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Conference Program

17th September

9:00-9:55 The set theoretic Yang-Baxter equation and the structure of skew braces
Paul Truman, Keele University
Skew braces provide an algebraic framework for studying bijective nondegenerate solutions of the set-theoretic Yang-Baxter equation (or braid equation). We will give an overview of this framework and explore tools for analysing and classifying finite skew braces. In particular, we will discuss recent progress on Sylow-type theorems and apply them to obtain a streamlined classification of skew braces of order $pq$, where $ p,q $ are distinct primes.
10:00-10:30 Coffee & Tea Break
10:30-11:30 On quasi-lisse vertex algebras and conformal embeddings
Dražen Adamović, University of Zagreb
In this talk, we discuss some constructions and examples of quasi-lisse vertex algebras and their relations to conformal embeddings. We first present the family of vertex algebras $\mathcal C_p$, obtained in joint work with A. Milas, which is related to regular representations of $SL_2$. For small values of $p$, these algebras can be identified with affine or $W$-algebras and give interesting decompositions of conformal embeddings. We also discuss some recent results on conformal embeddings and collapsing levels for affine W-algebras, and application to tensor categories.
11:30-12:00 Ito’s conjecture and supersymmetry
Arim Song, Friedrich-Alexander Universität Erlangen-Nürnberg
For a simple Lie algebra $\mathfrak{g}$, the principal $\mathcal{W}$-algebra admits a coset realization in terms of affine vertex algebras. In 2020, Creutzig and Linshaw showed that these coset constructions fit into a broader framework of trialities, which also generalizes the well-known Feigin-Frenkel duality. For Lie superalgebras, analogous duality phenomena remain largely open. For $\mathfrak{g}=\mathfrak{sl}_{n+1|n}$, the coset realization of the corresponding principal $\mathcal{W}$-algebra was conjectured by Ito in 1992, with the relation between the levels taking a form analogous to Feigin-Frenkel duality. The case $n=1$ was known earlier, where both sides are isomorphic to the $\mathcal{N}=2$ Neveu-Schwarz vertex algebra. More recently, in joint work with my collaborators, we proved Ito's conjecture, making essential use of its $\mathcal{N}=2$ supersymmetric structure. In this talk, I will introduce Ito's conjecture and explain $\mathcal{N}=2$ supersymmetry. If time permits, I will also give a brief idea of the proof.
Noon Lunch at Pugin Hall
14:00-14:55 From the pentagon equation to Hopf algebras with combinatorial bases
Ilaria Colazzo, University of Leeds
Hopf algebras give rise to linear solutions of the pentagon equation. When can these solutions be realised by bijections of a basis? This question connects set-theoretic solutions of the pentagon equation with the structure of Hopf algebras. I will explain the role of matched pairs of finite groups in the classification of finite bijective set-theoretic solutions, and use this to characterise the finite-dimensional Hopf algebras that admit a basis on which the associated pentagon operator acts as a permutation. In characteristic zero, these are precisely the Hopf algebras arising from matched pairs of finite groups through the bismash product construction, linking this characterisation to the theory of Hopf algebras with positive bases. The talk is based on joint work with Jan Okniński and Arne Van Antwerpen, and with Geoffrey Janssens.
15:00-15:10 $q$-series invariants of knots and characters of VOAs
Matthias Storzer, University College of Cork
I will discuss a connection between tails of the coloured Jones polynomials for knots and characters of VOAs.
15:10-15:20 From indecomposable solutions to the Yang--Baxter equation to irreducible representations
Silvia Properzi, Vrije Universiteit Brussel
What does the indecomposability of a set-theoretical solution of the Yang--Baxter equation have to do with representation theory? In joint work with C. Dietzel and E. Feingesicht, we explore this connection for involutive, non-degenerate solutions by studying monomial representations of their associated structure groups and their Coxeter-like quotients. Using the brace structure of these groups, we show that, in most cases, the irreducibility of these representations is equivalent to the indecomposability of the solution, while in the remaining cases we obtain sufficient conditions for irreducibility. We also obtain explicit descriptions of the representations as induced representations from one-dimensional characters of point stabilisers.
