Lie algebras and Higher torsion in p-groups

TL;DR

This paper investigates the exceptional torsion phenomena in the integral cohomology of p-groups linked to p-adic Lie algebras, revealing significant differences between characteristic zero and characteristic p cases.

Contribution

It introduces a spectral sequence model for the Bockstein spectral sequence of p-groups associated with Lie algebras and analyzes its behavior across different characteristics.

Findings

Spectral sequence E_r^{*,*}[g] models the Bockstein spectral sequence for p-groups.

In characteristic zero, the cohomology algebra is generated by two elements.

In characteristic p, the algebra requires at least 17 generators, indicating a phase transition.

Abstract

We study exceptional torsion in the integral cohomology of a family of p-groups associated to p-adic Lie algebras. A spectral sequence E_r^{*,*}[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 sl_n(C), and the results there are transferred to the corresponding p-group via the intermediary arithmetic Lie algebra defined over Z. Over C, it is shown that E_1^{*,*}[g]=H^*(g,U(g)^*)=H^*(\Lambda BG) where U(g)^* is the dual of the universal enveloping algebra of g and \Lambda BG is the free loop space of the classifying space of a Lie group G associated to g. In characteristic p, a phase transition is observed. For example, it is shown that the algebra E_1^{*,*}[sl_2[F_p]] requires at least 17 generators unlike its characteristic zero counterpart which only requires two.