Non-representability of cohomology classes by bi-invariant forms (gauge and Kac-Moody groups)

We give a necessary topological condition on a cohomology class of any Lie group, modelled on a Fréchet space, to be representable by a bi-invariant form on. As a corollary, we show that if for somed>0, then there exists a cohomology class in which cannot be represented by any bi-invariant form. In particular, we conclude that there are many cohomology generators, in general, in the case of gauge groups and also Kac-Moody groups which cannot be represented by bi-invariant forms, although, very often, they are representable by left invariant forms.