 
Summary: RESOLUTIONS OF ASSOCIATIVE AND LIE ALGEBRAS
RON ADIN AND DAVID BLANC
Abstract. Certain canonical resolutions are described for free associative and free
Lie algebras in the category of nonassociative algebras. These resolutions derive
in both cases from geometric objects, which in turn reflect the combinatorics of
suitable collections of leaflabeled trees.
1. Introduction
We here describe certain explicit canonical resolutions for free associative and free
(graded) Lie algebras, in the category of nonassociative algebras. Both resolutions
are based on the combinatorics of suitable collections of leaflabeled trees.
The Lie case was needed for the second author's description of higher homotopy
operations in rational homotopy theory, in [B2]: it turns out that in order to describe
all such higher operations, one must resolve the differential graded Lie algebra L \Lambda
over Q (representing the rational homotopy type of a given space X) simplicially,
by suitable free (differential) graded Lie algebras. The higher homotopy operations
correspond to relations and syzygies for these free graded Lie algebras, thought of
as nonassociative algebras over Q. Since we must replace all the Lie algebras by
the corresponding free differential algebras in a functorial manner (to preserve the
simplicial structure of the original resolution of L \Lambda ), we need canonical resolutions
of free Lie algebras in the category of nonassociative algebras, as described in this
