Brackets by any other name

Abstract

<p style='text-indent:20px;'>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 <i>sketch</i> 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.</p> <p style='text-indent:20px;'>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; unless particularized, <i>bracket</i> will be the generic term including product and brace. The path leads beyond binary to multi-linear <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-ary operations, either for a single <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula> or for whole coherent congeries of such assembled into what is known now as an <inline-formula><tex-math id="M3">\begin{document}$ \infty $\end{document}</tex-math></inline-formula>-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.</p>