European Quantum Algebra Lectures (EQuAL)

Spring 2024

January 18th: Yuri Bazlov (University of Manchester) – Cocycle and Galois cocycle twists of algebras, representations and orders

Abstract: In a construction known as Drinfeld twist, a 2-cocycle on a Hopf algebra H is used to modify the coproduct on H as well as the associative product in any H-module algebra A. I am interested to know to what extent the representation theory of the twist of A can be recovered from that of A; the A#H-module category, unchanged under the twist, plays a role here. I will talk about an application of this idea to rational Cherednik-type algebras, which led, in a joint work with E. Jones-Healey, to establishing nontrivial isomorphisms between braided and classical versions of these algebras. Twists also help to approach representation theory of the so-called mystic reflection groups

 defined by the Chevalley-Serre-Shephard-Todd property of their action on a quantum polynomial ring. An important source of twists, motivated by torsors in geometry, should be cocycles arising from (Hopf-)Galois extensions of algebras, and I will discuss this in the context of constructing orders and normal integral bases in central simple algebras over a number field.

February 1st: Fiona Torzewska (University of Bristol) – Topological quantum field theories and homotopy cobordisms

Abstract: I will begin by explaining the construction of a category CofCos, whose objects are topological spaces and whose morphisms are cofibrant cospans. Here the identity cospan is chosen to be of the form $X\to X\times [0,1]\rightarrow X$, in contrast with the usual identity in the bicategory $Cosp(V)$ of cospans over a category $V$. The category $CofCos$ has a subcategory $HomCob$ in which all spaces are homotopically 1-finitely generated. There exist functors into HomCob from a number of categorical constructions which are potentially of use for modelling particle trajectories in topological phases of matter: embedded cobordism categories and motion groupoids for example. Thus, functors from HomCob into Vect give representations of the aforementioned categories.

I will also construct a family of functors $Z_G\colon HomCob\to Vect$, one for each finite group $G$, and show that topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf-Witten, generalise to functors from HomCob. I will construct this functor in such a way that it is clear the images are finite dimensional vector spaces, and the functor is explicitly calculable. I will also give example calculations throughout.

February 15th: Jacob Bridgeman (Ghent University) – Invertible Bimodule Categories and Generalized Schur Orthogonality

Abstract: The Schur orthogonality relations are a cornerstone in the representation theory of groups. We utilize a generalization to weak Hopf algebras to provide a new, readily verifiable condition on the skeletal data for deciding whether a given bimodule category is invertible and therefore defines a Morita equivalence. Ultimately, the condition arises from Schur orthogonality relations on the characters of the annular algebra associated to a module category. As a first application, we provide an algorithm for the construction of the full skeletal data of the invertible bimodule category associated to a given module category, which is obtained in a unitary gauge when the underlying categories are unitary. As a second application, we show that our condition for invertibility is equivalent to the notion of MPO-injectivity, thereby closing an open question concerning tensor network representations of string-net models exhibiting topological order. Work with Laurens Lootens and Frank Verstraete. Based on arXiv: 2211.01947

February 29th: Martina Balagovic (University of Newcastle) - Braided Module Categories

Abstract: I will explain a construction leading to the structure of a braided module category over the braided category of finite dimensional representations of a quantum group, and discuss what we can hope to say about such a category. Joint work with Stefan Kolb.

March 14th: Tudor Dimofte (University of Edinburgh) - Finding quantum groups in QFT

Abstract: In three-dimensional topological quantum field theories, line operators carry the structure of a braided tensor category. Mathematically, one might hope for such categories to arise as modules for quasi-triangular Hopf algebras, a.k.a. `quantum groups'; in some classic examples, such as Chern-Simons QFT, this is famously borne out. However, it is a surprisingly difficult problem to see/construct these quantum groups in 3d QFT directly. Motivated by Tannakian duality, as well as some recent work in Chern-Simons theory by  N. Aamand, I'll present a general approach to finding quantum groups in cases where a QFT admits topological boundary conditions. (Joint with T. Creutzig and W. Niu.)

May 9th: Alexis Langlois-Rémillard (Hausdorff Center for Mathematics, Universität Bonn) - Quotients of the affine Temperley-Lieb algebras with a view towards generalised Deligne interpolation categories

Abstract: The affine (and periodic) Temperley-Lieb algebras appeared in the study of conformal field theories as useful tools to study the continuum scaling limits of critical statistical models. The fusion of their modules is believed to be connected to the fusion of bulk fields in CFT. However, the connection is not obvious. In part to seek the ideal structure to investigate the scaling limit, we study certain quotients of the affine Temperley-Lieb algebras, which we name uncoiled algebras, and we study their Jones-Wenzl idempotents. In this talk, we will present the uncoiled algebras, the construction of their Jones-Wenzl idempotents and investigate the traces of these, relating it to the extremal weight projectors of Queffelec and Wedrich. Time permitting, we will investigate a generalisation of these structures related to Deligne interpolation categories. 

