Some non-abelian phase spaces in low dimensions. (English) Zbl 1151.17309
Summary: 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 para-Kähler 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.
MSC:
| 17B81 | Applications of Lie (super)algebras to physics, etc. |
| 53D05 | Symplectic manifolds (general theory) |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
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