The structure of homotopy Lie algebras

Abstract

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 $d$, $N$, $q$, $\dim V_k\le d{\sum }_{i=k}^{k+q}\dim W_i$, $i\ge N$. We define an equivalence relation on the subspaces of $V$ by $U\sim W$ if $U$ and $W$ are full in $U+W$. Two subspaces $V,W$ in $L$ are then called $L$-equivalent ($V{\sim }_{L}W$) if for all ideals $K\subset L$, $V\cap K\sim W\cap K$. Then our main result asserts that the set $\mathcal{L}$ 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 $E\subset L$ such that $E\cap I=0$, $E\oplus I$ is full in $L$, and $\mathrm{depth}E+\mathrm{depth}I\le \mathrm{depth}L$.