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Boundary Solutions of the Classical Yang-Baxter Equation

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 Added by Tony Giaquinto
 Publication date 1996
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and research's language is English




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We define a new class of unitary solutions to the classical Yang-Baxter equation (CYBE). These ``boundary solutions are those which lie in the closure of the space of unitary solutions to the modified classical Yang-Baxter equation (MCYBE). Using the Belavin-Drinfeld classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer-Gervais solution to the MCYBE, we explicitly construct for all n > 2 a boundary solution based on the maximal parabolic subalgebra of sl(n) obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to $n$. We also give examples of non-boundary solutions for the classical simple Lie algebras.



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Let $r:X^{2}rightarrow X^{2}$ be a set-theoretic solution of the Yang-Baxter equation on a finite set $X$. It was proven by Gateva-Ivanova and Van den Bergh that if $r$ is non-degenerate and involutive then the algebra $Klangle x in X mid xy =uv mbox{ if } r(x,y)=(u,v)rangle$ shares many properties with commutative polynomial algebras in finitely many variables; in particular this algebra is Noetherian, satisfies a polynomial identity and has Gelfand-Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions $r_B$ that are associated to a left semi-brace $B$; such solutions can be degenerate or can even be idempotent. In order to do so we first describe such semi-braces and we prove some decompositions results extending results of Catino, Colazzo, and Stefanelli.
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209 - Seung-moon Hong 2012
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175 - J. Scott Carter 2005
We construct solutions to the set-theoretic Yang-Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.
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