Talks

The Yang–Baxter equation is a fundamental equation connecting algebra, topology, and mathematical physics. Its set-theoretic solutions naturally give rise to actions of braid groups, providing a bridge between algebraic structures and the topology of braids. In this talk, we will explore what happens when we replace classical braids with welded braids, which admit a topological interpretation in terms of motions of circles in three-dimensional space. A natural question is whether the braid group actions arising from solutions of the Yang–Baxter equation can be extended to welded braid groups. We will discuss the compatibility conditions needed for such extensions, how the resulting representations can be simplified, and the role played by algebraic structures such as skew braces and bi-skew braces. This is based on joint work with Victoria Lebed, João Faria Martins, and Senne Trappeniers.

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.

The pentagon equation appears in several areas of mathematics, from the coherence of associativity to topology and Hopf algebras. In this talk, I will introduce these connections before turning to set-theoretic solutions, where the equation becomes an identity for a map on the Cartesian square of a set. Through examples, I will explain how groups and matched products of groups enter the picture and describe the classification of finite bijective solutions. I will conclude by discussing the connection with Hopf algebras. No prior familiarity with the pentagon equation will be assumed.

Skew braces are a powerful algebraic source of set-theoretic solutions to the Yang–Baxter equa- tion. On the other hand, welded braid groups arise naturally in low-dimensional topology, for instance as motion groups of unlinked circles in three-space, and welded solutions give rise to colouring invariants of welded knots and links. In this talk, I will discuss how these two worlds meet. Starting from the solution associated to a skew brace, I will ask when it is compatible with the twist solution in the welded sense. This leads to a notion of welded skew brace. I will present a characterisation: a skew brace is welded precisely when it is bi-skew, together with an additional commutativity condition on the right action. I will illustrate the result through examples, including braces arising from nilpotent rings and from group actions, and explain how these recover and generalise known welded biquandle constructions. This is joint work with Jo˜ao Faria Martins, Victoria Lebed, and Senne Trappeniers.

In this talk I’ll report on joint work with Janssens. Building on a characterisation of set-theoretic solutions to the pentagon equation obtained in earlier joint work with Okninski and van Antwerpen, and on the connection between Hopf algebras and pentagon solutions, we show that such set-theoretic solutions give rise to Hopf algebras with a positive basis in the sense of Lu–Yan–Zhu. I will explain the characterisation, present the construction, highlight its structural consequences, and illustrate it with examples.

In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions. Next, we will focus on the fundamental components: the irretractable solutions, and we will examine how these solutions relate to matched pairs of groups. Finally, we will show how each irretractable solution can be lifted to classify all bijective solutions.

Skew braces are algebraic structures introduced to study set-theoretic solutions to the Yang–Baxter equation. They provide a framework that connects group-theoretic and ring-theoretic methods with the combinatorial properties of these solutions. This talk will offer an introduction to skew braces and their main properties, with an emphasis on their role in constructing and analysing solutions to the Yang–Baxter equation. As an application, we will briefly outline how skew braces can be used to describe simple solutions.

Introduction to GAP: - Basics of Computation Topics: Basic arithmetic, programming, linear algebra, and an introduction to group theory concepts. - Group Theory in GAP: Permutation and Finite Groups Topics: Permutation groups, cyclic groups, dihedral groups, symmetric groups, and alternating groups. - Group Theory in GAP: Matrix Groups and Homomorphisms Topics: Matrix groups, group homomorphisms, and automorphism groups, with computational applications.

This talk will focus on the set-theoretic version of the pentagon equation. After discussing some basic properties of such solutions, we will focus on bijective solutions and provide a complete description of all bijective solutions. As a surprising corollary, we will see that skew braces produce solutions to the pentagon equation. Based on a joint work with E.Jespers and Ł. Kubat and on a joint work with J. Okninski and A. Van Antwerpen.

Recently, several algebraic structures have emerged for studying the set-theoretic version of the Yang–Baxter equation. Skew braces are particularly important among them. This talk will give a brief introduction to skew braces and discuss their properties, as well as their connection to solutions. Additionally, we'll explore computer tools for constructing different families of finite solutions and skew braces.

An introduction to skew braces and solutions to the Yang-Baxter equation

In the last years, various algebraic structures have been introduced to produce and classify combinatorial solutions to the Yang–Baxter equation. Among them, skew braces play a central role. In this talk, we will provide a brief introduction to this algebraic structure and an overview of its properties. In addition, we will delve into the connection between skew braces and solutions.

