Set-theoretic Solutions Research Articles

On quadrirational pentagon maps

Abstract We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps R : ( x , y ) ↦ ( u ( x , y ) , v ( x , y ) ) , where u , v are two rational functions on two arguments, that serve as solutions of the pentagon equation. Furthermore, provided a pentagon map that admits a partial inverse, we obtain genuine entwining pentagon set theoretical solutions. Finally, we show how to obtain Yang–Baxter maps from entwining pentagon maps.

Indecomposable involutive set-theoretical solutions to the Yang–Baxter equation of size p 2

This article focuses on indecomposable involutive non-degenerate set-theoretical solutions to the Yang–Baxter equation. More specifically, we give a full classification of those solutions which are of size p 2 , for p a prime. We do this through a thorough analysis of their associated permutation braces and using the language of cycle sets.

When the Tracy-Singh product of matrices represents a certain operation on linear operators

Given two linear transformations, with representing matrices A and B with respect to some bases, it is not clear, in general, whether the Tracy-Singh product of the matrices A and B corresponds to a particular operation on the linear transformations. Nevertheless, it is not hard to show that in the particular case that each matrix is a square matrix of order of the form n 2, n > 1, and is partitioned into n 2 square blocks of order n, then their Tracy-Singh product, A ⊠ B , is similar to A ⊗ B , and the change of basis matrix is a permutation matrix. In this note, we prove that in the special case of linear operators induced from set-theoretic solutions of the Yang-Baxter equation, the Tracy-Singh product of their representing matrices is the representing matrix of the linear operator obtained from the direct product of the set-theoretic solutions.

Relative Rota–Baxter groups and skew left braces

Abstract Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and have been introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota–Baxter groups with skew left braces, which are well known to give bijective non-degenerate set-theoretical solutions of the Yang–Baxter equation. We prove that every relative Rota–Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota–Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota–Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota–Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces.

Electric network and Hirota type 4-simplex maps

Bazhanov-Stroganov (4-simplex) maps are set-theoretical solutions to the 4-simplex equation, namely the fourth member of the family of n-simplex equations, which are fundamental equations of mathematical physics. In this paper, we develop a method for constructing Bazhanov-Stroganov maps as extensions of tetrahedron maps which are set-theoretical solutions to the Zamolodchikov tetrahedron (3-simplex) equation. We employ this method to construct birarional Bazhanov-Stroganov maps which boil down to the famous electric network and Hirota tetrahedron maps at a certain limit.

Schur multiplier and Schur covers of relative Rota–Baxter groups

Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang–Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota–Baxter groups. We prove an analogue of the Hochschild–Serre exact sequence for central extensions of relative Rota–Baxter groups. We introduce the Schur multiplier MRRB(A) of a relative Rota–Baxter group A=(A,B,β,T), and prove that the exponent of MRRB(A) divides |A||B| when both A and B are finite. We define weak isoclinism of relative Rota–Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota–Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces.

Quantization of locally compact groups associated with essentially bijective 1-cocycles

Given an extension [Formula: see text] of locally compact groups, with [Formula: see text] abelian, and a compatible essentially bijective [Formula: see text]-cocycle [Formula: see text], we define a dual unitary [Formula: see text]-cocycle on [Formula: see text] and show that the associated deformation of [Formula: see text] is a cocycle bicrossed product defined by a matched pair of subgroups of [Formula: see text]. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of [Formula: see text] on [Formula: see text] and a unitary quantization map [Formula: see text] of Kohn–Nirenberg type.

Set-theoretical solutions of the pentagon equation on Clifford semigroups

Given a set-theoretical solution of the pentagon equation s:S×S→S×S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s:S\ imes S\\rightarrow S\ imes S$$\\end{document} on a set S and writing s(a,b)=(a·b,θa(b))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s(a, b)=(a\\cdot b,\\, \ heta _a(b))$$\\end{document}, with ·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cdot $$\\end{document} a binary operation on S and θa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _a$$\\end{document} a map from S into itself, for every a∈S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in S$$\\end{document}, one naturally obtains that S,·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( S,\\,\\cdot \\right) $$\\end{document} is a semigroup. In this paper, we focus on solutions defined in Clifford semigroups S,·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( S,\\,\\cdot \\right) $$\\end{document} satisfying special properties on the set of all idempotents E(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{E}\\,}}(S)$$\\end{document}. Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which θa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _a$$\\end{document} remains invariant in E(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{E}\\,}}(S)$$\\end{document}, for every a∈S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in S$$\\end{document}. Moreover, we construct a family of idempotent-fixed solutions, i.e., those solutions for which θa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta _a$$\\end{document} fixes every element in E(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{E}\\,}}(S)$$\\end{document} for every a∈S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in S$$\\end{document}, from solutions given on each maximal subgroup of S.

Solutions of the Yang–Baxter Equation and Strong Semilattices of Skew Braces

We prove that any set-theoretic solution of the Yang–Baxter equation associated with a dual weak brace is a strong semilattice of non-degenerate bijective solutions. This fact makes use of the description of any dual weak brace S we provide in terms of strong semilattice Y of skew braces Bα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_\\alpha $$\\end{document}, with α∈Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in Y$$\\end{document}. Additionally, we describe the ideals of S and study its nilpotency by correlating it to that of each skew brace Bα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_\\alpha $$\\end{document}.

Dehornoy’s class and Sylows for set-theoretical solutions of the Yang–Baxter equation

We explain how the germ of the structure group of a cycle set decomposes as a product of its Sylow-subgroups, and how this process can be reversed to construct cycle sets from ones with coprime classes. We study Dehornoy’s class associated to a cycle set, and conjecture a bound that we prove in a specific case. We combine the use of braces and a monomial representation, in particular to answer a question by Dehornoy on retrieving the Garside structure without a theorem of Rump, while also retrieving said theorem.