15:20-15:30 A lightning introduction to inverse hamiltonian reduction
Chris Raymond, Universität Hamburg
Inverse hamiltonian reduction has become a key tool in the representation theory of W-algebra VOAs in recent years. I will give a sketch of the fundamental ideas and highlight some current areas in the literature where these ideas are being applied.
15:30-15:40 Chain Conditions on Skew Braces and Solutions of the Yang-Baxter Equation
Massimiliano Di Matteo, Università degli Studi della Campania "Luigi Vanvitelli"
The Yang--Baxter Equation (YBE) is a fundamental equation in quantum and statistical mechanics, and the study of its solutions has become a point of convergence across different disciplines. In particular, the study of its set-theoretical solutions has led to skew braces, a new algebraic structure with a dual nature oscillating between groups and rings. Set-theoretical solutions are associated with infinite skew braces, and therefore, it is natural to analyse chain conditions in skew braces: maximal and minimal conditions on subbraces and ideals. We will present their interrelations within the framework of the structure theory of skew braces. We relate our results to classical results in the theory of chain conditions in groups. Additionally, chain conditions are also defined to deal with infinite set-theoretical solutions. These could play a decisive role in the analysis of solutions to the equation in infinite-dimensional vector spaces. This is a joint work with Ramon Esteban-Romero, Maria Ferrara and Vicent Pérez-Calabuig.
15:45-16:15 Coffee & Tea Break
16:15-17:10 Vertex algebras and the rich structures enjoyed by their categories of modules
Simon Wood, Cardiff University
Vertex algebras encode the symmetries of two-dimensional conformally invariant quantum field theories. They can be thought of as a generalisation of unital commutative rings. Hence there is a natural theory of modules over vertex algebras. In this talk I will give a bird's-eye view of vertex algebras, and why their categories of modules admit tensor products with braiding, twist and duality structures. No prior knowledge of vertex algebras will be assumed.
17:15-18:10 More on Grothendieck—Verdier categories
Christoph Schweigert & Max Demirdilek, Universität Hamburg
We show that Grothendieck—Verdier categories (and module categories over them) have interesting subcategories and admit a good theory of Frobenius algebras. To explore these structures, we use a three-dimensional graphical calculus based on surface diagrams. We also introduce Grothendieck—Verdier functors as a means to relate different Grothendieck—Verdier categories.

18th September

9:00-9:55 Non-combinatorial involutive braidings: the quantum algebra $\mathfrak{gl}_{k,m}$
Anastasia Doikou, Heriot-Watt University
We consider involutive, non-combinatorial solutions of the braid equation as special deformations of the permutation map. Utilizing these solutions, we identify the associated quantum algebra and introduce it as the $\mathfrak{gl}_{k,m}$ Yangian. This newly derived Yangian is distinct from the known Yangian of the general linear Lie superalgebra. More importantly as a Hopf algebra, it possesses the standard tensor product algebra structure. The underlying algebra $\mathfrak{gl}_{k,m}$ is also introduced as a novel structure. We also construct specific highest-weight modules of $\mathfrak{gl}_{k,m}$ and explicitly study the highest-weight representations and the corresponding combinatorial bases for $\mathfrak{gl}_{1,1}$, linking them to specific shapes of Young tableaux.
10:00-10:30 Coffee & Tea Break
10:30-11:30 NIM-representations of Tambara-Yamagami generalizations
Ana Ros Camacho, Universitat de València
Tambara-Yamagami categories are ubiquitous in mathematical physics, playing a relevant role in topological and conformal field theory. There have been several suggestions for extending these categories, and we focus on two, given by Jordan-Larsson and Galindo-Möller-Lentner. In this work, we compute and classify the non-negative, integral matrix representations (NIM-reps) of these two fusion rings, and detect potential algebra objects associated to these NIM-reps. This is joint work with Agustina Czenky, Emily McGovern, Melody Molander and Monique Müller.
