A classification of left-invariant symplectic structures on some Lie groups (Q6103604)

The classifications of left-invariant symplectic structures on Lie groups in low dimensions are known, see, e.g., [\textit{B.-Y. Chu}, Trans. Am. Math. Soc. 197, 145--159 (1974; Zbl 0261.53039); \textit{J. R. Gómez} et al., J. Pure Appl. Algebra 156, No. 1, 15--31 (2001; Zbl 0966.17006); \textit{M. Goze} and \textit{A. Bouyakoub}, Rend. Semin. Fac. Sci. Univ. Cagliari 57, No. 1, 85--97 (1987; Zbl 0679.17002); \textit{G. Ovando}, Beitr. Algebra Geom. 47, No. 2, 419--434 (2006; Zbl 1155.53042)]. These classifications are important because in geometry one would like to know whether a given manifold admits some nice geometric structures. In the setting of Lie groups, it is natural to ask about the existence of left-invariant structures. A symplectic Lie group is a Lie group $G$ endowed with a left-invariant symplectic form $\omega$ (that is, a nondegenerate closed 2-form). Although, there are many interesting results on the structure of symplectic Lie groups and some classifications in low dimensions, the general picture is far from complete. The authors of the paper use the method of [\textit{T. Hashinaga} et al., J. Math. Soc. Japan 68, No. 2, 669--684 (2016; Zbl 1353.53058)] to find left-invariant Riemannian metrics, to establish a new approach to classify (up to automorphism and scale) left-invariant symplectic structures on Lie groups. Their approach is based on the moduli space of left invariant nondegenerate 2-forms. They apply their method for two particular Lie groups of dimension $2n$ and give classifications of left-invariant symplectic structures on them.