Abstract
Lu has shown that any dynamical r-matrix for the pair (\(\mathfrak{g}\), \(\mathfrak{u}\)) naturally induces a Poisson homogeneous structure on G/U. She also proved that if \(\mathfrak{g}\) is complex simple, \(\mathfrak{u}\) is its Cartan subalgebra and r is quasitriangular, then this correspondence is in fact one-to-one. In this Letter we find some general conditions under which the Lu correspondence is one-to-one. Then we apply this result to describe all triangular Poisson homogeneous structures on G/U for a simple complex group G and its reductive subgroup U containing a Cartan subgroup.
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Karolinsky, E., Stolin, A. Classical Dynamical r-Matrices, Poisson Homogeneous Spaces, and Lagrangian Subalgebras. Letters in Mathematical Physics 60, 257–274 (2002). https://doi.org/10.1023/A:1016231526203
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DOI: https://doi.org/10.1023/A:1016231526203