Is it possible to associate a skew bracoid with a solution to the Yang- Baxter equation? In this talk, we will answer the question for skew bracoids which are quotients of skew braces by strong left ideals

We will mention combinatorial results about the structure of finite indecomposable solutions to the Yang-Baxter. In particular, we will focus on the differences between the involutive and non-involutive cases. A crucial difference between these two classes of solutions is that inovlutive ones naturally embed into solutions associated with the structure group. In contrast, it is well known that this is not always the case for non-involutive solutions. Therefore, our first aim will be to connect the decomposability of solutions with the decomposability of their injectivisation. As a consequence, we will obtain some criteria for decomposability. The talk will also include concrete examples to understand the construction and some open questions.

The problem of finding set-theoretic solutions to the Yang-Baxter equation goes back to Drienfeld. A first attempt to tackle this problem is studying indecomposable solutions. This talk, based on joint work with M. Ferrara and M. Trombetti, will introduce skew braces in which every element has finitely many conjugates, which might be considered the brace-theoretical counterpart of FC-groups. Moreover, I will show that fundamental examples of such skew braces are structure groups of derived-indecomposable solutions (i.e. solutions whose derived is indecomposable).

(joint work with M. Ferrara and M. Trombetti) Studying indecomposable solutions is a first attempt to tackle the problem of funding all set-theoretic solutions to the Yang-Baxter equation. This talk will focus on a class of indecomposable solutions called derived-indecomposable, i.e. solutions, as the name suggests, such that their derived solution is indecomposable. I will show that the study of derived-indecomposable solutions naturally leads to the study of skew braces that behave like $FC$-groups since this captures the features of the structure skew brace (i.e. the structure group regarded as a skew brace) associated with such solutions. I will also present some structure results for this class of skew braces and state some open problems.

The study of large classes of set-theoretic solutions can be reduced to the study of some associative algebraic objects, such as braces, skew braces, and YB-semitrusses. A YB-semitruss is an algebraic associative structure that forms a sub-category of semitrusses, allowing one to determine and analyse left non-degenerate solutions to the Yang-Baxter equation. Based on joint work with E. Jespers, A. Van Antwerpen and C. Verwimp, this talk will describe the algebraic structure of YB-semitrusses and its relation with solutions to the Yang-Baxter equation. Then, we will show that skew braces are examples of YB-semitrusses.Finally, we will use YB-semitrusses as a tool to prove that for a finite left non-degenerate solution being right non-degenerate is equivalent to being bijective.

A YB-semitruss is an algebraic associative structure that forms a sub-category of semitrusses, allowing one to determine and analyse left non-degenerate solutions to the Yang-Baxter equation. Based on joint work with E. Jespers, A. Van Antwerpen and C. Verwimp, this talk will focus on YB-semitrusses with bijective associated solutions. First, we will show that skew braces are examples of YB-semitrusses. Then, we will use YB-semitrusses as a tool to prove that any finite left non-degenerate set-theoretic solution of the Yang-Baxter equation is right non-degenerate if and only if it is bijective, extending a result obtained by Castelli,Catino and Stefanelli.

The Yang-Baxter equation, named after the authors of the seminal papers in which the equation arose, is a fundamental tool in several research fields such as statistical mechanics, quantum group theory, and low-dimensional topology. In 1992, Drinfel’d suggested focusing on set-theoretical solutions of the Yang-Baxter equation as a first step to studying solutions of the Yang-Baxter equation. These solutions were initially approached in a combinatorial fashion by Etingof, Soloviev and Schedler and Gateva-Ivanova and Van den Bergh. More recently, Rump introduced braces, an algebraic structure that generalises radical rings, to describe involutive non-degenerate solutions of the Yang-Baxter equation. To study non-involutive solutions, one needs skew braces, a non-commutative analogue of braces. In this talk, we briefly discuss some basic properties of skew braces and how these structures are related to solutions of the Yang-Baxter equation. We also introduce a useful construction technique that allows one to characterise a subclass of skew braces. (The talk will be mainly based on joint work with F. Catino and P. Stefanelli.)

