Lie algebra models for unstable homotopy theory

Abstract

Quillen showed how to describe the homotopy theory of simply-connected rational spaces in terms of differential graded Lie algebras. Here we survey a generalization of Quillen's results that describes the $v_n$-periodic localizations of homotopy theory (where rational corresponds to $n=0$) in terms of spectral Lie algebras. The latter form an extension of the theory of Lie algebras to the setting of stable homotopy theory. This is a chapter written for the Handbook of Homotopy Theory edited by Haynes Miller.