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The structure of homotopy Lie algebras

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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 Vk ≤ d ∑k + qi = k Wi, 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 2m 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.

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https://hal.univ-angers.fr/hal-03031602
Contributeur : Okina Université d'Angers <>
Soumis le : lundi 30 novembre 2020 - 15:20:46
Dernière modification le : mardi 1 décembre 2020 - 03:25:54

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Yves Felix, Steve Halperin, Jean-Claude Thomas. The structure of homotopy Lie algebras. Commentarii Mathematici Helvetici, 2009, 84 (4), pp.807 - 833. ⟨10.4171/CMH/182⟩. ⟨hal-03031602⟩

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