Infinitely many left-symmetric structures on nilpotent Lie algebras

2025/10/16 by Kato, Naoki
#17B05 #17B30 #53D05 #FOS: Mathematics #Rings and Algebras (math.RA) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2510.14610

Abstract

Dekimpe and Ongenae constructed infinitely many pairwise non-isomorphic complete left-symmetric structures on ℝⁿ for n≥ 6. In this paper, we construct a family of complete left-symmetric structures on the cotangent Lie algebra T^*\mathfrakg of a certain n-dimensional almost abelian nilpotent Lie algebra \mathfrakg and give a condition under which two left-symmetric structures in this family are isomorphic. As a consequence of this result, we obtain infinitely many pairwise non-isomorphic left-symmetric structures on T^*\mathfrakg. As an application of this construction, we also obtain infinitely many symplectic structures on T^*\mathfrakg which are pairwise non-symplectomorphic up to homothety.

Citations

• A classification of left-invariant symplectic structures on some Lie groups
• Lie algebras with ad-invariant metrics. A survey
• Symplectic structures on quadratic Lie algebras
• Symplectic or contact structures on Lie Groups
• Affine Structures on abelian Lie Groups

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