Poisson homogeneous spaces and Lie algebroids associated to Poisson actions
This paper gives a detailed investigation of Poisson homogeneous spaces, motivated by \textit{V. G. Drinfeld}'s results [Theor. Math. Phys. 95, 524-525 (1993; Zbl 0852.22018); translation from Teor. Mat. Fiz. 95, 226-227 (1993)]. Namely, for each \(p\) belonging to a Poisson homogeneous \(G\)-space, \(P\) V.G. Drinfeld constructed a special Lie algebra \(l_p\). The author defines the main purposes of the present paper as (1) to find an invariant setting for \(l_p\) and (2) to describe some Poisson geometrical properties of \(P\) in terms of \(l_p\). A new point here is that for every Poisson manifold \(P\) with Poisson \(G\)-action (that may not be \(G\)-homogeneous), the author constructs a special Lie algebroid \(A\) over \(P\), whose kernel of anchor map at each \(p\) is (in Drinfeld's case) naturally related to \(l_p\). The created machinery is applied to studying symplectic leaves of a Poisson homogeneous \(G\)-space, Poisson cohomologies, etc. The Lie groupoid corresponding to \(A\) is also described.
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