On compact semisimple Lie groups as 2-plectic manifolds

Abstract

In this paper a compact and semisimple Lie group G is considered endowed with a 2-plectic structure ω, induced by the Killing form. We show that the Lie group of 2-plectomorphisms of G is finite dimensional and compact, and hence the Darboux’s theorem fails to be true for this 2-plectic structure. Also it is shown that ω induces a left-invariant \({\mathfrak{g}^{*}}\) valued 2-form which is proportional to dΘ, where Θ is the Cartan–Maurer 1-form on G. At last we consider the action of G on its tangent bundle which is furnished with the 2-plectic structure ωc, the complete lift of ω, and calculate covariant momentum map of this action.