Group Orbits and Regular Partitions of Poisson Manifolds

Abstract

Abstract. We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties L of Lagrangian subalgebras of reductive quadratic Lie algebras d with Poisson structures defined by Lagrangian splittings of d. In the special case of g ⊕ g, where g is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on L defined by arbitrary Lagrangian splittings of g ⊕ g. Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin-Drinfeld splittings as special cases.

Symplectic and Poisson Geometry of the Moduli Spaces of Flat Connections Over Quilted Surfaces

“…As a result, we are able to construct of a number of well known spaces including: Lu's symplectic double groupoid integrating a Poisson Lie group [32], Boalch's Fission spaces [11,12], Poisson Lie groups [17,36], and Poisson homogeneous spaces [34], among others. Our approach builds upon the results and ideas of various authors including Fock and Rosly, Boalch, and the second author [7-12, 20, 38-40].…”

“…The symplectic manifold (M red , ω) is the symplectic groupoid integrating the Lu-Yakimov Poisson structure on the homogeneous space G/C [34]. The source and target maps are s(e, f, g, c) = g, t(e, f, g, c) = f g, and the multiplication is (e , f , g , c ) · (e, f, g, c) = (ee , f f, g, c c), g = f g.…”

Courant Algebroids and Poisson Geometry

“…with the Lagrangian subalgebra l m = ran(a * m ) + ker(a m As remarked in the introduction, the Poisson structure π on M given by Theorem 5.2, in the case that g i are Lagrangian sub-algebras, is due to Lu-Yakimov [19]. The anti-diagonal is not a subalgebra unless g is Abelian.…”

“…Manin triples were introduced by Drinfeld [10], and are of fundamental importance in his theory of Poisson Lie groups and Poisson homogeneous spaces. In their paper [12], S. Evens and J.-H. Lu found that every Manin triple defines a Poisson structure on the variety X of all Lagrangian subalgebras d. If g is a complex semisimple Lie algebra, and d = g ⊕ g (where g carries the Killing form and g indicates the same Lie algebra with the opposite bilinear form), then one of the irreducible components of X is the de Concini-Procesi 'wonderful compactification' of the adjoint group G integrating g. More recently, J.-H. Lu and M. Yakimov [19] studied Poisson structures on homogeneous spaces of the form D/Q, where D is a Lie group integrating d, and Q is a closed subgroup whose Lie algebra is co-isotropic for the inner product on d. Their results show that if M is a D-manifold such that all stabilizer algebras are co-isotropic, then the Manin triple (d, g 1 , g 2 ) defines a Poisson structure on M .…”

A note on Poisson homogeneous spaces

“…See [8,9,16] for studies of certain varieties which can serve as moduli spaces of Poisson homogeneous spaces. In forthcoming papers, we will use results from this note to compute explicitly the Poisson cohomology and study their symplectic groupoids for certain examples of Poisson homogeneous spaces treated in [8,9,16]. Such examples included flag varieties of complex semi-simple groups [8] and semi-simple Riemannian symmetric spaces [10] (see Example 5.14).…”