The Quantum Yang-Baxter equation, named after the authors of the seminalpapers in which the equation arose, Yang and Baxter , is a fundamental toolin several research fields such as statistical mechanics, quantum group theory,and low-dimensional topology. In 1992, Drinfel’d [Dd92] suggested focusing onset-theoretical solutions of the Yang-Baxter equation as a first step to study-ing solutions of the Quantum Yang-Baxter equation. These solutions were ini-tially approached in a combinatorial fashion by Etingof, Soloviev and Schedler[ESS99] and Gateva-Ivanova and Van den Bergh [GIVdB98]. More recently,Rump [Rum07] showed that involutive non-degenerate set-theoretic solutionscould be constructed using braces, an algebraic structure generalizing radicalrings. Furthermore, it turns out that such structure answers the question posedby Liebeck, Praeger, and Saxl [LPS00] of finding regular subgroups of the affinegroup: indeed, a particular instance of braces called brace over a field is inone-to-one correspondence with all regular subgroups of a specific affine group[CR09]. In this talk we expose the connection between braces and regular subgroups,then we present some constructions [CCS15, CCS16] of braces that lead to describing particular classes of regular subgroups. References [CCS15] F. Catino, I. Colazzo, and P. Stefanelli. On regular subgroups of the affine group. Bull. Aust. Math. Soc., 91(1):76–85, 2015. [CCS16] F. Catino, I. Colazzo, and P. Stefanelli. Regular subgroups of the affine groupand asymmetric product of radical braces. J. Algebra, 455:164–182, 2016. [CR09] F. Catino and R. Rizzo. Regular subgroups of the affine group and radical circle algebras. Bull. Aust. Math. Soc., 79(1):103–107, 2009. [Dd92] V. G. Drinfel′d. On some unsolved problems in quantum group theory. InQuantum groups (Leningrad, 1990), volume 1510 of Lecture Notes in Math., pages 1–8. Springer, Berlin, 1992. [ESS99] P. Etingof, T. Schedler, and A. Soloviev. Set-theoretical solutions to the quantum Yang-Baxter equation. Duke Math. J., 100(2): 169–209, 1999. [GIVdB98] T. Gateva-Ivanova and M. Van den Bergh. Semigroups of I-type. J. Algebra, x206(1):97–112, 1998. [LPS00] M. W. Liebeck, C. E. Praeger, and J. Saxl. Transitive subgroups of primitive permutation groups. J. Algebra, 234(2): 291–361, 2000. Special issue in honor of Helmut Wielandt. [Rum07] W. Rump. Braces, radical rings, and the quantum Yang-Baxter equation. J.Algebra, 307(1):153–170, 2007.

The Yang-Baxter equation is one of the fundamental equations in mathematical-physics. The problem of finding and classifying the set theoretical solutions of the Yang-Baxter equation has originally been posed by Drinfeld in [4]. In particular, the class of non-degenerate solutions has received considerable attention [5, 6, 7, 8, 3]. Although interesting and remarkable results on classifying non-degenerate solutions have been presented, there are still many open related problems. One of these problems is how to construct new families of solutions. Initial contribution to this aspect has been given by Etingof, Schedler, and Soloviev, who present a method to obtain a new solution via retraction. Recently, Vendramin in [9] and Bachiller, Cedó, Jespers, Okniński in [1] provide methods to obtain families of solutions starting from others, which are known. In this talk we introduce a novel construction technique, called matched product of solutions [2], which allows one to obtain new solutions from two generic (not necessarily non-degenerate) solutions. In addition, we prove that the matched product of two left non-degenerate involutive solutions is still left non-degenerate and involutive. References [1] D. Bachiller, F. Cedó, E. Jespers, J. Okniński: A family of irretractable square-free solutions of the Yang-Baxter equation, Forum Math., 29 (2017), 1291-1306. [2] F. Catino, I. Colazzo, P. Stefanelli: The matched product of set-theoretic solutions of the Yang-Baxter equation, in preparation. [3] F. Cedó, E. Jespers, J. Okniński: Braces and the Yang-Baxter equation, Comm. Math. Phys., 327 (2014), 101-116. [4] V. G. Drinfeld: On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Math. 1510, Springer, Berlin, (1992), 1-8. [5] P. Etingof, T. Schedler, A. Soloviev: Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169-209. [6] T. Gateva-Ivanova, M. Van den Bergh: Semigroups of I-type, J. Algebra, 206 (1998), 97-112. [7] J.-H. Lu, M. Yan, Y.-C. Zhu: On the set-theoretical Yang-Baxter equation, Duke Math. J., 104 (1) (2000) 1-18. [8] W. Rump: Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra 307 (2007), 153-170. [9] L. Vendramin: Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova, J. Pure Appl. Algebra, 220 (2016) 2064-2076.

The pentagon equation is a fundamental relation appearing in several areas of algebra. In this talk, I will discuss bijective set-theoretic solutions and some of the algebraic structures underlying them. I will focus in particular on their relationship with matched pairs of groups, and indicate connections with other algebraic settings, including Hopf algebras.

