A note on 2-plectic vector spaces

Among the 2-plectic structures on vector spaces, the canon-ical ones and the 2-plectic structures induced by the Killing form on semisimple Lie algebras are more interesting. In this note, we show that the group of linear preservers of the canonical 2-plectic structure is non-compact and disconnected and its dimension is computed. Also, we show that the group of automorphisms of a compact semisimple Lie algebra preserving its 2-plectic structure, is compact. Furthermore, it is shown that the 2-plectic structure on a semisimple Lie algebra g is canonical if and only if it has an abelian Lie subalgebra whose dimension satisfies in a special condition. As a consequence, we conclude that the 2-plectic structures induced by the Killing form on some important classical Lie algebras are not canonical.