Lie models for nilpotent spaces

Abstract

Let $(L,d)$ be a differential graded Lie algebra, where $L=L(V)$ is free as graded Lie algebra and $V=V_{\ge 0}$ is a finite type graded vector space. We prove that the injection of $(L,d)$ into its completion $(\hat{L},d)$ is a quasi-isomorphism if and only if $H(L,d)$ is a finite type pronilpotent graded Lie algebra. As a consequence, we obtain an equivalence between graded Lie models for nilpotent spaces in rational homotopy theory.