The pentagon equation appears naturally in several areas of algebra, geometry, and mathematical physics. For example, it arises in the theory of Hopf algebras and fusion operators in monoidal categories, and it also appears as the algebraic shadow of the 2–3 Pachner move, a local transformation of triangulations that plays a central role in the construction of invariants of manifolds. In this talk, I will introduce the pentagon equation from a set-theoretic point of view. Instead of starting with vector spaces or linear structures, one starts with a set and a map satisfying an algebraic identity that is simple to write down but rich in consequences. After discussing some basic examples and elementary properties, I will explain how the study of these solutions naturally leads to algebraic structures, in particular to groups and matched products of groups. The talk is intended as an introduction to the topic and will not assume prior familiarity with the pentagon equation. The main goal is to show how an apparently simple identity can lead to interesting classification problems and unexpected connections with other areas of algebra and geometry.

In this talk I will report on joint work with Janssens. We investigate the connection between bijective set-theoretic solutions of the pentagon equation and Hopf algebras. In the finite setting, we show that such solutions correspond to Hopf algebras with the positive basis property, and as a corollary we generalise the Lu–Yan–Zhu classification to arbitrary characteristic 0 fields. We also address the more general question of when a Hopf algebra admits a basis that yields a set-theoretic solution. If time permits, we will discuss classification results in the (co)commutative case. I will explain the main ideas of the constructions and illustrate the results with examples.

In this talk, I will discuss finite bijective set-theoretic solutions of the Pentagon Equation and present their complete classification. The main result reveals an unexpected connection between these solutions and matched pairs of groups. After reviewing the necessary definitions, I will turn to the basic building blocks of the theory, the irretractable solutions, and describe their relationship with matched pairs of groups. I will then explain how all bijective solutions arise as lifts of irretractable ones, thereby obtaining the full classification. If times permits, I will also briefly comment on related connections with Hopf algebras.

In this talk I will report on joint work with Janssens. We investigate the connection between bijective set-theoretic solutions of the pentagon equation and Hopf algebras. In the finite setting, we show that such solutions correspond to Hopf algebras with the positive basis property, and as a corollary we generalise the Lu–Yan–Zhu classification to arbitrary characteristic 0 fields. We also address the more general question of when a Hopf algebra admits a basis that yields a set-theoretic solution. If time permits, we will discuss classification results in the (co)commutative case. I will explain the main ideas of the constructions and illustrate the results with examples.

Motivated by the appearance of the pentagon equation in the algebraic structures underlying integrable systems, and in the spirit of this mathematical physics seminar, my aim in this talk is also to explore what kind of interaction there can be between these very algebraic constructions and questions coming from integrable systems. I will give a complete description of finite bijective set-theoretic solutions to the pentagon equation in terms of matched products of groups, based on joint work with Okninski and Van Antwerpen. The emphasis will be on the underlying group-theoretic structure: I will explain the characterisation via matched products, describe the role of irretractable solutions as building blocks, and show how they can be used to obtain all bijective solutions. Based on ongoing joint work with Janssens, I will also briefly indicate how this description, together with the link between pentagon solutions and Hopf algebras, leads to constructions of Hopf algebras with a positive basis in the sense of Lu–Yan–Zhu.

I will describe a complete classification of finite bijective set-theoretic solutions to the pentagon equation in terms of matched products of groups, based on joint work with Okninski and Van Antwerpen. The emphasis will be on the underlying group-theoretic structure: I will explain the characterisation via matched products, describe the role of irretractable solutions as building blocks, and show how they can be used to obtain all bijective solutions. Based on ongoing joint work with Janssens, I will also briefly indicate how this description, together with the link between pentagon solutions and Hopf algebras, leads to constructions of Hopf algebras with a positive basis in the sense of Lu–Yan–Zhu.

I will describe a complete classification of finite bijective set-theoretic solutions to the pentagon equation in terms of matched products of groups, based on joint work with Okninski and Van Antwerpen. The emphasis will be on the underlying group-theoretic structure: I will explain the characterisation via matched products, describe the role of irretractable solutions as building blocks, and show how they can be used to obtain all bijective solutions. Based on ongoing joint work with Janssens, I will also briefly indicate how this description, together with the link between pentagon solutions and Hopf algebras, leads to constructions of Hopf algebras with a positive basis in the sense of Lu–Yan–Zhu.

