Growth and Lie brackets in the homotopy Lie algebra

Abstract

Let $L$ be an infinite dimensional graded Lie algebra that is either the homotopy Lie algebra $\pi_*(\Omega X)\otimes {\mathbb Q}$ for a finite $n$-dimensional CW complex $X$, or else the homotopy Lie algebra for a local noetherian commutative ring $R $ ($UL = Ext_R(I\! k,I\! k)$) in which case put $n =$ (embdim $-$ depth)$(R)$.

Theorem: (i) The integers $\lambda_k = \displaystyle\sum_{q=k}^{k+n-2} \mbox{dim} L_i$ grow faster than any polynomial in $k$.

(ii) For some finite sequence $x_1, \ldots , x_d$ of elements in $L$ and some $N$, any $y\in L_{\geq N}$ satisfies: some $[x_i,y] \neq 0$.