Abstract
The classical Koszul braces, sometimes also called the Koszul hierarchy, were introduced in 1985 by Koszul (Astérisque, (Numero Hors Serie):257-271, 1985). Their non-commutative counterparts came as a surprise much later, in 2013, in a preprint by Börjeson (A-infinity algebras derived from associative algebras with a non-derivation differential, Preprint arXiv:1304.6231, 2013).
In Part I we show that both braces are the twistings of the trivial L-infinity- (resp. A-infinity) algebra by a specific automorphism of the underlying coalgebra.
This gives an astonishingly simple proof of their properties. Using the twisting, we construct other surprising examples of A-infinity- and L-infinity-braces.
We finish Part 1 by discussing C-infinity-braces related to Lie algebras. In Part 2 we prove that in fact all natural braces are the twistings by unique automorphisms.
We also show that there is precisely one hierarchy of braces that leads to a sensible notion of higher-order derivations. Thus, the notion of higher-order derivations is independent of human choices.
The results of the second part follow from the acyclicity of a certain space of natural operations.