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Poisson cohomology of SU(2)-covariant 'necklace' Poisson structures on S^{2}

Dmitry Roytenberg
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Abstract

We compute the Poisson cohomology of the one-parameter family of SU(2)-covariant Poisson structures on the homogeneous space S^{2}=CP^{1}=SU(2)/U(1), where SU(2) is endowed with its standard Poisson--Lie group structure,thus extending the result of Ginzburg \cite{Gin1} on the Bruhat--Poisson structure which is a member of this family. In particular, we compute several invariants of these structures, such as the modular class and the Liouville class. As a corollary of our computation, we deduce that these structures are nontrivial deformations of each other in the direction of the standard rotation-invariant symplectic structure on S^{2}; another corollary is that these structures do not admit smooth rescaling.

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Dmitry Roytenberg. 2003-02-04. Poisson cohomology of SU(2)-covariant 'necklace' Poisson structures on S^{2}. https://arxiv.org/abs/math/0302036

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