Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds

TL;DR

This paper explores the relationship between Euler's equations on central extensions of Lie algebras and their original algebras, focusing on nilpotent Lie algebras and symplectic nilmanifolds, revealing new structural insights.

Contribution

It introduces a sequence of central extensions of nilpotent Lie algebras from formal vector fields and analyzes their coadjoint orbits and associated symplectic nilmanifolds.

Findings

Euler equations on extended and original Lie algebras are connected

A sequence of central extensions with specific properties is constructed

Covering Lie groups for symplectic nilmanifolds can have arbitrary rank

Abstract

The connections between Euler's equations on central extensions of Lie algebras and Euler's equations on the original, extended algebras are described. A special infinite sequence of central extensions of nilpotent Lie algebras constructed from the Lie algebra of formal vector fields on the line is considered, and the orbits of coadjoint representations for these algebras are described. By using the compact nilmanifolds constructed from these algebras by I.K. Babenko and the author, it is shown that covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.