A note on $2$-plectic vector spaces

Document Type : Research Paper

Author

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

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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References

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Journal of Mahani Mathematical Research
Volume 13, Issue 1 - Serial Number 26
November 2023
Pages 443-455

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History

  • Receive Date: 17 January 2023
  • Revise Date: 08 June 2023
  • Accept Date: 15 October 2023

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