Integral braces and flat affine manifolds associated with finite L-algebras

Abstract

The structure group $G_X$ of a finite non-degenerate involutive solution $(X;S)$ to the set-theoretic Yang-Baxter equation is a cofinite integral brace, or equivalently, a crystallographic group with an affine structure. Furthermore, $G_X$ is the structure group of a finite L-algebra associated with the solution $(X;S)$. As is well known, $G_X$ is torsion-free, hence the fundamental group of a complete flat affine manifold. It is proved that conversely, a wide class of finite L-algebras have an associated integral brace with a torsion-free affine crystallographic adjoint group. The braces arising from finite L-algebras are Jacobson radicals of rings with a natural coalgebra structure. A slight extension of our construction yields the braces (alias pregroups) recently found in connection with the Hopf algebra of rooted trees in the sense of Connes and Kreimer.