11:30 - 12:00 $W(2,2)$ tensor category
Gordan Radobolja, University of Split
We present the category of C1-cofinite, grading-restricted modules and show it has a braided tensor category structure. Furthermore, the fusion rules are governed by the sl_2 Clebsch-Gordan rule. This is a joint work with D. Adamović and J. Yang
Noon Sandwich lunch at Department Conference room
14:00-14:55 From Homotopy Finite Spaces to Modular Tensor Categories
João Faria Martins, University of Leeds
Each homotopy finite space $B$ gives rise to an extended TQFT and, in the (1,2,3)-TQFT case, to a modular tensor category. In this talk, I will report on ongoing joint work with Jack Romo on the explicit construction of modular tensor categories arising from homotopy finite spaces. I will also describe these categories concretely when the space $B$ is a 2-type and is therefore represented by a crossed module of finite groups. The talk will be accessible to non-specialists and will include applications to the construction of solutions of the Yang-Baxter equation.
15:00-15:10 Indecomposable solutions with permutation brace of size $p^3$
Andrew Darlington, Vrije Universiteit Brussel
Indecomposable solutions of the Yang-Baxter equation are important building blocks for studying several broader classes of solutions.  In this talk, we show how, using a construction of Bachiller, Cedo, and Jespers, indecomposable solutions can be classified with respect to related structures known as braces. We demonstrate this approach using the braces of size p^3, as well as a computer implementation of the algorithm.
15:10-15:20 Finite Group Algebras
Leo Creedon, Atlantic Technological University Sligo
A finite field F and a finite group G are used to define a finite group algebra FG. This beautiful algebraic structure can be viewed from many perspectives, as an abstract ring, an algebra and a vector space, or as a ring of functions from the group to the field. Some possible research directions will be mentioned, and topics discussed such as derivations of group algebras, partial algebraic structures, error correcting codes and connections with dynamical systems, and algebraic cellular automata.
15:30-16:00 Coffee & Tea Break
16:00-17:00 Computing multiplicity-free fusion rings and categories
Joost Slingerland, Maynooth University
I will give an introduction to some of the methods and algorithms we used to obtain explicit structure coefficients (known as F-symbols, R-symbols) of what we believe are all multiplicity-free pivotal complex fusion categories up to rank 7, up to gauge and automorphism equivalence, with their braided structures, when they exist. These have been implemented in Anyonica, a software package we developed for working with fusion categories. Results for many categories are also made available through Anyonwiki, an open-access database of fusion rings and fusion categories for mathematical physics research. I will also try to give some overview and highlights of the fusion rings and categories in the database. This talk is based on joint work with Gert Vercleyen
17:00-18:00 Inside Indecomposability - Some Old, Some New and Some Future Results on the Classification of Indecomposable Cycle Sets
Carsten Dietzel, Vrije Universiteit Brussel
Cycle sets provide a convenient framework to parametrize and investigate non-degenerate involutive solutions to the Yang--Baxter Equation. Therefore, one of the main goals in the theory of the Yang--Baxter Equation is their classification. However, computational results by Akgun, Mereb and Vendramin, together with more recent results of Van Caudenberg, Bogaerts and Vendramin, show that even for small given cardinalities, there is an incredibly large number of cycle sets. Thus, for now, a full classification of cycle sets seems to be out of reach. Whereas a classification of all cycle sets might be doomed to remain a Yang--Baxterist's eternal dream, a more feasible goal is the classification of finite indecomposable cycle sets. The latter can be considered elementary building blocks for general cycle sets, as those can be submitted to a repeated decomposition process, at the end of which is a collection of indecomposable cycle sets. It turns out that the structure of indecomposable cycle sets is heavily dependent on their size - indeed, for several cardinalities, a complete classification is known! The aim of this talk is to guide the listener through some of these results. After having introduced the most basic notions surrounding indecomposability, we explain the classification of indecomposable cycle sets of size $p^2$, achieved in joint work with Silvia Properzi and Senne Trappeniers. In this part of the talk, we highlight the importance of studying cycle sets of $p$-type, that is, indecomposable cycle sets whose permutation group is a $p$-group. Afterwards, we give an outline how the extension theory of cycle sets, combined with a recent result of Cedó and Okniński, leads to the classification of cycle sets of cardinality $pqr$, where $p,q,r$ are distinct primes. Finally, we explain how computational methods provide a strong tool in the classification of small cycle sets. While doing so, we give a small glimpse of work in progress with Andrew Darlington, Silvia Properzi and Magdalena Wiertel.