The Radical of the Homotopy Lie Algebra

Abstract

Introduction. Let T be a simply connected CW complex of finite type. Its homotopy Lie is the rational Lie 7r*(9 T) 0 Q, equipped with the Samelson product. Let A be a local (noetherian) ring with residue field k (of arbitrary characteristic). Its homotopy Lie algebra, 7r*(A), is the naturally defined Lie whose enveloping is ExtA(k, k)-cf. [Av]. These Lie algebras have finite type (i.e. are finite dimensional in each degree) and are concentrated in strictly positive degrees. Thus, for the sake of simplicity, we shall use graded Lie algebra to mean graded Lie of finite type concentrated in strictly positive degrees throughout this paper. On the other hand these Lie algebras are usually nonzero in infinitely many degrees. Thus, while we may (and do) form the sum, R, of all the solvable ideals in such a Lie and call it the radical, it may well happen that R, itself, is not solvable. Indeed, this does happen in the topological context, since by Quillen's result [Q] every rational Lie of finite type occurs as the homotopy Lie of a space. By contrast, in the case of local rings, or of CW complexes with finite Lusternik-Schnirelmann (L.S.) category, serious restrictions apply to the homotopy Lie algebra. Indeed, our first main result is the remarkable