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      On the Yang-Baxter equation and left nilpotent left braces

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          Abstract

          We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang-Baxter equation, we call them simply solutions. We show that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is known that the structure group G(X,r) of a finite multipermutation solution (X,r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G(X,r) which are poly-Z groups but not Engel groups. We also show that a finite solution of the Yang-Baxter equation can be embedded in a convenient way into a finite brace and into a finite braided group. For a left brace A, we explore the close relation between the multipermutation level of the solution associated with it and the radical chain \(A^{(n+1)}=A^{(n)}* A\) introduced by Rump.

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          Braces, radical rings, and the quantum Yang–Baxter equation

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            Binomial Semigroups

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              Nilpotent p-algebras and factorized p-groups

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                Author and article information

                Journal
                2016-01-26
                Article
                1601.07131
                298c9809-143a-4174-8351-5e8ad94a8957

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                math.GR

                Algebra
                Algebra

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