Eight-Dimensional Symplectic Nilpotent Lie Groups with Lagrangian Normal Subgroups: A Complete Classification
T. Aït Aissa, M. W. Mansouri
TL;DR
The paper addresses the classification of eight-dimensional symplectic nilpotent Lie groups with Lagrangian normal subgroups by developing a Lagrangian extension framework for flat Lie algebras, rooted in cotangent-extension and symplectic-reduction ideas. It defines the Lagrangian extension cohomology $H^2_{L,\rho}(\mathfrak{h}, \mathfrak{h}^*)$ and proves a bijection between isomorphism classes of such algebras and geodesically complete flat nilpotent algebras equipped with Lagrangian extension data, enabling an explicit eight-dimensional classification. The main result is a complete list of exactly $95$ eight-dimensional symplectic nilpotent Lie algebras with Lagrangian normal subgroups, along with the corresponding symplectic forms, from which the eight-dimensional filiform real Lie algebras follow. The work integrates cohomological invariants, affine-geometry perspectives, and automorphism-group analyses (in the Appendix) to achieve a comprehensive, highly structured classification with potential implications for geometric representation theory and symplectic Lie group theory.
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.