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

Source: Adin, Ron - Department of Mathematics, Bar Ilan University

Collections: Mathematics