Journal of Homotopy and Related Structures

, Volume 10, Issue 3, pp 637–667 | Cite as

On the origin of higher braces and higher-order derivations

  • Martin Markl


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 (\(A_\infty \)-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 \(L_\infty \)- (resp. \(A_\infty \)-) 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 \(A_\infty \)- and \(L_\infty \)-braces. We finish Part 1 by discussing \(C_\infty \)-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.


Koszul braces Börjeson braces Higher-order derivation 

Mathematics Subject Classification (2000)

13D99 55S20 



I would like to express my thanks to Maria Ronco for Remark 2.9 and Kaj Börjeson for spotting a mistake in Example 1.3. I am also indebted to Olga Kravchenko and the referee for many other useful suggestions.


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Copyright information

© Tbilisi Centre for Mathematical Sciences 2014

Authors and Affiliations

  1. 1.Mathematical Institute of the AcademyPrague 1Czech Republic
  2. 2.MFF UKPrague 8Czech Republic

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