A note on $2$-plectic vector spaces

Abstract

Among the $2$-plectic structures on vector spaces, the canonical 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 noncompact 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 $\mathfrak{g}$ 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.

Keywords

• $2$-plectic structure
• canonical $2$-plectic structure
• semisimple lie algebra