Commutators of A∞ Structures

This note will be for the most part of an expository nature; many of the results here have appeared elsewhere. We would like however, to highlight the connection between Jim Stasheff’s early work on A∞ algebras in the 1960’s and the notion of sh Lie algebras (L∞ algebras) which is currently popular in mathematical physics; see, for example, [1] and [7]. This connection involves some work of Victor Gugenheim and Larry Lambe who were two of Jim’s colleagues over the years. We will first review the concepts of cofree coalgebras and cofree cocommutative coalgebras; the next idea that we will require is that of a coderivation of such coalgebras. We will then relate these concepts to A∞ and L∞ algebras and show that “commutators on A∞ algebras yield L∞ algebras”. I would like to thank Jim Stasheff for igniting my interest in higher homotopy structures years ago and for encouraging a continuation of it. It has been an honor as well as a pleasure working with him and with some of his colleagues who are included in the bibliography of this note.