Hopf braces are the quantum analogues of skew braces and, as such, their cocommutative counterparts provide solutions to the quantum Yang-Baxter equation. We investigate various properties of categories related to Hopf braces. In particular, we will discuss the co(completeness) of the category of cocommutative Hopf braces and the existence of certain free objects. Based on joint work with Alexandru Chirvasitu.
The Yang-Baxter equation (YBE) and the reflection equation (RE)—or boundary Yang-Baxter equation—are fundamental symmetries in integrable systems, governing particle interactions on a line or half-line. While the YBE is deeply tied to quantum groups, solutions of the RE arise naturally in the theory of quantum symmetric pairs (QSPs), which provide an algebraic framework for boundary integrability.
In this talk, I will survey recent results from joint work with B. Vlaar, emphasizing the role of QSPs in constructing and classifying integrable boundary conditions.
We interpret the homogeneous solutions of the super classical Yang-Baxter equation, also called super r-matrices, in terms of O-operators by
a unified treatment. Furthermore, by a parity reversion of Lie superalgebra representations, a duality is established between the even and odd O-operators. This leads to a parity duality of the super r-matrices induced by the O-operators in semi-direct product Lie superalgebras. Therefore a pre-Lie superalgebra naturally defines an even O-operator, and hence an odd O-operator by the duality, thereby giving rise to a parity pair of super r-matrices. This is a joint work with Li Guo and Runxuan Zhang.
We present a structural study of skew braces which are a product (and sum) of two abelian subbraces. A brace theoretical analog of Itô's theorem about products of abelian subgroups and some results about factorised ideals are showed.
A. Ballester-Bolinches, R. Esteban-Romero, L. A. Kurdachenko, and V. Pérez-Calabuig have recently studied Dedekind braces, that is, those braces in which every subbrace is an ideal.
We report on ongoing work with Ilaria Del Corso on finite Dedekind skew braces.
In this overview talk, I will discuss how the analysis of soliton interactions in various integrable models of physical interest naturally gives rise to (parametric) Yang-Baxter maps or set-theoretic solutions of the Yang-Baxter equation. There is an underlying mechanism, common to the different models, that produces these solutions: a refactorisation problem for certain elements of a loop group. In turn, this is rooted in a nonlinear version of the Fourier transform which explains why all these maps are inherently parametric. I will also explain how a very natural problem physically, the presence of a spatial boundary, led to the introduction of the set-theoretical reflection equation and of classes of solutions for it, known as (parametric) reflection maps. An important ingredient in the construction involves an involution. Finally, I will try to venture into connections with some algebraic structures that have emerged as central facets of the Yang-Baxter and reflection equations, as well as more geometric aspects related to symplectic and Poisson properties of Yang-Baxter and reflection maps (time allowing).
Braided monoidal categories are governed by braid groups of type A and the associated Yang-Baxter equation. Given a simply connected compact Lie group G, the Kazhdan-Lusztig theorem gives a non-trivial equivalence between two particular braided monoidal categories: one constructed from a non-trivial associator on the category of finite-dimensional G-representations, using solutions to the so-called KZ-equation, and one constructed as the representation category of the quantization of G. In this talk, we will explain how the Kazhdan-Lusztig theorem has an analogue `in type B’, with quantum groups replaced by coideals and braided monoidal categories replaced by braided module categories, which are governed by braid groups of type B and the associated reflection equation. This is based on joint work with S. Neshveyev, L. Tuset and M. Yamashita.
A challenging question concerning a skew brace is to understand how the additive and multiplicative group structures influence each other. The results in the literature cover the case of braces of “small rank”.
Recently, we considered the same question in a subclass of braces that we call module braces, showing that in this class, the previous results can be reproduced for a wider range of ranks. In this talk, we present an overview of this topic and we will examine some examples of braces of non-small rank.
In the second part of the talk, we will present an application of these results to the study of Fuchs’ question on the classification of abelian groups that occur as the group of units of a ring.
Cabling is a method developed by Lebed, Ramìrez and Vendramin to deform involutive, non-degenerate solutions to the Yang-Baxter equations while keeping control over the diagonal maps of the resulting solutions. This powerful tool allows one to prove a plethora of decomposability results for involutive solutions and has recently been generalized by Colazzo and Van Antwerpen to obtain similar results for non-involutive solutions.
