The retraction relation for biracks

Abstract

In {\it Set-theoretical solutions to the quantum Yang-Baxter equation} (Duke Math. J. {\bf 100} (1999), 169--209), Etingof, Schedler and Soloviev introduced, for each non-degenerate involutive set-theoretical solution $(X,\sigma ,\tau )$ of the Yang-Baxter equation, the equivalence relation $\sim$ defined on the set $X$ and they considered a new non-degenerate involutive induced \emph{retraction} solution defined on the quotient set $X^{\sim }$. It is well known that translating set-theoretical non-degenerate solutions of the Yang-Baxter equation into the universal algebra language we obtain an algebra called a \emph{birack}. In the paper we introduce the \emph{generalized retraction} relation $\approx$ on a birack, which is equal to $\sim$ in an involutive case. We present a complete algebraic proof that the relation $\approx$ is a congruence of the birack. Thus we show that the retraction of a set-theoretical non-degenerate solution is well defined not only in the involutive case but also in the case of all non-involutive solutions.

arXiv comment15 pages

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Cited by in corpus (11)

• Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
• The construction of multipermutation solutions of the Yang-Baxter equation of level 2
• From Braces to Hecke algebras & Quantum Groups
• Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces
• Distributive biracks and solutions of the Yang-Baxter equation
• Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices
• A Universal algebraic approach to rack coverings
• Indecomposable involutive solutions of the Yang-Baxter equation of multipermutational level 2 with abelian permutation group
• Simplicity and finite primitive level of indecomposable set-theoretic solutions of the Yang-Baxter equation
• Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and dynamical extensions of q-cycle sets
• Self-distributive structures, braces & the Yang-Baxter equation