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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
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References:

[1] Bachiller, D., Classification of braces of order \(p^3\), J. Pure Appl. Algebra, 219, 8, 3568-3603 (2015) · Zbl 1312.81099
[2] Bachiller, D.; Cedó, F., A family of solutions of the Yang-Baxter equation, J. Algebra, 412, 218-229 (2014) · Zbl 1303.16036
[3] Baxter, R. J., Partition function of the eight-vertex lattice model, Ann. Physics, 70, 1, 193-228 (1972) · Zbl 0236.60070
[4] Catino, F.; Colazzo, I.; Stefanelli, P., On regular subgroups of the affine group, Bull. Aust. Math. Soc., 91, 1, 76-85 (2015) · Zbl 1314.20001
[5] Catino, F.; Colazzo, I.; Stefanelli, P., Regular subgroups of the affine group and asymmetric product of radical braces, J. Algebra, 455, 164-182 (2016) · Zbl 1348.20002
[6] Cedó, F.; Jespers, E.; Okniński, J., Braces and the Yang-Baxter equation, Comm. Math. Phys., 327, 1, 101-116 (2014) · Zbl 1287.81062
[7] Clifford, A. H.; Preston, G. B., The Algebraic Theory of Semigroups, vol. I, Mathematical Surveys, vol. 7 (1961), American Mathematical Society: American Mathematical Society Providence, R.I. · Zbl 0111.03403
[8] Drinfel’d, V. G., On some unsolved problems in quantum group theory, (Quantum Groups. Quantum Groups, Leningrad, 1990. Quantum Groups. Quantum Groups, Leningrad, 1990, Lecture Notes in Math., vol. 1510 (1992), Springer: Springer Berlin), 1-8 · Zbl 0765.17014
[9] Etingof, P.; Schedler, T.; Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 2, 169-209 (1999) · Zbl 0969.81030
[10] Gateva-Ivanova, T.; Van den Bergh, M., Semigroups of \(I\)-type, J. Algebra, 206, 1, 97-112 (1998) · Zbl 0944.20049
[11] Gu, P., Another solution of Yang-Baxter equation on set and “metahomomorphisms on groups”, Chin. Sci. Bull., 42, 22, 1852-1855 (1997) · Zbl 0974.17508
[12] Guarnieri, L.; Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comp. (2017), accepted for publication · Zbl 1371.16037
[13] Hall, M., The Theory of Groups (1959), The Macmillan Co.: The Macmillan Co. New York, N.Y.
[14] Lu, J.-H.; Yan, M.; Zhu, Y.-C., On the set-theoretical Yang-Baxter equation, Duke Math. J., 104, 1, 1-18 (2000) · Zbl 0960.16043
[15] Matsumoto, D. K., Dynamical braces and dynamical Yang-Baxter maps, J. Pure Appl. Algebra, 217, 2, 195-206 (2013) · Zbl 1266.81112
[16] Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307, 1, 153-170 (2007) · Zbl 1115.16022
[17] Soloviev, A., Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation, Math. Res. Lett., 7, 5-6, 577-596 (2000) · Zbl 1046.81054
[18] Yang, C.-N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett., 19, 23, 1312 (1967) · Zbl 0152.46301
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