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Two theorems on balanced braces

Part of: Rings and algebras with additional structure Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  05 May 2021

Wolfgang Rump
Wolfgang Rump*
Affiliation:
Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, D-70550Stuttgart, Germany (rump@mathematik.uni-stuttgart.de)
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Abstract

Two theorems of Gateva-Ivanova [Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups, Adv. Math. 338 (2018), 649–701] on square-free set-theoretic solutions to the Yang–Baxter equation are extended to a wide class of solutions. The square-free hypothesis is almost completely removed. Gateva-Ivanova and Majid's ‘cyclic’ condition ${\boldsymbol {\rm lri}}$ is shown to be equivalent to balancedness, introduced in Rump [A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math. 193 (2005), 40–55]. Basic results on balanced solutions are established. For example, it is proved that every finite, not necessarily square-free, balanced brace determines a multipermutation solution.

Keywords

Yang–Baxter equationcycle setbracemultipermutation solution

MSC classification

Primary: 16T25: Yang-Baxter equations
Secondary: 16W22: Actions of groups and semigroups; invariant theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 2 , May 2021 , pp. 262 - 278
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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Footnotes

Dedicated to B. V. M.

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