Set-theoretic solutions of the Yang–Baxter equation, braces and symmetric groups
نویسندگان
چکیده
منابع مشابه
Coxeter-like Groups for Groups of Set-theoretic Solutions of the Yang–baxter Equation
We attach with every finite, involutive, nondegenerate set-theoretic solution of the Yang–Baxter equation a finite group that plays for the associated structure group the role that a finite Coxeter group plays for the associated Artin–Tits group.
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It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presente...
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We extend our recent work on set-theoretic solutions of the YangBaxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutio...
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Building on a result by W.Rump, we show how to exploit the right-cyclic law (xy)(xz) = (yx)(yz) in order to investigate the structure groups and monoids attached with (involutive nondegenerate) set-theoretic solutions of the Yang–Baxter equation. We develop a sort of right-cyclic calculus, and use it to obtain short proofs for the existence both of the Garside structure and of the I-structure o...
متن کاملA Combinatorial Approach to the Set-theoretic Solutions of the Yang-baxter Equation
A bijective map r : X −→ X, where X = {x1, · · · , xn} is a finite set, is called a set-theoretic solution of the Yang-Baxter equation (YBE) if the braid relation r12r23r12 = r23r12r23 holds in X. A non-degenerate involutive solution (X, r) satisfying r(xx) = xx, for all x ∈ X, is called squarefree solution. There exist close relations between the square-free set-theoretic solutions of YBE, the...
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2018
ISSN: 0001-8708
DOI: 10.1016/j.aim.2018.09.005