This is based on joint work with Alexi Morin-Duchesne.

May 23th: Léo Schelstraete (Université Catholique de Louvain) - Odd Khovanov homology and higher representation theory

Abstract: Khovanov homology is a homological invariant of links categorifying the Jones polynomial. It is by now well-understood through the lens of higher representation theory, categorifying the relationship between the Jones polynomial and the representation theory of Uq(sl2). Surprisingly, there exists another categorification of the Jones polynomial, called odd Khovanov homology. Subsequently, higher odd (or “super”) analogues were discovered in representation theoretic and geometric contexts. In this talk, I will begin with a gentle introduction to the above, and then explain how odd Khovanov homology can be understood as stemming from a supercategorification of the representation theory of Uq(gl2). This is joint work with Pedro Vaz.

June 6th: Agustina Czenky (University of Oregon) - Title TBC

Abstract: TBC

June 20th: Fabio Calderón (Universidad de los Andes) - Title TBC

Abstract: TBC

Autumn 2023

October 5th: Tomoyuki Arakawa (RIMS, Kyoto University, Japan) – Hilbert Schemes of the points in the plane and quasi-lisse vertex superalgebras 

Abstract: For each complex reflection group Γ one can attach a canonical symplectic singularity M_Γ.  Motivated by the 4D/2D duality discovered by Beem et at., Bonetti, Menegheli and Rastelli conjectured the existence of a supersymmetric vertex operator algebra W_Γ whose associated variety is isomorphic to M_Γ.  We prove this conjecture when the complex reflection group Γ is the symmetric group SN, by constructing a sheaf of ħ-adic vertex algebras on the Hilbert schemes of N-points in the plane.  In physical terms,  the vertex operator algebra W_SN corresponds,  by the 4D/2D duality, to the 4-dimensional N=4 super Yang-Mills theory with gauge group SL(N).

This is a joint work with Toshiro Kuwabara and Sven Moller.

October 19th: Joost Vercruysse (Université Libre de Bruxelles) – Generalizations of Yetter-Drinfel'd modules and the center construction of monoidal categories

Abstract: This is joint work with Ryan Aziz. A Yetter-Drinfel'd 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-Drinfel'd 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 over the categories of H-modules and K-modules. Moreover, we also show how other variations of Yetter-Drinfel'd modules, such as anti-Yetter-Drinfel'd modules, arise as a particular case and we discuss the bicategorical structure that arises this way.

November 2nd (later time: 12pm UK time): Tobias Dyckerhoff (Universität Hamburg) – Complexes of stable ∞-categories

Abstract: Derived categories have come to play a decisive role in a wide range of topics. Several recent developments, in particular in the context of topological Fukaya categories, arouse the desire to study not just single categories, but rather complexes of categories. In this talk, we will discuss examples of such complexes in algebra, topology, algebraic geometry, and symplectic geometry, along with some results and conjectures involving them. Based on joint work with Merlin Christ and Tashi Walde.

November 16th: Frank Taipe (Université Paris-Saclay) – Quantum Transformation Groupoids

Abstract: We define quantum transformation groupoids, a class of multiplier Hopf algebroids generalizing transformation groupoids and algebraic quantum groups. An interesting characteristic of this algebraic class is that it admits a Pontryagin-like duality. In the first part of the talk, we will discuss how the study of quantum transformation groupoids appears in a Galois-type theory of inclusions of von Neumann algebras. Then in the second part, we will give the construction of a quantum transformation groupoid from a braided commutative measured Yetter-Drinfeld *-algebra on an algebraic quantum group in the sense of A. Van Daele.

November 30th: Azat Gainutdinov (Université de Tours, CNRS) – Non-semisimple link and manifold invariants for Symplectic Fermions

Abstract: I will talk about link and three-manifold invariants defined in terms of a non-semisimple finite ribbon category C together with a choice of tensor ideal and modified trace. If the ideal is all of C, these invariants agree with those defined by Lyubashenko in the 90’s, and as we show, they only depend on the Grothendieck class of the objects labelling the link. These invariants are therefore not able to determine non-split extensions. However, we observed an interesting phenomenon: if one chooses an intermediate proper ideal between C and the minimal ideal of projective objects, the invariants do distinguish non-trivial extensions. This is demonstrated in the case of C being the ribbon category of N pairs of symplectic fermions. This is a joint work with J. Berger and I. Runkel.

