Growth and Lie brackets in the homotopy Lie algebra

Yves Félix ; Stephen Halperin ; Jean-Claude Thomas

Homology, homotopy, and applications, Tome 4 (2002) no. 2, pp. 219-225.

Résumé

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 $[x_i,y] \neq 0$.

Détail

DOI : 10.4310/HHA.2002.v4.n2.a10

Classification : 16L99, 17B70, 55P35, 55P62
Keywords: finite CW complex, local ring, homotopy lie algebra, depth