), which axiomatises sets and their membership relation. Furthermore, ETCS augmented with the axiom schema of replacement can be shown to be equiconsistent with ZFC.

In this talk, I will present a categorification of ETCS which axiomatises the 2-category of small categories, functors and natural transformations; this is the elementary theory of the 2-category of small categories (ET2CSC) of the title. This extends Bourke’s characterisation of categories internal to a category E with pullbacks to the setting where E satisfies the extra properties of ETCS. Important 2-categorical definitions I will introduce are 2-well-pointedness, the full subobject classifier and the categorified axiom of choice. The main conclusion is that ET2CSC is 'Morita biequivalent’ with ETCS, meaning that the two theories have biequivalent 2-categories of models.

I will also describe how Shulman and Weber’s ideas on discrete opfibration classifiers can be used to incorporate replacement, in a way reminiscent of algebraic set theory.

Claudemir Fideles: Graded identities in Lie algebras with Cartan gradings: an algorithm

3rd June, 2024

The classification of finite-dimensional semisimple Lie algebras in characteristic 0 represents one of the significant achievements in algebra during the first half of the 20th century. This classification was developed by Killing and Cartan. According to the Killing-Cartan classification, the isomorphism classes of simple Lie algebras over an algebraically closed field of characteristic zero correspond one-to-one with irreducible root systems. In the infinite-dimensional case, the situation is more complicated, and the so-called algebras of Cartan type appear. It is somewhat surprising that graded identities for Lie algebras have been relatively few results to that extent. In this presentation, we will discuss some of the results obtained thus far and introduce an algorithm capable of generating a basis for all graded identities in Lie algebras with Cartan gradings. Specifically, over any infinite field, we will apply this algorithm to establish a basis for all graded identities of U1, the Lie algebra of derivations of the algebra of Laurent polynomials K[t,t-1]], and demonstrate that they do not admit any finite basis.

Cornelius Pillen: Proving and disproving conjectures by Humphreys, Verma and Donkin

31st May, 2024

Let G be a simple, simply connected algebraic group defined over a field of positive characteristic p, Gr be its rth Frobenius kernel, and, for q = pr, G(q) denotes the group of rational points over a field with q elements.

Motivated by work of Curtis and Steinberg, who showed that the simple Gr- and the simple G(q)-modules can be lifted to G, Humphreys and Verma conjectured that the projective covers of the simple Gr-modules also afford a G-module structure. Donkin later refined this conjecture by suggesting that these Gr-projectives lift uniquely to G in the form of tilting modules.

Ballard and Jantzen verified Donkin’s Tilting Module Conjecture for primes that are roughly twice the Coxeter number of the underlying root system or larger. But it was shown by Nakano and his collaborators that the conjecture fails in general. Counterexamples exist for all root systems with the exception of types B2, where the conjecture holds, and type A, where the conjecture remains completely open.

In this talk we delve into the rich history of these and closely related conjectures and report on their current status.

Emilie Wiesner: Representations of the Virasoro algebra

31st May, 2024

The Virasoro algebra is the central extension of derivations on Laurent polynomials. It plays an important role in mathematical physics and is itself a nice case study of an infinite-dimensional Lie algebra with triangular decomposition. I’ll give an overview of several families of representations of the Virasoro algebra and some connections between them.

Paul Sobaje: Combinatorics for Special Tilting Modules

31st May, 2024

The quest to find a character formula for the simple modules of a reductive algebraic group in positive characteristic took an unexpected turn roughly a decade ago when Williamson found a large number of counterexamples to the Lusztig Conjecture. Since then, the path to the simple characters has gone through the characters of the indecomposable tilting modules, thanks to the work of Riche and Williamson. However, the combinatorics required for determining all tilting characters are quite complicated, and the vast majority of these characters are not necessary to determine the simple characters. This talk is based on our pursuit of a more simplistic model in terms of what we’ve called the 'Steinberg quotient' of special tilting characters.

Aslak Bakke Buan: From exceptional to τ-exceptional sequences in module categories

30th May, 2024

Exceptional sequences and their mutations were first considered in triangulated categories by the Moscow school of algebraic geometers. In the early nineties, Crawley-Boevey and Ringel studied exceptional sequences for module categories of hereditary algebras. We first recall their definitions and their main results, and then proceed to discuss a natural generalization to all (not necessarily hereditary) finite-dimensional algebras. This is the theory of τ-exceptional sequences, which was developed in joint work with Marsh, motivated by τ-tilting theory, by Adachi-Iyama-Reiten, by Jasso's reduction techniques for such modules and corresponding torsion pairs, and by the introduction of signed exceptional sequences by Igusa-Todorov.

The interplay between theories for τ-rigid modules, torsion pairs, and wide subcategories is central to our discussions.

Pramod Achar: Cleanness in Springer theory

30th May, 2024

In Lusztig's papers from 1985-1986 that invented the theory of character sheaves, he proved (in nearly all cases) a remarkable property of cuspidal perverse Q-sheaves on the nilpotent variety: they are 'clean', meaning that their stalks vanish outside a single orbit. This property is crucial to making character sheaves computable by an algorithm, and it is a precursor of various 'block decompositions' of the derived category studied by various authors (Gunningham, Rider, Russell, and others) later. About 10 years ago, Mautner conjectured that these perverse sheaves remain clean after reduction modulo p (with some exceptions for small p). In this talk, I will discuss the history and context of the cleanness phenomenon, along with recent progress on Mautner’s conjecture.

