Higher braces via formal (non)commutative geometry

TL;DR

This paper translates the main results on braces from noncommutative to formal geometry, showing they are pullbacks of linear vector fields and are essentially the same object in different geometric contexts.

Contribution

It provides a formal geometric interpretation of Koszul and Borjeson braces, demonstrating their equivalence in commutative and noncommutative settings.

Findings

Braces are pullbacks of linear vector fields over formal automorphisms.

Koszul and Borjeson braces are equivalent in formal geometry.

Braces can be viewed as the same object in different geometric frameworks.

Abstract

We translate the main result of author's arXiv:1309.7744 to the language of formal geometry. In this new setting we prove directly that the Koszul resp. Borjeson braces are pullbacks of linear vector fields over the formal automorphism exp(a) -1 in the Koszul, resp. a/(1-a) in the Borjeson case. We then argue that both braces are versions of the same object, once materialized in the world of formal commutative geometry, once in the non-commutative one.