arXiv subjects

P. Antunes

Publications and source records attributed to P. Antunes.

4 recordsLinked to original sources

arXiv2022

Higher Multi-Courant Algebroids

The binary bracket of a Courant algebroid structure on $(E,\langle \cdot,\cdot \rangle)$ can be extended to a $n$-ary bracket on $Γ(E)$, yielding a multi-Courant algebroid. These $n$-ary brackets form a Poisson algebra and were defined, in an algebraic setting, by Keller and Waldmann. We construct a higher geometric version of Keller-Waldmann Poisson algebra and define higher multi-Courant algebroids. As Courant algebroid structures can be seen as degree $3$ functions on a graded symplectic manifold of degree $2$, higher multi-Courant structures can be seen as functions of degree $n\geq 3$ on that graded symplectic manifold.

math.DG↗

arXiv2016

Hypersymplectic structures with torsion on Lie algebroids

Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one correspondence between hypersymplectic structures with torsion and hyperkähler structures with torsion. We show that given a Lie algebroid with a hypersymplectic structure with torsion, the deformation of the Lie algebroid structure by any of the transition morphisms does not affect the hypersymplectic structure with torsion. We also show that if a triplet of $2$-forms is a hypersymplectic structure with torsion on a Lie algebroid $A$, then the triplet of the inverse bivectors is a hypersymplectic structure with torsion for a certain Lie algebroid structure on the dual $A^*$, and conversely. Examples of hypersymplectic structures with torsion are included.

math.DG↗

arXiv2014

Hypersymplectic structures on Courant algebroids

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 simpler way to define a hyperkähler structure on a Courant algebroid. We show that hypersymplectic structures on Courant algebroids encompass hyperkähler and hyperkähler 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.

math.DG↗

arXiv2013

Hyperstructures on Lie algebroids

We define hypersymplectic structures on Lie algebroids recovering, as particular cases, all the classical results and examples of hypersymplectic structures on manifolds. We prove a 1-1 correspondence theorem between hypersymplectic structures and (pseudo-)hyperkähler structures. We show that the hypersymplectic framework is very rich in already known compatible pairs of tensors such as Poisson-Nijenhuis, $ΩN$ and $P Ω$ structures.

math.SG↗