A NOTE ON 2-PLECTIC VECTOR SPACES

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Shafiee Mohammad

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Journal Paper

Paper Information

Journal: Journal of Mahani Mathematical Research Year:2024 | Volume:13 | Issue:1 Page(s): 443-455

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Information Journal Paper

Title

A NOTE ON 2-PLECTIC VECTOR SPACES

Author(s)

Shafiee Mohammad | Issue Writer Certificate

Keywords

2-plectic structure

Canonical 2-plectic structure

Semisimple Lie algebr

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.

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