Bilinear forms and 2-dimensional cohomology

Abstract

AbstractThis paper calculates the central Borel 2 cocycles for certain 2-step nilpotent Lie groups G with values in the injectives A of the category of 2nd countable locally compact abelian groups. The G's include, among others, all groups locally isomorphic to a Heisenberg group. The A's are direct sums of vector groups and (possibly infinite dimensional) tori, and in particular include R, T, and Cx. The main results are as follows.(4.1) Every symmetric central 2 cocycle is trivial.(4.2) Every central 2 cocycle is cohomologous with a skew symmetric bimultiplicative one (which is necessarily jointly continuous by [7]).(4.3) The corresponding cohomology group H2cent (G, A) is calculated as the skew symmetric jointly continuous bimultiplicative maps modulo Homcont ([G, G]–, A).These results generalize the case when G is a connected abelian Lie group and A = T, due to Kleppner [3]. Using standard facts of the cohomology of groups they can be interpreted as classifying all continuous central extensions (1) → A → E → G → (1) of the group G by the abelian group A. Finally some counterexamples are given to extending these results.