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Set theoretic Yang-Baxter equation, braces and Drinfeld twists

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arxiv 2102.13591 v3 pith:T4DRXRG5 submitted 2021-02-26 math-ph hep-thmath.MPmath.QAmath.RA

Set theoretic Yang-Baxter equation, braces and Drinfeld twists

classification math-ph hep-thmath.MPmath.QAmath.RA
keywords solutionstwistsequationtheoreticyang-baxteradmissiblebracescertain
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We consider involutive, non-degenerate, finite set theoretic solutions of the Yang-Baxter equation. Such solutions can be always obtained using certain algebraic structures that generalize nil potent rings called braces. Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent preliminary results. We first identify the generic form of the twists associated to set theoretic solutions and we show that these twists are admissible, i.e. they satisfy a certain co-cycle condition. These findings are also valid for Baxterized solutions of the Yang-Baxter equation constructed from the set theoretical ones.

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    Lyubashenko solutions of the Yang-Baxter equation produce Markov processes equivalent to a twisted SSEP, whose stationary sectors are labeled exactly by a species profile and a total charge.