The structure of homotopy Lie algebras

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.