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 Ext A ∗(M,A)≠0 for some A-module M of at most polynomial growth. Theorem 1: If f :X→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 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, ∑ i= k+1 k+ d dim L i ⩾ k r , k⩾ some k( r).