Homology, Homotopy and Applications

Growth and Lie brackets in the homotopy Lie algebra

Yves Félix, Stephen Halperin, and Jean-Claude Thomas

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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 $[x_i,y] \neq 0$.

Article information

Homology Homotopy Appl., Volume 4, Number 2 (2002), 219-225.

First available in Project Euclid: 13 February 2006

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P62: Rational homotopy theory
Secondary: 17B70: Graded Lie (super)algebras 55P35: Loop spaces 55Q15: Whitehead products and generalizations


Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude. Growth and Lie brackets in the homotopy Lie algebra. Homology Homotopy Appl. 4 (2002), no. 2, 219--225. https://projecteuclid.org/euclid.hha/1139852463

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