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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 Li. A subspace W in a graded vector space V is called full if for some integers d, N, q, dim Vkd k + qi = k Wi, iN. We define an equivalence relation on the subspaces of V by UW if U and W are full in U + W. Two subspaces V, W in L are then called L-equivalent (VL W) if for all ideals KL, VK WK. Then our main result asserts that the set ℒ of L-equivalence classes of ideals in L is a distributive lattice with at most 2m elements. To establish this we show that for each ideal I there is a Lie subalgebra EL such that EI = 0, EI 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