December 14th: Emily Norton (University of Kent) – Decomposition numbers for unipotent blocks with small sl_2-weight in finite classical groups

Abstract: There are many familiar module categories admitting a categorical action of a Lie algebra. The combinatorial shadow of such an action often yields answers to module-theoretic questions, for instance via crystals. In proving a conjecture of Gerber, Hiss, and Jacon, it was shown by Dudas, Varagnolo, and Vasserot that the category of unipotent representations of a finite classical group has such a categorical action. In this talk I will explain how we can use the categorical action to deduce closed formulas for certain families of decomposition numbers of these groups. This is joint work with Olivier Dudas.

Spring 2023

February 13th: Angela Tabiri (African Institute for Mathematical Sciences, Ghana) – Plane Curves which are Quantum Homogeneous Spaces

Abstract: Plane Curves which are Quantum Homogeneous Spaces Abstract: In this talk, we will discuss the construction of examples of quantum homogeneous spaces using the equation of a plane curve. The Hopf algebras we construct are isomorphic to the quantum plane and down-up algebras when the degree of the equation is two or three respectively. Interesting properties and open problems about these Hopf algebras will be discussed. 

February 27th: Ana Kontrec (RIMS, Kyoto University)

Title: Representation theory and duality properties of some affine W-algebras 

Abstract: One of the most important families of vertex algebras are affine vertex algebras and their associated $\mathcal{W}$-algebras, which are connected to various aspects of geometry and physics. Among the simplest examples of $\mathcal{W}$-algebras is the Bershadsky-Polyakov vertex algebra $\mathcal{W}^k(\mathfrak{g}, f_{min})$, associated to $\mathfrak{g} = sl(3)$  and the minimal nilpotent element $f_{min}$.

In this talk we are particularly interested in the Bershadsky-Polyakov algebra $\mathcal W_k$  at positive integer levels, for which we obtain a complete classification of irreducible modules.

In the case  $k=1$, we show that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple affine vertex superalgebra $L_{k'} (osp(1 \vert 2))$ for $k'=-5/4$. This is joint work with D. Adamovic. 

March 13th: Marco De Renzi (University of Zurich)

Title: Algebraic presentation of cobordisms and quantum invariants in dimensions 3 and 4 

Abstract: The category 2Cob of 2-dimensional cobordisms is freely generated by a commutative Frobenius algebra: the circle. This yields a complete classification of 2-dimensional TQFTs (Topological Quantum Field Theories). In this talk, I will discuss some consequences of analogous algebraic presentations in dimensions 3 and 4, due to Bobtcheva and Piergallini. In both cases, the fundamental algebraic structures are provided by certain Hopf algebras called BPH algebras. In dimension 3, I will consider the category 3Cob of connected cobordisms between connected surfaces with connected boundary. I will explain that an algebraic presentation conjectured (or rather announced without proof) by Habiro is in fact equivalent to the one established by Bobtcheva and Piergallini. In dimension 4, I will focus on a category denoted 4HB, whose morphisms are 2-deformation classes of 4-dimensional 2-handlebodies. I will show that any unimodular ribbon category contains a BPH algebra, which can be characterized very explicitly. This result proves the existence of a very large family of TQFT functors on 4HB. Finally, I will explain that a unimodular ribbon category has the potential to detect exotic phenomena in dimension 4 only if it is neither semisimple nor factorizable. This is a joint work with A. Beliakova, I. Bobtcheva, and R. Piergallini.

March 27th: Bojana Femic (Serbian Academy of Sciences and Arts)

Title: Categorical centers and Yetter Drinfel`d-modules as 2-categorical (bi)lax structures

Abstract: Joint work with Sebastian Halbig. See file below.

April 24th: Leandro Vendramin (Vrije Universiteit Brussel) – Nichols algebras

Abstract: Nichols algebras appear in several branches of mathematics, going from Hopf algebras and quantum groups, to Schubert calculus and conformal field theories. In this talk, we review the main problems related to Nichols algebras and I discuss some classification theorems and some applications.

May 22nd: Paolo Saracco (Université Libre de Bruxelles) – Closed categories, modules and (one-sided) Hopf algebras

Abstract: A well-known characterization of Hopf algebras, that I always found fascinating and elegant, states that an algebra A over a field k is a Hopf algebra if and only if its category of modules is a closed monoidal category in such a way that the forgetful functor to vector spaces preserves the closed monoidal structure. We usually split this result into two steps: the lifting of the monoidal structure corresponds to the bialgebra structure, and then the further lifting of the closed structure as adjoint to the monoidal one corresponds to the existence of an antipode. However, closed structures can be defined independently of monoidal ones and have their own dignity and importance. Which new structure on our algebra A would correspond to lifting the closed structure of vector spaces alone? How would this relate with the familiar bialgebra and Hopf algebra structures coming from lifting the monoidal and closed monoidal ones? It turns out that lifting the closed structure corresponds to the existence of algebra maps 𝛿 : A -> A⊗A^op and ε : A -> k satisfying appropriate conditions. Moreover, a quite unexpected source of examples is provided by certain one-sided Hopf algebras, i.e. bialgebras with a morphism which is just a one-sided convolution inverse of the identity. In this seminar, based on an ongoing collaboration with Johannes Berger and Joost Vercruysse which is continuing discussions with Gabriella Böhm, I will present our progresses in the study of these new algebraic structures.

