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Zbl 1066.17019
Felix, Yves; Halperin, Stephen; Thomas, Jean-Claude
Graded Lie algebras with finite polydepth. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 36, No. 5, 793-804 (2003). ISSN 0012-9593

The paper defines a new invariant for a graded connected algebra $A$: the polynomial depth polydepth\,$A$ which is finite when $\text{Ext}^*_A(M,A)\ne 0$ for some $A$-module $M$ having at most polynomial growth i.e. $\sum_{i\le n}\dim M_i\le Cn^d$. The polynomial bound polybd\,$M$ being the least such~$d$.\par For a fibration $f: X\to Y$ of path connected spaces with fibre $F$, one has $\operatorname{polydepth}H_*(\Omega Y)\le \operatorname{polybd}H_*(F)+ \operatorname{cat}(f)$ where $\text{cat}(f)$ is the Lusternik-Schnirelmann category of $f$.\par If $L$ is a graded Lie algebra and if the polydepth of its universal enveloping algebra $UL$ is finite, then either $UL$ grows at most polynomially ($L$ is solvable) or for some integer $d$ and all $r$, one has $\sum^{k+d}_{i=k+1}\dim L_i\ge k^r$ for $k\ge$ some $k(r)$.
[Georges Hoff (Villetaneuse)]

MSC 2000:: *17B70 Graded Lie algebras
55P62 Rational homotopy theory

Keywords: graded Lie algebra; polynomial depth; polynomial bound; polynomial growth

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