Let r : X 2 → X 2 be a set-theoretic solution of the Yang-Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive then the algebra K x ∈ X | xy = uv if r(x, y) = (u, v) shares many properties with commutative polynomial algebras in finitely many variables; in particular this algebra is Noetherian, satisfies a polynomial identity and has Gelfand-Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary nondegenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions r B that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so we first describe such semi-braces and we prove some decompositions results extending results of Catino, Colazzo, and Stefanelli.
For a finite involutive non-degenerate solution (X, r) of the Yang-Baxter equation it is known that the structure monoid M (X, r) is a monoid of I-type, and the structure algebra K[M (X, r)] over a field 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) and the algebra 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) as a regular submonoid in the semidirect product A(X, r) Sym(X), where A(X, r) is the structure monoid of the rack solution associated to (X, r), we prove that the structure algebra K[M (X, r)] is a module-finite normal extension of a commutative affine subalgebra. In particular, K[M (X, r)] is a Noetherian PI-algebra of finite Gelfand-Kirillov dimension bounded by |X|. We also characterize, in ring-theoretical terms of K[M (X, r)], when (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).These results allow us to control the prime spectrum of the algebra K[M (X, r)] and to describe the Jacobson radical and prime radical of K[M (X, r)]. Finally, we give a matrix-type representation of the algebra K[M (X, r)]/P for each prime ideal P of K[M (X, r)]. As a consequence, we show that if K[M (X, r)] is semiprime then there exist finitely many finitely generated abelian-by-finite groups, G 1 , . . . , Gm, each being the group of quotients of a cancellative subsemigroup of M (X, r) such that the algebra, a direct product of matrix algebras. Vendramin [33] studied the structure group G(X, r) for arbitrary finite bijective non-degenerate solutions (i.e., not necessarily involutive). In [33,35,47] they associate, via a bijective 1-cocycle, to the structure group G(X, r) the structure group G(X, r ) of the structure rack (X, r ) of (X, r). As a consequence, it follows that again the groups G(X, r) are abelian-by-finite. Recall that a set X with a self-distributive operation is called a rack if the map y → y x is bijective, for any x ∈ X (cf. [29]). In contrast to the involutive case, the set X is not necessarily canonically embedded into G(X, r), the reason being that M (X, r) need not be cancellative in general (i.e., it is not necessarily embedded in a group). Hence, for an arbitrary solution (X, r) the structure monoid M (X, r) contains more information on the original solution. However, it is in general not true that two set-theoretic solutions (X, r) and (Y, s) are isomorphic if and only if the monoids M (X, r) and M (Y, s) are isomorphic. This does hold if one of both solutions (and thus both) is assumed to be an involutive non-degenerate set-theoretic solution.In this paper we give a structural approach of the study of the structure monoid ...
We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of Itô's theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang-Baxter equation. Finally, we classify skew braces that contain no non-trivial proper ideals.2010 Mathematics Subject Classification. Primary:16T25; Secondary: 81R50.
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