List of talks

Autumn 2022

October 10th: Shahn Majid (Queen Mary University) – Quantum Riemannian Geometry of the A_n graph 

Abstract: We solve for quantum Riemannian geometries on the finite lattice interval • − • − · · · − • with n nodes (the Dynkin graph of type A_n) and find that they are necessarily q-deformed with q a root of unity. This comes out of the intrinsic geometry and not by assuming any quantum group in the picture. Specifically, we discover a novel ‘boundary effect’ whereby, in order to admit a quantum-Levi Civita connection, the ‘metric weight’ at any edge is forced to be greater pointing towards the bulk compared to towards the boundary, with ratio given by (i + 1_)q/(i)_q at node i, where (i)_q is a q-integer. The Christoffel symbols are also q-deformed. The limit q → 1 is the quantum Riemannian geometry of the natural numbers N with rational metric multiples (i + 1)/i in the direction of increasing i. In both cases there is a unique metric up to normalisation with zero Ricci scalar curvature. Elements of QFT and quantum gravity are exhibited for n = 3 and for the continuum limit of the geometry of N. The Laplacian for the scaler-flat metric becomes the Airy equation operator (1/ x) d^2/ dx^2 in so far as a limit exists. The talk is based on joint work with J. Argota-Quiroz available on arXiv: 2204.12212 (math.QA).

October 24th: Lorant Szegedy (University of Vienna) – Parity and Spin conformal field theory with boundaries and defects

Abstract: Rational conformal field theory (CFT) on oriented surfaces is well understood in terms of 3-dimensional topological field theory (TFT). We extend these notions to surfaces with spin structures using defects in oriented CFT and a modified TFT taking values in super vector spaces.

November 7th: Catherine Meusburger (University of Erlangen-Nuremberg) – Turaev-Viro-Barrett-Westbury invariants with defects

Abstract: Turaev-Viro-Barrett-Westbury state sum models are concrete constructions of TQFTs based on triangulated 3-manifolds and spherical fusion categories. Introducing defects in these models is of interest for defect TQFTs and for applications in condensed matter physics.

In the talk we explain how to construct Turaev-Viro-Barrett-Westbury state sums with defects in terms of generalised 6j symbols. Defect surfaces are labeled with bimodule categories over spherical fusion categories, defect lines and points form graphs on these surfaces and are labeled with bimodule functors and bimodule natural transformations.

We show that the resulting state sums are triangulation independent, compute examples and interpret them.

Based on https://arxiv.org/abs/2205.06874

November 21st: Lukas Woike (University of Burgundy) – Quantum representations of mapping class groups and factorization homology

Abstract: Quantum representations of mapping class groups are finite-dimensional representations of mapping class groups that have their origin in quantum algebra (e.g. the representation theory of Hopf algebras) and that often has strong ties to three-dimensional topological field theory. After explaining the interest in these representations from the perspectives of algebra, topology and mathematical physics and how they can be formally described through modular functors, I will give an idea of the classical construction procedures. I will then present a new and more general construction procedure using cyclic and modular operads, as well as factorization homology. The main result of this approach is a classification of modular functors. This is based on different joint works with Lukas Müller and Adrien Brochier.

Slides from Lukas Woike's talk can be found on this subpage.

December 5th: Lukas Müller (Perimeter Institute) – Reflection Structures and Spin Statistics in Low Dimensions 

Abstract: In physics the spin of a particle determines its statistics.

Furthermore, physical systems (in Euclidean signature) usually have a reflection structure, i.e. an identification of orientation reversal with complex conjugation. Neither of these two structures is part of Atiyah's original definition of topological quantum field theories.

They can be formulated in the setting of functorial field theories as equivariant symmetric monoidal functors from a bordism category to an appropriate target. Based on the cobordism hypothesis I will present a complete classification of such functors in dimension one and two. The answers can be formulated in terms of algebraic objects associated to an internal fermionic symmetry (2-)group. The talk is based on joint work in progress with Luuk Stehouwer.

December 19th: Vanessa Miemietz (University of East Anglia) – Symmetric bimodules and Hopf algebras 

Abstract: I will explain the basics of finitary 2-representation theory and explain a reduction theorem that motivates the study of certain types of 2-categories. I will then explain two examples of such, associated to Hopf algebras and symmetric bimodules, and explain the connection between the two.

If you would like to give a talk in the seminar, and most especially if you belong to an underrepresented minority in mathematics (female+, BAME, LGBTQIA+, etc), please contact the organisers with a proposed title and abstract.