# 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

supp { vertical-align: 1.8ex; font-size:90%; } subb { vertical-align: -0.8ex; font-size:90%; } .ions { line-height: 1.8; } .ions subb {position: relative; top:1; left 18; } .ions supp {position: absolute; top: 10; left:5; } 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 dim *L__i*. A subspace *W* in a graded vector space *V* is called full if for some integers *d*, *N*, *q*, dim *V__k* ≤ *d* ∑_k_ + *q__i* = *k* *W__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 depth *E* + depth *I* ≤ depth *L*.