On the origin of higher braces and higher-order derivations

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• 作者：Martin Markl
• 关键词：Koszul braces ; B?rjeson braces ; Higher ; order derivation ; 13D99 ; 55S20
• 刊名：Journal of Homotopy and Related Structures
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：10
• 期：3
• 页码：637-667
• 全文大小：816 KB
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• 作者单位：Martin Markl (1) (2)
  
  1. Mathematical Institute of the Academy, ?itná 25, 115?67?, Prague 1, Czech Republic
  2. MFF UK, Sokolovská 83, 186?75?, Prague 8, Czech Republic
  
• 刊物主题：Algebraic Topology; Algebra; Functional Analysis; Number Theory;
• 出版者：Springer Berlin Heidelberg
• ISSN：1512-2891

文摘

The classical Koszul braces, sometimes also called the Koszul hierarchy, were introduced in 1985 by Koszul (Astérisque, (Numero Hors Serie):257-71, 1985). Their non-commutative counterparts came as a surprise much later, in 2013, in a preprint by B?rjeson (\(A_\infty \)-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. Keywords Koszul braces B?rjeson braces Higher-order derivation