Tag - Yang-Baxter equations

Raschid Abedin: Classification of D-bialgebras via algebraic geometry

28th October, 2024

In a now classic paper, Belavin and Drinfeld categorized solutions to the classical Yang-Baxter equation (CYBE), an equation crucial to the theory of integrable systems, into three classes: elliptic, trigonometric and rational. It is possible to reproduce this result by geometrizing solutions of the CYBE and then applying algebro-geometric methods. In this talk, we will explain how this approach can be used to categorize Lie bialgebra structures on power series Lie algebras, as well as non-associative generalizations of these structures: D-bialgebra structures on more general power series algebras.

Ana Agore: Solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability (FS) type

7th October, 2024

We investigate a special class of solutions of the set-theoretic Yang-Baxter equation, called Frobenius-Separability (FS) type solutions. In particular, we show that the category of solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability (FS) type is equivalent to the category of pointed Kimura semigroups. As applications, all involutive, idempotent, non-degenerate, surjective, finite order, unitary, or indecomposable solutions of FS type are classified. For instance, if |X| = n, then the number of isomorphism classes of all such solutions on X that are (a) left non-degenerate, (b) bijective, (c) unitary or (d) indecomposable and left-non-degenerate is: (a) the Davis number d(n), (b) Σm|n p(m), where p(m) is the Euler partition number, (c) τ(n) + Σd|n[d/2], where τ is the number of divisors of n, or (d) the Harary number. The automorphism groups of such solutions can also be recovered as automorphism groups Aut(f) of sets X equipped with a single endo-function f : X → X. We describe all groups of the form Aut(f) as iterations of direct and (possibly infinite) wreath products of cyclic or full symmetric groups, characterize the abelian ones as products of cyclic groups, and produce examples of symmetry groups of FS solutions not of the form Aut(f).

Arne Van Antwerpen: Indecomposable and simple solutions of the Yang-Baxter equation

30th September, 2024

Recall that a set-theoretic solution of the Yang-Baxter equation is a tuple (X,r), where X is a non-empty set and r: X × X → X × X a bijective map such that

(r × idX) (idX × r) (r × idX) = (idX × r) (r × idX) (idX × r),

where one denotes r(x,y)=(lx(y), ry(x)). Attention is often restricted to so-called non-degenerate solutions, i.e. lx and ry are bijective. We will call these solutions for short in the remainder of this abstract. To understand more general objects, it is an important technique to study 'minimal' objects and glue them together. For solutions both indecomposable and simple solutions fit the bill for being a minimal object. In this talk, we will report on recent work with I. Colazzo, E. Jespers and L. Kubat on simple solutions. In particular, we will discuss an extension of a result of M. Castelli that allows to identify whether a solution is simple, without having to know or calculate all smaller solutions. This method employs so-called skew braces, which were constructed to provide more examples of solutions, but also govern many properties of general solutions. In the latter part of the talk, we discuss the extension of a method to construct new indecomposable or simple solutions from old ones via cabling, originally introduced by V. Lebed, S. Ramirez, and L. Vendramin to unify the known results on indecomposability of solutions.

Isabel Martin-Lyons: Skew Bracoids

2nd September, 2024

The skew brace was devised by Guanieri and Vendramin in 2017, building on Rump's brace. Since then, the skew brace has been central to the study of solutions to the Yang-Baxter equation, with connections to many other areas of mathematics including Hopf-Galois theory. We introduce the skew bracoid, a generalization of the skew brace which can arise as a partial quotient thereof. We explore the connection between skew bracoids and Hopf-Galois theory, as well as the more recent connection to solutions of the Yang-Baxter equation.

Łukasz Kubat: On Yang-Baxter algebras

5th August, 2024

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-theoretical 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.

Anastasia Doikou: Parametric set-theoretic Yang-Baxter equation: p-racks, solutions and quantum algebras

20th May, 2024

The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. We introduce generalizations of the familiar shelves and racks named parametric (p)-shelves and racks. These objects satisfy a 'parametric self-distributivity' condition and lead to solutions of the Yang-Baxter equation. Novel, non-reversible solutions are obtained from p-shelf/rack solutions by a suitable parametric twist, whereas all reversible set-theoretic solutions are reduced to the identity map via a parametric twist. The universal algebras associated to both p-rack and generic parametric set-theoretic solutions are next presented and the corresponding universal R-matrices are derived. By introducing the concept of a parametric coproduct we prove the existence of a parametric co-associativity. We show that the parametric coproduct is an algebra homomorphism and the universal R-matrices intertwine with the algebra coproducts.

Paola Stefanelli: Płonka sums of set-theoretical solutions of the Yang-Baxter equation

15th April, 2024

The Płonka sum is one of the most significant composition methods in Universal Algebra introduced by Jerzy Płonka in 1967. In particular, Clifford semigroups have turned out to be the first instances of Płonka sums of groups. In this talk, we illustrate a method for constructing set-theoretical solutions of the Yang-Baxter equation that is inspired by the notion of the Płonka sums. Moreover, we will show how to obtain solutions of this type by considering dual weak braces, algebraic structures recently studied and described in a joint work with Francesco Catino and Marzia Mazzotta.

Yanyong Hong: Novikov bialgebras, infinite-dimensional Lie bialgebras, and Lie conformal bialgebras

29th January, 2024

In this talk, I will introduce a bialgebra theory for the Novikov algebra, namely the Novikov bialgebra, which is characterized by the fact that its affinization (by a quadratic right Novikov algebra) gives an infinite-dimensional Lie bialgebra. A Novikov bialgebra is also characterized as a Manin triple of Novikov algebras. The notion of Novikov Yang-Baxter equation is introduced, whose skew-symmetric solutions can be used to produce Novikov bialgebras and hence Lie bialgebras. These solutions also give rise to skewsymmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras from the Novikov algebras. Moreover, a similar connection between Novikov bialgebras and Lie conformal bialgebras will be introduced.

Senne Trappeniers: The interplay between skew braces, the Yang-Baxter equation and Hopf-Galois structures

27th November, 2023

In 2007, Wolfgang Rump introduced algebraic objects called braces, these generalize Jacobson radical rings and are related to involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation (YBE). These objects were subsequently generalized to skew braces by Leandro Guarnieri and Leandro Vendramin in 2017, and a similar relation was shown to hold for non-degenerate set-theoretic solutions of the YBE which are not necessarily involutive. In this talk, we will describe this interplay between skew braces and the YBE. We will also discuss their relation to Hopf-Galois structures and see how this extends the classical Galois theory in an elegant way.

Fabienne Chouraqui: Connections between the Yang-Baxter equation and Thompson’s group F

22nd June, 2023

The quantum Yang-Baxter equation is an equation in mathematical physics and it lies in the foundation of the theory of quantum groups. One of the fundamental problems is to find all the solutions of this equation. Drinfeld suggested the study of a particular class of solutions, derived from the so-called set-theoretic solutions. A set-theoretic solution of the Yang-Baxter equation is a pair (X,r), where X is a set and

r : X ⨯ X → X ⨯ X     r(x,y)=(σx(y),γy(x))

is a bijective map satisfying r12r23r12 = r23r12r23, where r12 = r ⨯ IdX and r23 = IdX ⨯ r. We define non-degenerate involutive partial solutions as a generalization of non-degenerate involutive set-theoretical solutions of the quantum Yang-Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and the Thompson's group F. This raises the question of whether there are further connections between partial solutions and Thompson's groups in general.