Skew left braces of nilpotent type. (English) Zbl 1432.16031

The authors study series of left ideals of skew left braces and a class of skew left braces in relation with non-degenerate set-theoretical solutions \(R: X\times X \to X\times X\) of the Yang-Baxter equation \[ (R\times id_X)\circ (id_X \times R) \circ (R\times id_X) = (id_X \times R) \circ (R\times id_X)\circ (id_X \times R), \] where \(X\) is a non-empty set and \(R\) is a map. We recall that the solution \(R: X\times X \to X\times X\) is called non-degenerate if \(R(x,y) = R(\sigma_x(y), \sigma_y(x))\), for all \(x,y\in X\), where \(\sigma_x, \sigma_y:X \to X\) are permutations of the set \(X\).
We also recall that a skew left brace is a triple \((A, +, \circ)\), where \((A, +)\) and \((A, \circ)\) are groups (not necessarily abelian) such that the compatibility equation holds \(a\circ(b+c) = a\circ b - a + a\circ c\), for all \(a,b,c\in A\). The skew left brace \((A, +, \circ)\) is defined to be abelian, if its additive group \((A, +)\) is commutative. This means that a skew left brace \((A, +, \circ)\) of abelian type is just the brace in the sense of W. Rump [J. Algebra 307, No. 1, 153–170 (2007; Zbl 1115.16022)].
Braces provide a useful algebraic framework to work with set-theoreitical solutions of the Yang-Baxter equation. To see it, we recall from [the last two authors, J. Comb. Algebra 2, No. 1, 47–86 (2018; Zbl 1416.16037)] that, given a skew left brace \((A, +, \circ)\), the map \[ R_A: A\times A \to A\times A, \quad R_A(a,b)= (-a + a\circ b, \, (-a + a\circ b)' \circ a \circ b) \] is a non-degenerate set-theoretical solution of the Yang-Baxter equation, where \((-a + a\circ b)' \) means the inverse of \( -a + a\circ b \) with respect to the circle operation \(\circ\).
The authors define and study left and right nilpotent left braces by means of series of their left ideals, that are analogs of upper central series of groups. Among others, they study the connection between left and right nilpotency and the structure of skew left braces. They also study the connection between right nilpotent skew left braces and multipermutation solutions of the Yang-Baxter equation. In particular, it is proved in the paper that a finite skew left brace \((A, +, \circ)\) with nilpotent additive group \((A, +)\) is left nilpotent if and only if the multiplicative group \((A, \circ)\) is nilpotent.
The results are applied to the study of infinite skew left braces \((A, +, \circ)\) with infinite cyclic multiplicative group \((A, \circ)\). One should add that indecomposable solutions of the Yang-Baxter equation are explored in the paper using the structure of skew left braces.


16T25 Yang-Baxter equations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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