Some Non-Abelian Phase Spaces in Low Dimensions

TL;DR

This paper explores non-abelian phase spaces associated with Lie algebras, revealing their symplectic isomorphisms and providing explicit classifications in low dimensions to aid future research.

Contribution

It introduces a classification of low-dimensional non-abelian phase spaces and demonstrates their symplectic properties, expanding understanding of their geometric and algebraic structures.

Findings

All such phase spaces in dimension 4 are classified.

Examples of phase spaces in dimension 6 are provided.

Non-abelian phase spaces are symplectically isomorphic under certain conditions.

Abstract

A non-abelian phase space, or a phase space of a Lie algebra is a generalization of the usual (abelian) phase space of a vector space. It corresponds to a parak\"ahler structure in geometry. Its structure can be interpreted in terms of left-symmetric algebras. In particular, a solution of an algebraic equation in a left-symmetric algebra which is an analogue of classical Yang-Baxter equation in a Lie algebra can induce a phase space. In this paper, we find that such phase spaces have a symplectically isomorphic property. We also give all such phase spaces in dimension 4 and some examples in dimension 6. These examples can be a guide for a further development.