Communications in Mathematical Physics

, Volume 283, Issue 3, pp 729–748 | Cite as

Group Orbits and Regular Partitions of Poisson Manifolds

  • Jiang-Hua Lu
  • Milen YakimovEmail author


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 \({\mathcal {L}}\) of Lagrangian subalgebras of reductive quadratic Lie algebras \({\mathfrak {d}}\) with Poisson structures defined by Lagrangian splittings of \({\mathfrak {d}}\) . In the special case of \({\mathfrak {g} \oplus \mathfrak {g}}\) , where \({\mathfrak {g}}\) is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on \({\mathcal {L}}\) defined by arbitrary Lagrangian splittings of \({\mathfrak {g} \oplus \mathfrak {g}}\) . Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin–Drinfeld splittings as special cases.


Poisson Structure Closed Subgroup Poisson Manifold Symplectic Leave Regular Partition 
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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Hong KongPokfulamHong Kong
  2. 2.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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