Skew braces are algebraic structures introduced to study set-theoretic solutions to the Yang–Baxter equation. They provide a framework that connects group-theoretic and ring-theoretic methods with the combinatorial properties of these solutions. This talk will offer an introduction to skew braces and their main properties, with an emphasis on their role in constructing and analysing solutions to the Yang–Baxter equation. As an application, we will briefly outline how skew braces can be used to describe simple solutions.

In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, revealing an unexpected connection with matched pairs of groups. We will begin by introducing the necessary definitions and concepts. The focus will then shift to the core components: the irretractable solutions. We will explore how these solutions are related to matched pairs of groups. Finally, I will explain how each irretractable solution can be extended to recover all bijective solutions.

In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions. Next, we will focus on the fundamental components: the irretractable solutions, and we will examine how these solutions relate to matched pairs of groups. Finally, we will show how each irretractable solution can be lifted to classify all bijective solutions.

In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions. Next, we will focus on the fundamental components: the irretractable solutions, and we will examine how these solutions relate to matched pairs of groups. Finally, we will show how each irretractable solution can be lifted to classify all bijective solutions.

In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions. Next, we will focus on the fundamental components: the irretractable solutions, and we will examine how these solutions relate to matched pairs of groups. Finally, we will show how each irretractable solution can be lifted to classify all bijective solutions.

In this talk, we introduce a definition of cabling for bijective non-degenerate set-theoretic solutions to the Yang–Baxter equation, with a focus on indecomposability. We examine fundamental differences between involutive and non-involutive solutions, particularly that involutive solutions naturally embed into solutions associated with their structure group, while this embedding often fails for non-involutive cases. To address this, we connect the decomposability of solutions with that of their injectivization. A core result we prove is that cabling preserves morphisms, ensuring that simplicity and indecomposability are maintained under suitable cabling. Our methods not only extend the theorems of Lebed, Ramírez, and Vendramin to non-involutive solutions but also provide numerical criteria for indecomposability. This talk is based on joint work with A. Van Antwerpen.

In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions, including the notion of irretractable solutions, and explore how these solutions correspond with matched pairs of groups. Finally, I will show how each irretractable solution lifts to provide the full classification of all bijective solutions.

In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions, including the notion of irretractable solutions, and explore how these solutions correspond with matched pairs of groups. Finally, I will show how each irretractable solution lifts to provide the full classification of all bijective solutions.

The Yang–Baxter equation is a fundamental equation in mathematical physics. Following Drienfeld's suggestion, the set-theoretic version of such an equation gained a lot of attention, and in recent years, many authors found a connection between this equation and various areas of pure mathematics such as knot theory, factorisation of groups, group rings, Hopf–Galois theory, nil, and nilpotent rings, and pre-Lie algebras Describing all solutions is a challenging task. A strategy to tackle such a problem is to focus on “building blocks,” such as simple solutions. A solution is simple if any epimorphic image is either isomorphic or is a singleton. Recently, Rump, and Guarnieri and Vendramin introduced skew braces to provide an algebraic framework for studying set-theoretic solutions to the Yang–Baxter equation. In this talk, we will discuss this connection between skew braces and solutions and we will provide a characterisation of simple solutions in terms of algebraic properties of the associated skew braces. The talk is based on a recent joint work with Eric Jespers, Łukasz Kubat ,and Arne Van Antwerpen.

In the last years, various algebraic structures have been introduced to produce and classify combinatorial solutions to the Yang–Baxter equation. Among them, skew braces play a central role. In this talk, we will provide a brief introduction to this algebraic structure and an overview of its properties. In addition, we will delve into the connection between skew braces and solutions. Applying this connection, we will see how skew braces can be exploited to study specific classes of solutions, with a particular focus on indecomposable ones.

The Yang-Baxter equation is a fundamental equation in theoretical physics and pure mathematics, with many applications in different fields of mathematics. The importance of this equation led Drinfeld to propose the following problem: studying combinatorial solutions. A key tool for exploring this family of solutions is the concept of a skew brace, introduced by Rump and Guarnieri and Vendramin. In this talk, we will focus on combinatorial solutions to the YBE and skew braces, providing a variety of examples and discussing their interplay.

The first part of the talk will connect the topological problem of distinguishing mathematical knots with combinatorial solutions to the Yang-Baxter equation. The second one will be devoted to a new algebraic structure (skew braces) that provides the perfect tool to study and classify such solutions.