Karin Erdmann: The Hemmer-Nakano Theorem and relative dominant dimension

30th May, 2024

Let ℋq(d) be the Iwahori-Hecke algebra of the symmetric group where q is a primitive ℓ-th root of unity, and let A = Sq(n,d) be the q-Schur algebra. Hemmer and Nakano proved amongst others that for ℓ ≥ 4, the Schur functor gives an equivalence between the category of A-modules with Weyl filtration, and the category of ℋq(d)-modules with dual Specht filtration, and that certain extension groups get identified. This has been a surprise and has inspired further research. In this talk we discuss some extensions of this result.

Dave Benson: The nucleus and the singularity category of cochains on the classifying space

30th May, 2024

The definition of the nucleus was originally formulated in joint work with Carlson and Robinson, to capture the supports of modules with no cohomology. This definition works in various contexts such as finite groups, restricted Lie algebras, and more generally, suitable triangulated categories of modules. In the finite group context it has a characterization in terms of subgroups whose centralizer is not p-nilpotent. In the restricted Lie algebra context, it is described in terms of the Richardson orbit, at least for large primes. Recent work with Greenlees has highlighted a connection with the singularity category of the cochains on the classifying space, in the group theoretic context. My plan is to give an introduction to these ideas.

Vera Serganova: Sylow theorems and support varieties for supergroups

30th May, 2024

We discuss support variety theory for quasireductive algebraic supergroups, i.e. supergroups with reductive even part over complex numbers. The corresponding categories of representations are Frobenius and share many properties of representations of finite groups in positive characteristic. It is desirable to describe Balmer spectrum of related triangulated symmetric monoidal categories. Our approach involves so called homological odd elements and certain tensor functors associated to them. On the way we encounter analogues of p-groups and Sylow subgroups for supergroups. We prove projectivity detection for our support theory and present other related results. We also explore connections with homological support theory developed by B. Boe, J. Kujawa and D. Nakano.

Shrawan Kumar: A new diagrammatic categorical setting for Schur dualities

29th May, 2024

Let Γ be a finite group acting on a simple Lie algebra 𝔤 and acting on an s-pointed projective curve (Σ, p = {p1, . . . ,ps}) faithfully (for s ≥ 1). Also, let an integrable highest weight module Hc(λi) of an appropriate twisted affine Lie algebra determined by the ramification at pi with a fixed central charge c is attached to each pi. We prove that the space of twisted conformal blocks attached to this data is isomorphic to the space associated to a quotient group of Γ acting on 𝔤 by diagram automorphisms and acting on a quotient of Σ. Under some mild conditions on ramification types, we prove that calculating the dimension of twisted conformal blocks can be reduced to the situation when Γ acts on 𝔤 by diagram automorphisms and covers of ℙ1 with 3 marked points. Assuming a twisted analogue of Teleman's vanishing theorem of Lie algebra homology, we derive an analogue of the Kac-Walton formula and the Verlinde formula for general Γ-curves (with mild restrictions on ramification types). In particular, if the Lie algebra 𝔤 is not of type D4, there are no restrictions on ramification types.

Chun-Ju Lai: Quantum Wreath Products

29th May, 2024

A quantum wreath product is the algebra produced from a given (not necessarily commutative) algebra B, a positive integer d, and a choice of certain coefficients in B ⊗ B. Important examples include variants of the Hecke algebras, such as (1) affine Hecke algebras and their degenerate version, (2) Wan-Wang’s wreath Hecke algebras, (3) Kleshchev-Muth’s affinization algebras, (4) Rosso-Savage’s (affine) Frobenius Hecke algebras, (5) endomorphism algebras arising from Elias’s Hecke-type categories, (6) Mathas-Stroppel’s Rees affine Frobenius Hecke algebras, and (7) Hu algebra, which quantizes the wreath product Sm ≀ S2 between the symmetric groups. Our goal is to develop a uniform approach to the structure and representation theory in order to encompass known results which were proved in a case by case manner. In this talk, I’ll focus on the Schur-Weyl duality and the Clifford theory. Our theory is motivated by (and has application to) the Ginzburg-Guay-Opdam-Rouquier problem on quasi-hereditary covers of Hecke algebras for complex reflection groups.

Jon Brundan: Isomeric Heisenberg and Kac–Moody categorification

29th May, 2024

The isomeric Heisenberg category acts naturally on a number of abelian categories appearing in the representation theory of the isomeric supergroup Q(n), and also on representations of Sergeev’s algebra which is related to the double covers of symmetric groups. I will explain an efficient way to convert an action of the isomeric Heisenberg category on these and other abelian categories into an action of a corresponding super Kac–Moody 2-category. To properly understand the odd simple root indexed by the element zero of the ground field requires the theory of odd symmetric functions developed by Ellis, Khovanov and Lauda, the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka, and the covering quantum groups defined and studied by Clark and Wang.

Weiqiang Wang: A new diagrammatic categorical setting for Schur dualities

29th May, 2024

The classical Schur duality is a simple yet powerful concept which relates the representations of the symmetric group and general linear Lie algebra, as well as combinatorics of symmetric functions. This admits a quantum deformation to a duality between a quantum group and Hecke algebra of type A. In this talk, we will describe several new simple diagrammatic (monoidal/quotient) categories, where old and new algebras behind (affine/cyclotomic) Schur duality emerge naturally. Our construction has new combinatorial implications on symmetric functions and RSK correspondence.