Non-Hamiltonian Actions and Lie-Algebra Cohomology of Vector Fields

Abstract

Two examples of ${\mathrm{Diff}}^{+}{S}^1$-invariant closed two-forms obtained from forms on jet bundles, which does not admit equivariant moment maps are presented. The corresponding cohomological obstruction is computed and shown to coincide with a nontrivial Lie algebra cohomology class on $H^2(\mathfrak{X}(S^1))$.