RESEARCH RESULTS SO FAR

LUIS PEDRO CASTELLANOS MOSCOSO

1. Introduction
The problem of determining whether a given manifold admits a symplectic structure isa classical and hard problem. In the setting of Lie groups, it is natural to ask about theexistence of left-invariant structures. A symplectic Lie group is a Lie group G endowedwith a left-invariant symplectic form ω (that is, a nondegenerate closed 2-form). The
study of symplectic Lie groups reduces to the study of symplectic Lie algebras (g, ω), thatis Lie algebras g endowed with nondegenerate closed 2-forms (or equivalently two-cocyclesω ∈ Z 2 (g)). Still the problem of determining if a given Lie algebra admits a symplecticstructre remains difficult in general and the picture seems far from complete. Only someclassifications in low dimensions and some special cases in higher dimensions are known.
In [4] we can find a novel method to find nice (e.g., Einstein or Ricci soliton) leftinvariant Riemannian metrics. The method is based on the moduli space of left-invariant
Riemannian metrics on a Lie group G (the orbit of space of certain group action). In [5]and [6] the authors adapted the same ideas in the pseudo-Riemannian case succesfully. Itwas natural then, to try to use the same ideas for symplectic Lie groups.
2. Results(1) Inspired by those previous studies in [3] we developed a similar approach forthe study of symplectic Lie groups. We study the moduli space of left-invariant
nondegenerate 2-forms on Lie algebras g.
(2) As a first application of these ideas also in [3] we studied two particular Lie
algebras: the Lie algebra of the real hyperbolic space gRH2n , and the direct sumof the 3-dimensional Heisenberg Lie algebra and the abelian Lie algebra h3 ⊕R2n−3 .
We obtained a classification of symplectic structures on both of them.
(3) These two Lie algebras belong to a special family of Lie algebras: they are bothalmost abelian Lie algebras. In [2] we studied a particular family of almost abelian
Lie algebras whose adjoint homomorphism is diagonalizable, and obtained a classification of symplectic structures.
(4) We obtained an improvement on two decomposition theorems of symplectic matrices: the symplectic QR decomposition and the SR decomposition.
References[1] Baues, O. and Cortés, V. : Symplectic Lie groups. Astérisque 379 (2016).
[2] Castellanos Moscoso, L. P.: Left-invariant symplectic structures on diagonal almost abelian Liegroups. Submitted.
[3] Castellanos Moscoso, L. P. and Tamaru, H.: A classification of left-invariant symplectic structures
on some Lie groups. arXiv e-prints, 2020.
[4] Hashinaga, T., Tamaru, H. and Terada, K.: Milnor-type theorems for left-invariant Riemannianmetrics on Lie groups J. Math. Soc. Japan 68 (2016), 669-684.
[5] Kondo, Y., and Tamaru, H.: A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups. Tohoku Math. J. To appear.
[6] Kubo, A., Onda, K., Taketomi, Y., Tamaru, H.: On the moduli spaces of left-invariant pseudoRiemannian metrics on Lie groups. Hiroshima Math. J. 46 (2016), 357–374.
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2L. CASTELLANOS

Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto,Sumiyoshi-ku, Osaka City, Japan 585-8585
Email address: caste3.1416@gmail.com