On the rational homotopy Lie algebra of spaces with finite dimensional rational cohomology and homotopy (Q1110862)

The author studies finite dimensional rational graded Lie algebras which are isomorphic to the rational homotopy Lie algebra \(\pi_*(\Omega X)\otimes {\mathbb{Q}}\) of some simply connected space X having finite- dimensional rational cohomology. The central result says that, for a given graded \({\mathbb{Q}}\)-vector space \(W_*\), the set f\({\mathcal L}(W)\) of all graded Lie algebra structures on \(W_*\) which are realizable in the above sense, is either empty or Zariski-open and dense in the algebraic variety \({\mathcal L}(W)\) of structure constants of graded Lie algebras on \(W_*\), each of these cases being characterized by a condition similar to the ``strong arithmetic condition'' of \textit{J. B. Friedlander} and \textit{S. Halperin} [An arithmetic characterization of the rational homotopy groups of certain spaces, Invent. Math. 53, 117-133 (1979; Zbl 0408.55010)]. This result is then used to prove some criteria for the realization of concrete graded Lie algebras.