A note on set-theoretic solutions of the Yang–Baxter equation

The independence assumption for a series or parallel system when component lifetimes are exponential : John P. Klein and M. L. Moeschberger. IEEE Trans. Reliab.R-35 (3), 330 (1986)

For a finite involutive non-degenerate solution ( X , r ) (X,r) of the Yang–Baxter equation it is known that the structure monoid M ( X , r ) M(X,r) is a monoid of I-type, and the structure algebra K [ M ( X , r ) ] K[M(X,r)] over a field K K shares many properties with commutative polynomial algebras; in particular, it is a Noetherian PI-domain that has finite Gelfand–Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid M ( X , r ) M(X,r) and the algebra K [ M ( X , r ) ] K[M(X,r)] is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of M ( X , r ) M(X,r) as a regular submonoid in the semidirect product A ( X , r ) ⋊ Sym ⁡ ( X ) A(X,r)\rtimes \operatorname {Sym} (X) , where A ( X , r ) A(X,r) is the structure monoid of the rack solution associated to ( X , r ) (X,r) , we prove that K [ M ( X , r ) ] K[M(X,r)] is a finite module over a central affine subalgebra. In particular, K [ M ( X , r ) ] K[M(X,r)] is a Noetherian PI-algebra of finite Gelfand–Kirillov dimension bounded by | X | |X| . We also characterize, in ring-theoretical terms of K [ M ( X , r ) ] K[M(X,r)] , when ( X , r ) (X,r) is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of M ( X , r ) M(X,r) . These results allow us to control the prime spectrum of the algebra K [ M ( X , r ) ] K[M(X,r)] and to describe the Jacobson radical and prime radical of K [ M ( X , r ) ] K[M(X,r)] . Finally, we give a matrix-type representation of the algebra K [ M ( X , r ) ] / P K[M(X,r)]/P for each prime ideal P P of K [ M ( X , r ) ] K[M(X,r)] . As a consequence, we show that if K [ M ( X , r ) ] K[M(X,r)] is semiprime, then there exist finitely many finitely generated abelian-by-finite groups, G 1 , … , G m G_1,\dotsc ,G_m , each being the group of quotients of a cancellative subsemigroup of M ( X , r ) M(X,r) such that the algebra K [ M ( X , r ) ] K[M(X,r)] embeds into M v 1 ⁡ ( K [ G 1 ] ) × ⋯ × M v m ⁡ ( K [ G m ] ) \operatorname {M}_{v_1}(K[G_1])\times \dotsb \times \operatorname {M}_{v_m}(K[G_m]) , a direct product of matrix algebras.

On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang–Baxter equation

Set-theoretic solutions of the Yang–Baxter equation, braces and symmetric groups

To every involutive non-degenerate set-theoretic solution [Formula: see text] of the Yang–Baxter equation on a finite set [Formula: see text] there is a naturally associated finite solvable permutation group [Formula: see text] acting on [Formula: see text]. We prove that every primitive permutation group of this type is of prime order [Formula: see text]. Moreover, [Formula: see text] is then a so-called permutation solution determined by a cycle of length [Formula: see text]. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.

We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang–Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists, we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions, we also construct admissible Drinfeld twists similar to the set-theoretic ones, subject to certain extra constraints dictated by the q-deformation. These findings greatly generalize recent relevant results on set-theoretic solutions and their q-deformed analogues.

Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation are discussed and many consequences are derived. In particular, for each positive integer n a finite square-free multipermutation solution of the Yang–Baxter equation with multipermutation level n and an abelian involutive Yang–Baxter group is constructed. This answers a problem of Gateva-Ivanova and Cameron. It is proved that finite non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation whose associated involutive Yang–Baxter group is abelian are multipermutation solutions. Earlier the authors proved this with the additional square-free hypothesis on the solutions. It is also proved that finite square-free non-degenerate involutive set-theoretic solutions associated to a left brace are multipermutation solutions.

This paper explores the structure groupsG(X,r)of finite non-degenerate set-theoretic solutions (X,r) to the Yang–Baxter equation. Namely, we construct a finite quotient$\overline {G}_{(X,r)}$ofG(X,r), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: ifXinjects intoG(X,r), then it also injects into$\overline {G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization ofG(X,r). We show that multipermutation solutions are the only involutive solutions with diffuse structure groups; that only free abelian structure groups are bi-orderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: bi-orderable, left-orderable, abelian, free abelian and torsion free.

On the Yang–Baxter equation and left nilpotent left braces

Given a skew left brace [Formula: see text], a method is given to construct all the non-degenerate set-theoretic solutions [Formula: see text] of the Yang–Baxter equation such that the associated permutation group [Formula: see text] is isomorphic, as a skew left brace, to [Formula: see text]. This method depends entirely on the brace structure of [Formula: see text]. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.

We introduce the notion of non-degenerate solution of the braid equation on the incidence coalgebra of a locally finite order. Each one of these solutions induces by restriction a non-degenerate set-theoretic solution over the underlying set. So, it makes sense to ask if a non-degenerate set-theoretic solution on the underlying set of a locally finite order extends to a non-degenerate solution on its incidence coalgebra. In this paper we begin the study of this problem.

In 1992 Drinfeld posed the question of finding the set-theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group-theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate solutions on finite sets and groups of I I -type. A group G \mathcal {G} of I I -type is a group isomorphic to a subgroup of F a n ⋊ S y m n \mathrm {Fa}_n\rtimes \mathrm {Sym}_n so that the projection onto the first component is a bijective map, where F a n \mathrm {Fa}_n is the free abelian group of rank n n and S y m n \mathrm {Sym}_{n} is the symmetric group of degree n n . The projection of G \mathcal {G} onto the second component S y m n \mathrm {Sym}_n we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfeld’s problem for involutive non-degenerate set-theoretic solutions. First classify the IYB groups and second, for a given IYB group G G , classify the groups of I I -type with G G as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups of I I -type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group.

Solutions of the Yang–Baxter equation associated with a left brace

Braces were introduced by Rump as a promising tool in the study of the set-theoretic solutions of the Yang-Baxter equation. It has been recently proved that, given a left brace $B$, one can construct explicitly all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation such that the associated permutation group is isomorphic, as a left brace, to $B$. It is hence of fundamental importance to describe all simple objects in the class of finite left braces. In this paper we focus on the matched product decompositions of an arbitrary finite left brace. This is used to construct new families of finite simple left braces.

In this paper, we produce a method to construct quasi-linear left cycle sets A with Rad (A) ⊆ Fix (A). Moreover, among these cycle sets, we give a complete description of those for which Fix (A) = Soc (A) and the underlying additive group is cyclic. Using such cycle sets, we obtain left non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation which are different from those obtained in [P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100 (1999) 169–209; P. Etingof, A. Soloviev and R. Guralnick, Indecomposable set-theoretical solutions to the quantum Yang–Baxter equation on a set with a prime number of elements, J. Algebra 242 (2001) 709–719].