nLab symplectic dual pair




A kind of correspondence space between Poisson manifolds.



For (X 1,π 1)(X_1, \pi_1) and (X 2,π 2)(X_2, \pi_2) two Poisson manifolds, a symplectic dual pair between them is a correspondence symplectic manifold (Z,π Z=ω 1)(Z,\pi_Z = \omega^{-1})

(X 1,π 1)i 1(Z,ω)i 2(X 2,π 2) (X_1, \pi_1) \stackrel{i_1}{\leftarrow} (Z,\omega) \stackrel{i_2}{\to} (X_2, \pi_2)

such that for all f 1C (X 1)f_1 \in C^\infty(X_1) and f 2C (X 2)f_2 \in C^\infty(X_2) the Poisson bracket on ZZ vanishes:

{i 1 *f 1,i 2 *f 2} ω=0. \{i_1^\ast f_1, i_2^\ast f_2\}_{\omega} = 0 \,.



A Lagrangian correspondence between symplectic manifolds, regarded as Poisson manifolds, is a symplectic dual pair.

Morita Equivalence

One may also impose additional conditions on i 1i_1 and i 2i_2. Suppose i 1i_1 and i 2i_2 are Poisson and anti-Poisson maps that are complete, constant rank with connected, simply-connected fibers satisfying the symplectic orthogonality of i 1 *C (X 1)i_1^\ast C^\infty (X_1) and i 2 *C (X 2)i_2^\ast C^\infty (X_2). Then this gives the notion of Morita equivalence between X 1X_1 and X 2X_2 as Poisson manifolds.


The notion was introduced in

  • M.V. Karasev, (1989), The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds I, II Selecta Math. Soviet. 8, 213–234, 235–258.


  • Alan Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18, 523–557 (1983)

A textbook accounts are in

  • Ch. 4 Dual pairs, in: A. Cannas da Silva, Alan Weinstein, Geometric models for noncommutative algebras, Berkeley Math. Lec. Notes Series, AMS 1999, pdf
  • J.-P. Ortega, T.S. Ratiu, Momentum maps and Hamiltonian reduction, Progress in Math. 222, Birkhauser 2004


  • Paul Skerritt, Cornelia Vizman, Dual pairs for matrix Lie groups, arxiv/1805.01519

A review with an eye towards geometric quantization with codomain KK-theory (geometric quantization by push-forward) is in section 3 of

Last revised on June 26, 2020 at 14:39:55. See the history of this page for a list of all contributions to it.