In this talk, I will give an outline of classical cabling in the style of Lebed, Ramìrez and Vendramin, and explain some standard applications of the method. Afterwards, I will demonstrate how classical cabling can be generalized to endocabling, where involutive solutions are deformed by means of endomorphisms of the module structure of permutation braces which is given by the λ-action. Finally, I will give a rough sketch how endocabling can be applied to provide insights into the structure of solutions whose diagonal map is a cyclic permutation.
In this talk we will use trifactorised groups to encode the information about skew left braces, their structure, their quotients, and their homomorphisms. This encoding can be used to improve the efficiency of algorithms for skew left braces in computer algebra systems and can provide a natural tool to present a definition of representations of skew left braces that can reduce problems on representations of skew left braces to problems on group representations.
This work is the result of a collaboration with Adolfo Ballester-Bolinches, Pedro Pérez-Altarriba, and Vicent Pérez-Calabuig. It has been supported by the grant CIAICO/2023/007 from the Conselleria d’Educació, Universitats i Ocupació, Generalitat Valenciana.
In this talk, we will introduce a class of infinite solutions to the Yang–Baxter equation that can be approached in an algorithmic way. This is a joint work with A. Ballester-Bolinches, R. Esteban-Romero, V. Pérez-Calabuig, and M. Trombetti.
In this talk, we establish a bialgebra theory for averaging Lie algebras. First we introduce the notion of a quadratic averaging Lie algebra and show that it induces an isomorphism from the adjoint representation to the coadjoint representation. Then we introduce the notion of matched pairs, Manin triples and bialgebras for averaging Lie algebras, and show that Manin triples, bialgebras and certain matched pairs of averaging Lie algebras are equivalent. In particular, we introduce the notion of an averaging operator on a quadratic Rota-Baxter Lie algebra which can induce an averaging Lie bialgebra naturally. Finally, we introduce the notion of the classical Yang-Baxter equation in an averaging Lie algebra whose solutions give rise to averaging Lie bialgebras. We also introduce the notion of relative Rota-Baxter operators on an averaging Lie algebra and averaging pre-Lie algebras, and construct solutions of the classical Yang-Baxter equation in terms of relative Rota-Baxter operators and averaging pre-Lie algebras. This work is joint with Professor Yunhe Sheng and Yanqiu Zhou.
We characterize and construct all indecomposable solutions of multipermutation level 2. Although the result looks like a straightforward generalization of the same result obtained for involutive solutions, the approach is slightly different and the description of the solutions is based on several permutation groups that act on them.
A set-theoretic solution of the Yang-Baxter equation is said to be simple if any epimorphic image is either isomorphic or a solution of cardinality 1. We present a characterization of simple finite non-degenerate bijective solutions in terms of algebraic properties of the associated permutation skew left brace. As an application one obtains an abundance of new simple solutions complementary to all the work done in the context of involutive simple solutions and derived solutions.
The talk is based on joint work with Ilaria Colazzo, Łukasz Kubat and Arne Van Antwerpen.
We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, up to Möbius transformations, we find quadrirational one-component maps of rational functions in two arguments that serve as solutions of the pentagon equation. Furthermore, provided a pentagon map admits at least one partial inverse, we obtain genuine entwining pentagon set theoretical solutions.
Finally, we show how to obtain Yang–Baxter and tetrahedron maps from pentagon maps.
K-theory is the study of certain abelian rings which are generated from vector bundles over some algebraic variety. Their quantum deformations were first considered by Givental and Lee and these deformations are intimately linked to integrable systems. More recently, through works of Maulik and Okounkov, there has been a flurry of activity linking the quantum K-theory of Nakajima varieties to quantum groups and solutions of the quantum Yang-Baxter equation. I will report on ongoing joint work with V. Gorbounov and L. Mihalcea where we study an extended affine Weyl group action on the quantum K-theory rings of Grassmannians which leads to a theory of quantum localisation for these rings, a q-deformed version of the Atiyah-Bott fixed-point theorem. I will emphasise the representation theoretic aspects in my talk focussing on the underlying Yang-Baxter algebra and its action on the quantum K-rings.
To each solution of the Yang–Baxter equation one may associate a quadratic algebra over a field, called the YB-algebra, encoding certain information about the solution. It is known that YB-algebras of finite non-degenerate solutions are (two-sided) Noetherian, PI and of finite Gelfand–Kirillov dimension. If the solution is additionally involutive then the corresponding YB-algebra shares many other properties with polynomial algebras in commuting variables (e.g., it is a Cohen–Macaulay domain of finite global dimension).
