Some Non-Abelian Phase Spaces in Low Dimensions

Details Some Non-Abelian Phase Spaces in Low Dimensions Some Non-Abelian Phase Spaces in Low Dimensions

2008-08-11

arxiv.org/abs/0808.1454v1

Journal of Geometry and Physics 58 (2008) 1752-1761

Physics

Mathematical Physics

16 pages, appear in Journal of Geometry and Physics

Scientific paper

10.1016/j.geomphys.2008.08.001

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.

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