Set-theoretic solutions of the Yang–Baxter equation, braces and symmetric groups
By Tatiana Gateva-Ivanova. Advances in Mathematics, 2015
135 citations recorded by the source corpus.
Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. In this work we involve simultaneously the matched pairs of groups theory and the theory of braces to study set-theoretic solutions of YBE. We show the intimate relation between the notions of a symmetric group (a braided involutive group) and a left brace, and find new results on symmetric groups of finite multipermutation level and the corresponding braces. We prove that a square-free solution $(X,r)$ has finite multipermutation level $\mpl X = m$ \emph{iff} the associated symmetric group $G(X,r)$ does so. We find new criteria sufficient to claim that a symmetric set $(X, r)$ is a multipermutation solution. In particular, we show that if $(X,r)$ is a finite solution with permutation group $\mathcal{G} = \mathcal{G}(X,r)$ then each of the conditions (i) $\mathcal{G}$ is a two-sided brace; (ii) $(X, r)$ is square-free and $\mathcal{G}$ has mutually inverse left and right actions upon itself (condition lri) is sufficient to claim that $(X, r)$ is a multipermutation solution. Moreover, if $(X, r)$ is a finite square-free solution, and $\mathcal{G}$ is a two-sided brace, then $\mpl X$ equals the nilpotency degree of the radical ring associated with $\mathcal{G}$. We find an identity satisfied by each two-sided brace and prove that a solution $(X,r)$ whose associated brace $G=G(X,r)$ is two-sided must be a trivial solution.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/05adf2936fc49b4e9e248a73321e2e10868a1c95
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Citations in the corpus
- On skew braces (with an appendix by N. Byott and L. Vendramin) (2017, background): “In [44], Rump introduced braces, a new algebraic structure that turns out to be equivalent to bijective 1-cocycles; see [13, 27, 47].”
- Trusses: Between braces and rings (2017, methodology): “with the Yang-Baxter equation and, more fundamentally, group theory, braces attracted a lot of attention and the literature devoted to them and their generalisations is growing rapidly; see e.g. [15]–[31] to mention but a few. The questions the author of these notes would like to address are these: ”