Let M be a set. A set-theoretical solution of the pentagon equation on M is a map s : M × M −→ M × M such that s 23 s 13 s 12 = s 12 s 23 , where s 12 = s × id M , s 23 = id M ×s and s 13 = (id M ×τ )s 12 (id M ×τ ), and τ is the flip map, i.e., the permutation on M × M given by τ (x, y) = (y, x), for all x, y ∈ M .In this paper we give a complete description of the set-theoretical solutions of the form s(x, y) = (x · y, x * y) when either (M, ·) or (M, * ) is a group; moreover, we raise some questions.
The Yang-Baxter and pentagon equations are two well-known equations of Mathematical Physic. If S is a set, a map s : S × S → S × S is said to be a set theoretical solution of the Yang-Baxter equation if s23 s13 s12 = s12 s13 s23,where s12 = s × idS, s23 = idS ×s, and s13 = (idS ×τ ) s12 (idS ×τ ) and τ is the flip map, i.e., the map on S × S given by τ (x, y) = (y, x). Instead, s is called a set-theoretical solution of the pentagon equation if s23 s13 s12 = s12 s23.The main aim of this work is to display how solutions of the pentagon equation turn out to be a useful tool to obtain new solutions of the Yang-Baxter equation. Specifically, we present a new construction of solutions of the Yang-Baxter equation involving two specific solutions of the pentagon equation. To this end, we provide a method to obtain solutions of the pentagon equation on the matched product of two semigroups, that is a semigroup including the classical Zappa product.where s 12 = s × id S , s 23 = id S ×s, and s 13 = (id S ×τ ) s 12 (id S ×τ ) and τ is the flip map, i.e., the map on S × S given by τ (x, y) = (y, x). Set-theoretical solutions of the pentagon equation are used in the pioneering work of Baaj and Skandalis [1] to obtain multiplicative unitary operators on a Hilbert space. Later, there have been appeared several papers on this topic: Zakrzewski [38], Baaj and Skandalis [2], Kashaev and Sergeev [25], Jiang and Liu [22], Kashaev and Reshetikhin [24], Kashaev [23], and Catino, Mazzotta, and Miccoli [9]. On the other hand, a map s : S × S → S × S is a set-theoretical solution of the quantum Yang-Baxter equation [14] on a set S if the relation s 23 s 13 s 12 = s 12 s 13 s 23is satisfied, with the same notation adopted for the pentagon equation. Just to give an example, if f, g are idempotent maps from S into itself such that f g = gf , then the map s defined as s(x, y) = (f (x), g(y)) is a solution to the both equations. These examples are provided by Militaru [31, Examples 2.4] and in particular they lie in the class of the well-known Lyubashenko solutions [14]. Finding set-theoretical solutions of the Yang-Baxter equation is equivalent to determining set-theoretical solutions of the braid equation, i.e., maps r : S × S → S × S such that the relation r 23 r 12 r 23 = r 12 r 23 r 13 holds. In particular, a map s is a set-theoretical solution of the quantum Yang-Baxter equation if and only if r = τ s is a set-theoretical solution of the braid equation. This map r is usually written as r(x, y) = (λ x (y), ρ y (x)) with λ x , ρ y maps from S into itself. Since the late 1990s a large number of works related to this equation has been produced, including the seminal papers of Gateva-Ivanova and Van den Bergh [17], Etingof, Schedler, and Soloviev [15], and Lu, Yan, and Zhu [29]. In particular, the class of involutive nondegenerate solutions, i.e., r 2 = id and the maps λ x , ρ x are bijective for every x ∈ S, has been the most studied. Some algebraic structures related to the braid equation have been introduced and investigated over the yea...
We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang–Baxter equation. Specifically, a weak (left) brace is a non-empty set S endowed with two binary operations $$+$$ + and $$\circ $$ ∘ such that both $$(S,+)$$ ( S , + ) and $$(S, \circ )$$ ( S , ∘ ) are inverse semigroups and $$\begin{aligned} a \circ \left( b+c\right) = \left( a\circ b\right) - a + \left( a\circ c\right) \qquad \text {and} \qquad a\circ a^- = - a + a \end{aligned}$$ a ∘ b + c = a ∘ b - a + a ∘ c and a ∘ a - = - a + a hold, for all $$a,b,c \in S$$ a , b , c ∈ S , where $$-a$$ - a and $$a^-$$ a - are the inverses of a with respect to $$+$$ + and $$\circ $$ ∘ , respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution r associated to an arbitrary weak brace S has a behavior close to bijectivity, namely r is a completely regular element in the full transformation semigroup on $$S\times S$$ S × S . In addition, we provide some methods to construct weak braces.
This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang–Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones, among which we especially study the class of simple solutions. In particular, we provide a group-theoretic characterization of these solutions, including their permutation groups. Finally, we deal with some open questions.
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