Involutive set-theoretic solutions to the Yang-Baxter equation have been widely studied in the last two decades, and various connections with algebraic structures have been introduced and explored. Among these structures, in 2005, Rump introduced cycle sets which are in one-to-one correspondence with left non-degenerate involutive solutions. One of the questions yet to be solved in the context of involutive set-theoretic solutions to the Yang-Baxter equation is the following: When is such a solution retractable? In this talk, we will focus on cycle sets with an underlying compatible abelian group structure called quasi-linear cycle sets, introduced by Rump. We will state basic properties of this structure and discuss some conjectures about the retraction problem for the class of solutions associated with a quasi-linear cycle set.

The Yang-Baxter equation is a fundamental equation in theoretical physics and pure mathematics, with many applications in different fields of mathematics. The importance of this equation led Drinfeld to propose the following problem: studying set-theoretical solutions. A key tool for exploring this family of solutions is the concept of a skew brace, introduced by Rump and Guarnieri and Vendramin. In this talk, we will review the basic theory of this algebraic structure and this family of solutions, describe their connection, discuss some problems, and give some applications.

The pentagon equation appears in various contexts: For example, any finite-dimensional Hopf algebra is characterised by an invertible solution of the Pentagon Equation, or an arrow is a fusion operator for a fixed braided monoidal category if it satisfies the Pentagon Equation. This talk, based on joint work with E.Jespers and Ł. Kubat, will introduce the basic properties of set-theoretic solutions of the Pentagon Equation. Furthermore, we will look at bijective solutions, focusing on the involutive case. In the latter case, we provide a complete description of all involutive solutions and discuss when two involutive solutions are isomorphic.

The study of large classes of set-theoretic solutions can be reduced to the study of some associative algebraic objects, such as braces, skew braces, and semi-trusses. This talk will discuss the basic properties of skew braces and how these structures are related to non-degenerate bijective set-theoretic solutions to the Yang-Baxter equation. We will also discuss a possible extension of this theory to left non-degenerate solutions, not necessarily right non-degenerate. These new tools will allow us to prove that for a finite left non-degenerate solution being right non-degenerate is equivalent to being bijective.

The pentagon equation appears in various contexts: for example, any finite-dimensional Hopf algebra is characterised by an invertible solution of the Pentagon Equation, or an arrow is a fusion operator for a fixed braided monoidal category if it satisfies the Pentagon Equation. This talk, based on joint work with E. Jespers and Ł. Kubat, will introduce the basic properties of set-theoretic solutions of the Pentagon Equation, present a complete description of all involutive solutions, and discuss when two involutive solutions are isomorphic.

To study set-theoretic solutions of the Yang-Baxter equation, several authors introduced algebraic structures. Rump, and Cedó, Jespers and Okniński introduced braces, Guarnieri and Vendramin introduced skew braces and Catino, Colazzo and Stefanelli, and Jespers and Van Antwerpen introduced semi-braces. All these objects are subclasses of (semi-)trusses, an algebraic structure introduced by Brzeziński to study the distributive law of (semi-)braces. In this talk, we focus on a subclass of left semi-trusses, the class of brace-like semi-trusses. First, we prove that in the finite case, the additive structure is a completely regular semigroup. Secondly, we apply this result on a specific instance of a left semi-truss called an almost left semi-brace, introduced by Miccoli to study its algebraic structure. In particular, we show that one can associate a left semi-brace to any almost left semi-brace. Furthermore, we show that the set-theoretic solutions of the Yang-Baxter equation originating from almost left semi-braces arise from this correspondence. This talk is based on joint work with Arne Van Antwerpen (arXiv:1908.11744)

The Yang-Baxter equation is one of the fundamental equations in mathematical-physics. The problem of finding and classifying the set-theoretical solutions of the Yang-Baxter equation has originally been posed by Drinfeld. Rump introduced braces, as a generalization of radical rings, in order to construct set-theoretical involutive solutions of the Yang-Baxter equation. Braces over a field have an unexpected connection with regular subgroups of an affine group. Finding all regular subgroups of an affine group is an open problem formalized by Liebeck, Praeger and Saxl in 2010. Last year Guarnieri and Vendramin obtained a generalization of braces, namely the skew braces, and Catino, Colazzo and Stefanelli obtained the semi-braces a further generalization. In this talk first we explore the link between braces over a field an regular subgroups of the affine group and we prove that with this connection we can construct a family of regular subgroups of the affine group such that they have trivial intersection with the translation group. Then we extend this result to the connection between skew braces and regular subgroups of the holomorph of a group. Finally, we introduce a suitable definition of the holomorph of a right group and extend the link between semi-braces and regular subgroups of this holomorph.

