On the origin of higher braces and higher-order derivations

0446764 - MÚ 2016 RIV GE eng J - Journal Article
Markl, Martin
On the origin of higher braces and higher-order derivations.
Journal of Homotopy and Related Structures. Roč. 10, č. 3 (2015), s. 637-667. ISSN 2193-8407. E-ISSN 1512-2891
Institutional support: RVO:67985840
Keywords : Koszul braces * Börjeseon braces * higher-order derivation
Subject RIV: BA - General Mathematics
Impact factor: 0.600, year: 2015
http://link.springer.com/article/10.1007/s40062-014-0079-2

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 (... -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 ... (resp. ...) 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 ... and ... braces. We finish Part 1 by discussing ... 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.
Permanent Link: http://hdl.handle.net/11104/0248739