Resolutions of Associative and Lie algebras
Ron Adin
Bar Ilan University
David Blanc
University of Haifa
We here describe certain explicit canonical resolutions for free
associative and free (graded) Lie algebras, in the category of non-associative
algebras. Both resolutions are based on the combinatorics of suitable
collections of leaf-labeled trees.
The Lie case was needed for the second author's description of higher homotopy
operations in rational homotopy theory: it turns out that in
order to describe all such higher operations, one must resolve the
rational differential graded Lie algebra L_* (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 non-associative algebras. 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_*
we need canonical resolutions of free Lie algebras in the category of
non-associative algebras, as described in this paper.
The construction is closely related to ``strongly homotopy Lie algebras''
Our main interest is indeed in the Lie case. The associative case, which is
based on work of Stasheff, is included mainly as a preliminary
illustration of the ideas involved, and to fix notation.