In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions. Next, we will focus on the fundamental components: the irretractable solutions, and we will examine how these solutions relate to matched pairs of groups. Finally, we will show how each irretractable solution can be lifted to classify all bijective solutions.

This talk is based on joint work with Arne Van Antwerpen. We will focus on bijective non-degenerate solutions to the Yang-Baxter equation (YBE). We will introduce the cabling for bijective non-degenerate solutions and show that this is a useful tool for dealing with indecomposability.

This talk is based on joint work with Arne Van Antwerpen. We will focus on bijective non-degenerate solutions to the Yang-Baxter equation (YBE). We will introduce the cabling for bijective non-degenerate solutions and show that this is a useful tool for dealing with indecomposability.

In this talk, we will introduce the structure monoid of a set-theoretic so- lution to the Yang-Baxtr equation and explore some combinatorial aspects. We will show that the structure monoid is useful for investigating bijective non-degenerate solutions to the Yang-Baxter equation. This talk is based on joint work with Jespers, Van Antwerpen and Verwimp and an ongoing project with Kubat, Jespers, and Van Antwerpen.

This talk sits at the intersection between two areas: the solutions of the Yang-Baxter equation and self distributive systems. In the last years, the connection between self distributive systems and the solutions of the Yang-Baxter equation was highlighted. In [3], Lebed and Vendramin studied the (co)homology of left non-degenerate solutions. Additionally, they showed how to associate to any left non-degenerate solution a shelf operation that captures many of its properties: such as invertibility and involutivity. In [2] Lebed further deepened the link between a left non-degenerate solution and its associated shelf. In this talk, we investigate the matched product of solutions associated with right and left shelves [1]. First, we focus on the matched product of left shelves, and we prove that we can simplify the requirements to provide the matched product of solutions related to left shelves. Moreover, we show that we obtain the left non-degeneracy if and only if we start with left racks. Later, we treat solutions associated with right shelves. Here again, we show that the requirements to obtain a matched product system of solutions are straightforward. Furthermore, we present the matched product of one solution associated with a left shelf and the other associated with a right shelf; and we obtain natural conditions to check the properties of the matched product system. In all three cases, we prove that the structure shelf associated with the matched product solution does not depend on the action maps that give the matched product. Finally, we show that this is a general result, i.e., that the structure shelf associated with the matched product of left non-degenerate solutions does not depend on the actions. References [1] F. Catino, I. Colazzo, P. Stefanelli, The matched product of self-distributive systems, In preparation. [2] V. Lebed, Applications of self-distributivity to Yang-Baxter operators and their cohomology, J. Knot Theory Ramifications 27 (11) (2018) 1843012, 20. [3] V. Lebed, L. Vendramin, Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation, Adv. Math. 304 (2017) 1219–1261.

Skew braces are strictly link with non-degenerate (bijective) solution of the Yang-Baxter equation. In this talk, I will present a construction, the extension of skew braces, that allow us to describe all skew braces with non trivial annihilator. In particular, this construction is based on classical tools like 2-cocycles of groups and central group extensions. Finally, I describe all skew braces in terms of Hochschild product of skew braces.

Semi-braces and the Yang-Baxter equationIlaria ColazzoThe Yang-Baxter equation is a basic equation of the statistical mechanicsthat arose from Yang’s work in 1967 and Baxter’s one in 1972. Drinfeld in [2] posed the question of classifying the solutions of the Yang-Baxter equation,in particular those called set-theoretical. This is a difficult task and manyauthors dealt with this problem. In particular, several algebraic structureswere studied to answer this problem, such as groups, cycle sets, braces (forinstance, see [3], [4], [5]). Recently, a new generalization of braces,semi-brace, was introduced in [1]. In this talk, we describe how to obtain a solution of the Yang-Baxter equation through semi-braces. Furthermore, we show which properties satisfythis kind of solutions. Finally, we present a construction of solution of theYang-Baxter equation that arises from the matched product of left semi-braces. References [1] F. Catino, I. Colazzo, P. Stefanelli: Semi-braces and the Yang-Baxter equation, J. Algebra (2017), http://dx.doi.org/10.1016/j.jalgebra.2017.03.035 [2] V. G. Drinfeld: On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Math.1510, Springer, Berlin, (1992), 1–8. [3] P. Etingof, T. Schedler, A. Soloviev:Set-theoretical solutions to the quantumYang-Baxter equation, Duke Math. J.100(1999), 169–209. [4] J.-H. Lu, M. Yan, Y.-C. Zhu:On the set-theoretical Yang-Baxter equation, DukeMath. J.,104(1) (2000) 1–18. [5] W. Rump:Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra 307 (2007), 153–170.