The aim of this talk is to explain the intriguing relationship between ring-theoretic and homological properties of YB-algebras and properties of the corresponding solutions of the Yang–Baxter equation. The main focus is on when such algebras are Noetherian, (semi)prime and representable. The talk is based on a joint work with I. Colazzo, E. Jespers and A. Van Antwerpen.
The vector fields on a Lie group, and more generally the smooth maps from a smooth manifold M into the Lie algebra g of a Lie group G acting on M, form a post-Lie algebra. After recalling the definition of a post-group (also known as skew-brace), we give the structure of weak post-group on C∞(M, G) and show how to recover the above mentioned post-Lie algebra from it. We also give the construction of the free post-group generated by a left-regular diagonal magma. This class of magmas includes quandles.
Joint work with Mahdi Al-Kaabi and Kurusch Ebrahimi-Fard.
I will present some recent results on quadrirational solutions of the parametric, set-theoretical Yang-Baxter equation associated with the refactorisation problems of 3×3 Lax matrices. I will present some various low dimensional reductions, as well as several degenerate limits with respect to the parameters of the obtained maps. The degenerate limits lead to binational Yang-Baxter maps and we discuss several of their integrability and dynamical properties. Time permitted, I will discuss several related open problems.
This is based on joint work with P. Adamopoulou (Heriot-Watt) and T. Kouloukas (London Metropolitan) (arXiv:2501.01210v1) to appear soon in Journal of Physics A: Theoretical and Mathematical.
In this talk, we initiate a detailed study of left nilpotent skew braces B of class 2. We show that if B is of nilpotent type, then B is centrally nilpotent. In particular, if B is of abelian type then B is right nilpotent of class 3. Consequences in the structure theory of skew braces and in the study of solutions of the Yang-Baxter equation are also presented.
A set-theoretical solution of the pentagon equation may be described as a pentagon algebra (S, +, ∗) with two binary operations such that (S,+) is a semigroup and the operations + and ∗ are related by some additional equations. It is well known that solutions of the pentagon equation are useful tool to obtain new solutions of the Yang-Baxter equation.
In the talk we will focus our attention on pentagon algebras in which the second operation is also associative. We will present some characterization of pentagon algebras in the case the semigroup (S, ∗) belongs to specific variety of right-normal semigroups.
This is a joint work with Marzia Mazzotta.
In 2007, Wolfgang Rump introduced the algebraic structure of a brace to study set-theoretic solutions of the Yang–Baxter equation, or equivalently, the braid equation. In this talk, I will discuss further generalisations of the relation studied by Rump. In particular, I will present a simple example demonstrating how different solutions can be obtained from a single brace by selecting different elements. This approach shows how multiple distinct solutions can arise from the same underlying structure. To illustrate the differences between the resulting solutions, I will also discuss the connection between the braid relation and the algebraic structure of shelves.
In this talk, first we recall post-groups, which are equivalent to skew-left braces, and give rise to set-theoretical solutions of the Yang-Baxter equation. Then We introduce the notion of post-groupoids, whose differentiations are post-Lie algebroids. The notion of skew-left bracoids are introduced as the "oid''-generalization of skew-left braces. We show that post-groupoids are equivalent to skew-left bracoids, and give quiver-theoretical solutions of the Yang-Baxter equation on the underlying quiver of the subadjacent groupoids. The talk is based on the joint work with Rong Tang and Chenchang Zhu.
Verbal and marginal subgroups of commutators are, respectively, related to the lower and upper central series.
In this talk, we consider their analogs in skew braces and introduce verbal and marginal sub-skew braces. We are particularly interested in the verbal and marginal sub-skew braces for star products and commutators. In relation, we also discuss some potential analogs of the notion of n-isoclinism in the setting of skew braces.
In this talk, I will revisit the finite-dimensional representations of quantum affine sl(2) that occur at q a root of unity which are not related to the standard integer-dimension representations of sl(2). These representations depend upon two points in two copies of a higher-genus algebraic curve. The R-matrix which is the intertwiner of two such representations factors into four term - each of which encodes a Boltzmann weight of the Chiral Potts model as has long been known.
In this work I will investigate this phenomenon further and show that, viewed as Borel subalgebra representations, the finite-dimensional representations themselves factorize into two very simple representations, each of which depends on a single point in the algebraic curve. This factorization will be used to both explain the R-matrix factorization and show that the transfer matrix factorises into two operators that play the role of Q-operators in the roots of unity case. The functional relations satisfied by these Q-operators will be obtained from the study of short-exact-sequences of representations. The situation parallels the generic q case, with the key difference being that only finite-dimensional representations appear in the roots-of-unity case. Based on arXiv:2412.14811.