On the Riemann-Lie Algebras and Riemann-Poisson Lie Groups
نویسندگان
چکیده
A Riemann-Lie algebra is a Lie algebra G such that its dual G∗ carries a Riemannian metric compatible (in the sense introduced by the author in C. R. Acad. Sci. Paris, t. 333, Série I, (2001) 763–768) with the canonical linear Poisson structure of G∗ . The notion of Riemann-Lie algebra has its origins in the study, by the author, of Riemann-Poisson manifolds (see Differential Geometry and its Applications, Vol. 20, Issue 3(2004), 279–291). In this paper, we show that, for a Lie group G , its Lie algebra G carries a structure of Riemann-Lie algebra iff G carries a flat left-invariant Riemannian metric. We use this characterization to construct examples of Riemann-Poisson Lie groups (a Riemann-Poisson Lie group is a Poisson Lie group endowed with a left-invariant Riemannian metric compatible with the Poisson structure).
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