# Exponential growth of Lie algebras of finite global dimension

Abstract : Let $X$ be a finite simply connected CW complex of dimension $n$. The loop space homology $H_*(\Omega X;\mathbb Q)$ is the universal enveloping algebra of a graded Lie algebra $L_X$ isomorphic with $\\pi_{*-1} (X)\otimes \mathbb Q$. Let $Q_X \subset L_X$ be a minimal generating subspace, and set $\alpha = \limsup_i \frac{\log\mbox{\scriptsize rk} \pi_i(X)}{i}$. Theorem: If $\mbox{dim}\, L_X = \infty$ and $\limsup (\mbox{dim} ( Q_X)_k)^{1/k} < \limsup (\mbox{dim} (L_X)_k)^{1/k}$ then $\sum_{i=1}^{n-1} \mbox{rk} \pi_{k+i}(X) = e^{(\alpha + \varepsilon_k)k} \hspace{1cm} \mbox{where} \varepsilon_k \to 0 \mbox{as} k\to \infty.$ In particular $\displaystyle\sum_{i=1}^{n-1} \mbox{rk} \pi_{k+i}(X)$ grows exponentially in $k$.
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Cited literature [9 references]

https://hal.archives-ouvertes.fr/hal-00008857
Contributor : Secrétariat Math. Angers <>
Submitted on : Monday, September 19, 2005 - 12:08:15 PM
Last modification on : Monday, March 9, 2020 - 6:15:51 PM
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### Citation

Yves Félix, Steve Halperin, Jean-Claude Thomas. Exponential growth of Lie algebras of finite global dimension. 2005. ⟨hal-00008857⟩

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