Set-theoretic type solutions of the braid equation

In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. More precisely, the linear solution associated to a set-theoretic solution on a set X can be regarded as coming from the coalgebra kX, where k is a field and the elements of X are grouplike. We introduce and study a broader class of linear solutions associated in a similar way to more general coalgebras. We show that the relationships between set-theoretical solutions, q-cycle sets, q-braces, skew-braces, matched pairs of groups and invertible 1-cocycles remain valid in our setting.

Corrigendum and addendum to "Structure monoids of set-theoretic solutions of the Yang-Baxter equation”

Corrigendum and addendum to "Structure monoids of set-theoretic solutions of the Yang-Baxter equation”

Mini-Workshop: Skew Braces and the Yang–Baxter Equation

The workshop was focused on the interplay between set-theoretic solutions to the Yang–Baxter equation and several algebraic structures used to construct and understand new solutions. In this vein, the YBE and properties of these algebraic structures are used as a source of inspiration to study other mathematical problems not directly related to the YBE.

Primes in coverings of indecomposable involutive set-theoretic solutions to the Yang-Baxter equation

Primes in coverings of indecomposable involutive set-theoretic solutions to the Yang-Baxter equation

A characterization of finite simple set-theoretic solutions of the Yang-Baxter equation

In this paper we present a characterization of finite simple involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation by means of left braces and we provide some significant examples.

A note on set theoretical solutions of the Yang-Baxter equation with trivial retraction

We show that every finite non-degenerate set theoretical solution to the YBE whose retraction is a flip linearizes to a twist of the flip solution by roots of unity. This generalizes a result of Gateva-Ivanova and Majid. To prove the result we use a new invariant associated to a solution, its Lie algebra. We show also that a solution retracts to a flip solution if and only if its Lie algebra is abelian.

Skew left braces and 2-reductive solutions of the Yang-Baxter equation

We study 2-reductive non-involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. We give a combinatorial construction of any such solution of any (even infinite) size. We also prove that solutions associated to a skew left brace are 2-reductive if and only if the skew left brace is nilpotent of class 2. Moreover, all such skew left braces are actually bi-skew left braces. We focus on these structures and we give several equivalent properties characterizing solutions associated to bi-skew left braces.

Idempotent set-theoretical solutions of the pentagon equation

AbstractA set-theoretical solution of the pentagon equation on a non-empty set X is a function $$s:X\times X\rightarrow X\times X$$ s : X × X → X × X satisfying the relation $$s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}$$ s 23 s 13 s 12 = s 12 s 23 , with $$s_{12}=s\times \,{{\,\textrm{id}\,}}_X$$ s 12 = s × id X , $$s_{23}={{\,\textrm{id}\,}}_X \times \, s$$ s 23 = id X × s and $$s_{13}=({{\,\textrm{id}\,}}_X\times \, \tau )s_{12}({{\,\textrm{id}\,}}_X\times \,\tau )$$ s 13 = ( id X × τ ) s 12 ( id X × τ ) , where $$\tau :X\times X\rightarrow X\times X$$ τ : X × X → X × X is the flip map given by $$\tau (x,y)=(y,x)$$ τ ( x , y ) = ( y , x ) , for all $$x,y\in X$$ x , y ∈ X . Writing a solution as $$s(x,y)=(xy,\theta _x(y))$$ s ( x , y ) = ( x y , θ x ( y ) ) , where $$\theta _x: X \rightarrow X$$ θ x : X → X is a map, for every $$x\in X$$ x ∈ X , one has that X is a semigroup. In this paper, we study idempotent solutions, i.e., $$s^2=s$$ s 2 = s , by showing that the idempotents of X have a crucial role in such an investigation. In particular, we describe all such solutions on monoids having central idempotents. Moreover, we focus on idempotent solutions defined on monoids for which the map $$\theta _1$$ θ 1 is a monoid homomorphism.

Finite Idempotent Set-Theoretic Solutions of the Yang–Baxter Equation

Abstract It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang–Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies of a group) and some maps $q$ and $\varphi _{x}$ on $X$, for $x\in X$. This structure turns out to be a group precisely when the associated Yang–Baxter monoid $M(X,r)$ is cancellative and all the maps $\varphi _{x}$ are equal to an automorphism of this group. Equivalently, the Yang–Baxter algebra $K[M(X,r)]$ is right Noetherian, or in characteristic zero it has to be semiprime. The Yang–Baxter algebra is always a left Noetherian representable algebra of Gelfand–Kirillov dimension one. To prove these results, it is shown that the Yang–Baxter semigroup $S(X,r)$ has a decomposition in finitely many cancellative semigroups $S_{u}$ indexed by the diagonal, each $S_{u}$ has a group of quotients $G_{u}$ that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that $X$ equals the diagonal is fully described by a single permutation on $X$.

Skew-braces and 𝑞-braces

Abstract Skew-braces are ring-like objects arising in connection with Hopf–Galois theory and set-theoretic solutions 𝑆 to the Yang–Baxter equation. Interactions between skew-braces are often related to 𝑞-braces. For example, every 𝑞-brace 𝐴 is given by a pair of skew-braces which induces a ℤ-indexed sequence of skew-braces. The sequence collapses if 𝐴 itself is a skew-brace. The free group over the underlying set of a solution 𝑆 is a 𝑞-brace. Bi-crossed products of skew-braces are shown to be 𝑞-braces, and criteria are developed when they are skew-braces. Two classes of skew-braces are put into a bijective correspondence with special 𝑞-braces, which themselves need not be skew-braces. Characterizations of 𝑞-braces, including one in terms of a graph and one as special modules over skew-braces, are given.