Brackets by any other name

TL;DR

This paper explores the development and applications of various brackets and n-ary operations in homotopy theory, algebra, and physics, highlighting their evolution from classical brackets to modern infinity-algebras.

Contribution

It provides a historical and conceptual overview of brackets and n-ary operations, emphasizing their role in homotopy theory, algebra, and mathematical physics, and discusses their progression to infinity-algebras.

Findings

Development of brackets from classical to infinity-algebras

Connections between brackets and physical theories

Introduction of multi-linear n-ary operations

Abstract

Brackets by another name, Whitehead or Samelson products, have a history parallel to that in Kosmann-Schwarzbach's From Schouten to Mackenzie: notes on brackets. Here I sketch the development of these and some of the other brackets and products and braces within homotopy theory and homological algebra and with applications to mathematical physics. In contrast to the brackets of Schouten, Nijenhuis and of Gerstenhaber, which involve a relation to another graded product, in homotopy theory many of the brackets are free standing binary operations. My path takes me through many twists and turns. The path leads beyond binary to multi-linear n-ary operations, either for a single n or for whole coherent congeries of such assembled into what is known now as an $\infty$-algebra, such as in homotopy Gerstenhaber algebras. It also leads to more subtle invariants. Along the way, attention will be called to interaction with `physics'; indeed, it has been a two-way street.