LAREMA Prépublication n° 158

Yves FÉLIX, Stephen HALPERIN, Jean-Claude THOMAS

### Graded Lie algebras with finite polydepth

If $A$ is a graded connected algebra then we define a new invariant, polydepth $A$, which is finite if $\mbox{Ext}_A^*(M,A) \neq 0$ for some $A$-module $M$ of at most polynomial growth.
Theorem 1: If $f : X \to Y$ is a continuous map of finite category, and if the orbits of $H_*(\Omega Y)$ acting in the homology of the homotopy fibre grow at most polynomially, then $H_*(\Omega Y)$ has finite polydepth.
Theorem 5: If $L$ is a graded Lie algebra and polydepth $UL$ is finite then either $L$ is solvable and $UL$ grows at most polynomially or else for some integer $d$ and all $r$, $\sum_{i=k+1}^{k+d} \mbox{dim}\, L_i \geq k^r$, $k\geq$ some $k(r)$.

Mots Clés
Graded Lie algebras ; loop space homology, depth, solvable Lie algebras

Codes MSC
55P35 Loop spaces
55P62 Rational homotopy theory
17B55 Homological methods in Lie (super)algebras

Date d'enregistrement : 12 septembre 2002

[
Accueil] [Autres publications]