The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order

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...

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“…By (7), (8), and [9, Lemma 11], it follows that (p2), (r2), (p3), and (r3) are trivially satisfied. Moreover, (p4) and (r4) become…”

Section: Ybe Solutions Derived From Pe Solutionsmentioning

confidence: 97%

“…First, we give the definitions of the index and the period of a map f as i (f ) := min j | j ∈ N 0 , ∃ l ∈ N f j = f l , j = l p (f ) := min k | k ∈ N, f i (r)+k = f i (r) , respectively. As observed in [8], these definitions are slightly different from the classical ones (cf. [20, p. 10]).…”

Section: Some Comments and Questionsmentioning

confidence: 99%

Set-theoretical solutions of the Yang–Baxter and pentagon equations on semigroups

2020

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“…Sufficient conditions so that it is can be found in [10,Theorem 7]. Furthermore, in the context of semi-braces, a characterization has been given in [7,Theorem 3]. In particular, if (S, +) is a left cancellative semigroup, the map r is a left nondegenerate solution [5,Theorem 9].…”

Section: Introductionmentioning

confidence: 99%

Set-theoretic solutions of the Yang–Baxter equation associated to weak braces

et al. 2022

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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.

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“…Recently generalized by Jorge A. Guccione, Juan J. Guccione and Christian Valqui [13] to non trivial actions. Various type of products like matched product, semi-direct product, asymmetric product has been defined for the solutions of Yang-Baxter equation [see [3], [5], [9], [10], , [12], [21]]. In [23], M. K. Yadav and author developed the theory of skew brace extensions for skew brace extensions by an abelian group and developed the Well's type exact sequence for skew braces.…”

Section: Introductionmentioning

confidence: 99%