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Commentarii Mathematici Helvetici

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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 + qi = 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