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# Commentarii Mathematici Helvetici

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**Volume 84, Issue 4, 2009, pp. 807–833**

**DOI: 10.4171/CMH/182**

Published online: 2009-12-23

The structure of homotopy Lie algebras

Yves Félix^{[1]}, Steve Halperin

^{[2]}and Jean-Claude Thomas

^{[3]}(1) Université Catholique de Louvain, Belgium

(2) University of Maryland, College Park, USA

(3) Université d'Angers, France

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*.

*Keywords: *Homotopy Lie algebra, graded Lie algebra, ideal, radical of a Lie algebra, log index, growth

Félix Yves, Halperin Steve, Thomas Jean-Claude: The structure of homotopy Lie algebras. *Comment. Math. Helv.* 84 (2009), 807-833. doi: 10.4171/CMH/182