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On para‐Kähler Lie algebroids and contravariant pseudo‐Hessian structures
- Source :
- Mathematical News / Mathematische Nachrichten, Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, 2019, 292 (7), pp.1418-1443. ⟨10.1002/mana.201700137⟩
- Publication Year :
- 2019
- Publisher :
- Wiley, 2019.
-
Abstract
- In this paper, we generalize all the results obtained on para‐Kahler Lie algebras in [3] to para‐Kahler Lie algebroids. In particular, we study exact para‐Kahler Lie algebroids as a generalization of exact para‐Kahler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra (A,.), the orbits of the action Φ of (A,+) on A∗ given by Φ(a,μ)=exp(La∗)(μ) are pseudo‐Hessian manifolds, where La(b)=a.b. We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.
- Subjects :
- Hessian matrix
Pure mathematics
010308 nuclear & particles physics
Generalization
General Mathematics
010102 general mathematics
01 natural sciences
Action (physics)
symbols.namesake
0103 physical sciences
Lie algebra
Associative algebra
Covariance and contravariance of vectors
symbols
Mathematics::Differential Geometry
[MATH]Mathematics [math]
0101 mathematics
Mathematics::Symplectic Geometry
Commutative property
ComputingMilieux_MISCELLANEOUS
Associative property
Mathematics
Subjects
Details
- ISSN :
- 15222616 and 0025584X
- Volume :
- 292
- Database :
- OpenAIRE
- Journal :
- Mathematische Nachrichten
- Accession number :
- edsair.doi.dedup.....9f7db934ee778c716827e322e87f2491