Rump, in [4], introduced braces to study non-degenerate involutive solutions of the Yang-Baxter equation. Many aspects of this algebraic structure were studied and developed (see, for instance [2] and its bibliography). Recently, Guarnieri and Vendramin in [3] obtained a generalization of braces, skew braces, in order to construct non-degenerate bijective solutions of the Yang-Baxter equation. In this talk, we focus on semi-braces, a further generalization of braces introduced in [1] that allows us to construct new solutions, not necessarily bijective. In particular, we describe the structural aspects of a semi-brace and provide a clear characterization of this structure. Finally, we introduce suitable concepts of ideal and quotient structure of a semi-brace. References [1] F. Catino, I. Colazzo and P. Stefanelli: Semi-braces and the Yang-Baxter equation, accepted for publication, J. Algebra. [2] F. Cedó, E. Jespers and J. Okniński: Brace and Yang-Baxter equation, Comm. Math. Phys. 327(1) (2014), 101–116. [3] L. Guarnieri and L. Vendramin: Skew braces and the Yang-Baxter equation, accepted for publication in Math. Comp. arXiv:1511.03171 (2015). [4] W. Rump: Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (2007), 153–170.

The Yang-Baxter equation is a basic equation of the statistical mechanics that arose from Yang’s work in 1967 and Baxter’s one in 1972. In 1992, Drinfeld [in Quantum Groups (Leningrad, 1990), Lecture Notes in Math.Springer, Berlin, 1992, 1–8, 1510] posed the question of finding all set-theoretic solutions of the Yang-Baxter equation. Later, in the seminal paper, Etingof, Schedler and Soloviev [Duke Math. J.,1999, 169–209, 100] laid the groundwork for the study of a particular class of these solutions, the non-degenerate involutive ones. From a radical ring we may construct a non-degenerate involutive solution of the Yang-Baxter equation. In 2007, Rump [J. Algebra, 2007, 153–170, 307] found a strict link between a generalization of radical rings, called radical braces, and the non-degenerate involutive solutions. Later in this talk, we introduce more general algebraic structures, the skew braces and the semi-braces that further generalize radical rings and have a link with solutions of the Yang-Baxter equation that are not necessary involutive (see [L. Guarnieri, L. Vendramin, Skew braces and the Yang-Baxter equation, Accepted for publication in Math. Comp., 2015] and [F. Catino, I. Colazzo, P. Stefanelli, Semi-braces and the Yang-Baxter equation, in preparation] )

Finding all regular subgroups of an affine group is an open problem formalized by Liebeck, Praeger and Saxl in 2010 (see [5]). A systematic way to obtain regular subgroups is obtained by Catino and Rizzo in [3] in terms of radical braces over a field F , a generalization of radical algebras. On the other hand, there is an unexpected connection between radical braces and involutive non-degenerate solutions of Yang-Baxter equation (see [6]). In this talk we are going to introduce constructions of radical braces over a field $F$ (see [1] and [2]) that allows us to obtain systematic constructions of regular subgroups of the affine group. In particular, this approach allows to put in a more general context the regular subgroups constructed by Hegedűs in [4] and by Tamburini Bellani in [7]. References [1] F. Catino, I. Colazzo, P. Stefanelli: On regular subgroups of the affine group, Bull. Aust. Math. Soc. 91 (1) (2015) 76–85. [2] F. Catino, I. Colazzo, P. Stefanelli: Regular subgroups of the affine group and asym- metric product of radical braces, J. Algebra, 455 (2016) 164–182. [3] F. Catino, R. Rizzo: Regular subgroups of the affine group and radical circle algebras, Bull. Aust. Math. Soc. 79 (2009), 103–107. [4] P. Hegedűs: Regular subgroups of the affine group, J. Algebra 225 (2) (2000) 740–742. [5] M. Liebeck, C. Praeger, J. Saxl: Regular subgroups of primitive permutation groups, Memoirs of the AMS 203 (2010), no. 952. [6] W. Rump:Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra 307 (2007), 153–170. [7] M.C. Tamburini Bellani: Some remarks on regular subgroups of the affine group Int. J. Group Theory, 1 (2012), 17–23.

In this talk, we are going to introduce a new radical brace construction which extended the semidirect product of two radical braces. In the case of braces over a field, we may obtain some regular subgroups of the affine group and this approach allows us to generalise the regular subgroups constructed by Hegedus in Regular Subgroups of the Affine Groups, J. Algebra 225 (2000), 740–742.