# The structure of homotopy Lie algebras

### Yves Félix

Université Catholique de Louvain, Belgium### Steve Halperin

University of Maryland, College Park, USA### Jean-Claude Thomas

Université d'Angers, France

## Abstract

In this paper we consider a graded Lie algebra, $L$, of finite depth $m$, and study the interplay between the depth of $L$ and the growth of the integers $dimL_{i}$. A subspace $W$ in a graded vector space $V$ is called full if for some integers $d$, $N$, $q$, $dimV_{k}≤d∑_{i=k}dimW_{i}$, $i≥N$. We define an equivalence relation on the subspaces of $V$ by $U∼W$ if $U$ and $W$ are full in $U+W$. Two subspaces $V,W$ in $L$ are then called $L$-equivalent ($V∼_{L}W$) if for all ideals $K⊂L$, $V∩K∼W∩K$. Then our main result asserts that the set $L$ of $L$-equivalence classes of ideals in $L$ is a distributive lattice with at most $2_{m}$ elements. To establish this we show that for each ideal $I$ there is a Lie subalgebra $E⊂L$ such that $E∩I=0$, $E⊕I$ is full in $L$, and $depthE+depthI≤depthL$.

## Cite this article

Yves Félix, Steve Halperin, Jean-Claude Thomas, The structure of homotopy Lie algebras. Comment. Math. Helv. 84 (2009), no. 4, pp. 807–833

DOI 10.4171/CMH/182