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

TL;DR

This paper introduces a new method for classifying left-invariant symplectic structures on Lie groups by analyzing the moduli space of nondegenerate 2-forms, and applies it to specific higher-dimensional cases.

Contribution

A novel approach to classify left-invariant symplectic structures on Lie groups using moduli space analysis, extending classifications to higher dimensions.

Findings

Classified symplectic structures on two specific Lie groups of dimension 2n

Established a systematic procedure for classification up to automorphism and scale

Extended known classifications to higher-dimensional Lie groups

Abstract

We are interested in the classification of left-invariant symplectic structures on Lie groups. Some classifications are known, especially in low dimensions. In this paper we establish a new approach to classify (up to automorphism and scale) left-invariant symplectic structures on Lie groups. The procedure is based on the moduli space of left-invariant nondegenerate $2$-forms. Then we apply our procedure for two particular Lie groups of dimension $2n$ and give classifications of left-invariant symplectic structures on them.