Graded Lie algebras with finite polydepth

Abstract

If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite if $Ex{t}_A^{\ast }(M,A)\ne 0$ 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_*(ΩY) acting in the homology of the homotopy fibre grow at most polynomially, then H_*(ΩY) has finite polydepth. Theorem 2: 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, ${\sum }_{i=k+1}^{k+d}dim{L}_i\ge k^r$, $k\ge$ some $k(r)$.