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The Journal of Geometric Mechanics (JGM)
 

Hypersymplectic structures on Courant algebroids

Pages: 255 - 280, Volume 7, Issue 3, September 2015      doi:10.3934/jgm.2015.7.255

 
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Paulo Antunes - CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal (email)
Joana M. Nunes da Costa - CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal (email)

Abstract: We introduce the notion of hypersymplectic structure on a Courant algebroid and we prove the existence of a one-to-one correspondence between hypersymplectic and hyperkähler structures. This correspondence provides a simple way to define a hyperkähler structure on a Courant algebroid. We show that hypersymplectic structures on Courant algebroids encompass hypersymplectic structures with torsion on Lie algebroids. In the latter, the torsion existing at the Lie algebroid level is incorporated in the Courant structure. Cases of hypersymplectic structures on Courant algebroids which are doubles of Lie, quasi-Lie and proto-Lie bialgebroids are investigated.

Keywords:  Hypersymplectic, hyperkähler, Courant algebroid, Lie algebroid.
Mathematics Subject Classification:  Primary: 53D17; Secondary: 53D18, 53C26.

Received: January 2015;      Revised: June 2015;      Available Online: July 2015.

 References