# Lie algebra models for unstable homotopy theory

## TL;DR

This paper surveys how spectral Lie algebras extend Lie algebra models to describe the $v_n$-periodic localizations in unstable homotopy theory, generalizing Quillen's rational case.

## Contribution

It generalizes Quillen's Lie algebra models from rational homotopy to $v_n$-periodic localizations using spectral Lie algebras, bridging algebra and stable homotopy theory.

## Key findings

- Spectral Lie algebras model $v_n$-periodic homotopy theory.
- Extension of Lie algebra models to unstable homotopy via spectral methods.
- Provides a comprehensive survey for the Handbook of 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.