## Semi-braces and the Yang-Baxter equation.(English)Zbl 1385.16035

A brace is a set $$B$$ together with two binary operations $$+$$ and $$\circ$$ such that $$(B,+)$$ is an abelian group, $$(B,\circ)$$ is a group, and $a\circ (b+c)= a\circ b - a +a \circ c,$ for all $$a,b,c\in B$$. The importance of this algebraic structure is that a brace produces a set-theoretic solution of the Yang-Baxter equation, and, by linearization, a solution of the Yang-Baxter equation on the vector space $$kV$$. The construction is due to W. Rump [J. Algebra 307, No. 1, 153–170 (2007; Zbl 1115.16022)]. A generalization was proposed by L. Guarnieri and L. Vendramin [Math. Comput. 86, No. 307, 2519–2534 (2017; Zbl 1371.16037)]. The definition is the same, but it is no longer assumed that $$B$$ is abelian. In the present paper, the authors further relax the conditions on $$(B,+)$$. $$(B,+)$$ is assumed to be a left cancellative semigroup, $$(B,\circ)$$ is still a group, and the additional condition now takes the form $a\circ (b+c)= a\circ b + (a\circ (a^{-}+c)),$ where $$a^{-}$$ is the inverse of $$a$$ with respect to $$\circ$$. Then $$B$$ is called a left semi-brace, and one of the main results tells that the map $$r:\;B\times B\to B\times B$$, given by the formula $r(a,b)=(a\circ (a^{-}+b), (a^{-}+b)^{-}\circ b,$ is a solution the set-theoretical Yang-Baxter equation that is left non-degenerate, but in general not right non-degenerate. Several properties of left semi-braces are presented. Two constructions allowing the production of new braces from old ones (and new solutions of the Yang-Baxter equation) are presented: the asymmetric product of left semi-braces is a new left semi-brace. Ideals of braces are introduced; a particular example is the socle of a left semi-brace, which is itself a left semi-brace.

### MSC:

 16T25 Yang-Baxter equations 16Y99 Generalizations 16N20 Jacobson radical, quasimultiplication 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory

### Citations:

Zbl 1115.16022; Zbl 1371.16037
Full Text:

### References:

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