Lie algebras and higher torsion in p-groups

Abstract

The primary aim of this paper is to study exceptional torsion in the integral cohomology of a family of p-groups associated to p-adic Lie algebras. A spectral sequence Er⁎,⁎[g] is defined for any Lie algebra g which models the Bockstein spectral sequence of the corresponding group in characteristic p.This spectral sequence is then studied for complex semisimple Lie algebras like sln(C) and the results there are transferred to the corresponding p-group via the intermediary arithmetic Lie algebra defined over Z. The results obtained this way for a fixed Lie algebra scheme like sln(−) hold in a range in the corresponding Bockstein spectral sequence for all but finitely many primes depending on the chosen range.Over C, it is shown that E1⁎,⁎[g]=H⁎(g,U(g)⁎)=H⁎(ΛBG) where U(g)⁎ is the (filtered) dual of the universal enveloping algebra of g equipped with the dual adjoint action and ΛBG is the free loop space of the classifying space of an associated compact, connected real form Lie group G to g.When passing to characteristic p, in the corresponding Bockstein spectral sequence, a char 0 to char p phase transition is observed. For example, it is shown that the algebra E1⁎,⁎[sl2[Fp]] requires at least 17 generators unlike its characteristic zero counterpart which only requires two.