Eight-Dimensional Symplectic Nilpotent Lie Groups with Lagrangian Normal Subgroups: A Complete Classification

TL;DR

This paper classifies all eight-dimensional symplectic nilpotent Lie groups with Lagrangian normal subgroups, establishing a bijection with certain flat, nilpotent Lie groups and identifying exactly ninety-five such groups.

Contribution

It provides a complete classification of eight-dimensional symplectic nilpotent Lie groups with Lagrangian normal subgroups, including a detailed enumeration of ninety-five cases.

Findings

Established a bijection with geodesically complete, flat, nilpotent Lie groups.

Identified exactly ninety-five such eight-dimensional groups.

Derived a classification of eight-dimensional symplectic filiform Lie groups.

Abstract

We investigate symplectic nilpotent Lie groups with Lagrangian normal subgroups. We show that there exists a bijection between the isomorphism classes of nilpotent Lie groups with Lagrangian normal subgroups and the isomorphism classes of geodesically complete, flat, nilpotent Lie groups with Lagrangian extension cohomology class. Finally, we provide a complete classification of eight-dimensional symplectic nilpotent Lie groups with Lagrangian normal subgroups, identifying exactly ninety-five such groups. As a consequence, we obtain a complete classification of eight-dimensional symplectic filiform real Lie groups.