Higher torsion in p-groups, Casimir operators and the classifying spectral sequence of a Lie algebra

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 sln(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(ΛBG) where U(g) is the dual of the universal enveloping algebra of g and ΛBG 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 [sl2[Fp]] requires at least 17 generators unlike its characteristic zero counterpart which only requires two.