symplectic dual pair

A kind of correspondence space between Poisson manifolds.

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

$(X_1, \pi_1) \stackrel{i_1}{\leftarrow} (Z,\omega) \stackrel{i_2}{\to} (X_2, \pi_2)$

such that for all $f_1 \in C^\infty(X_1)$ and $f_2 \in C^\infty(X_2)$ the Poisson bracket on $Z$ vanishes:

$\{i_1^\ast f_1, i_2^\ast f_2\}_{\omega} = 0
\,.$

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

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

and

- 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

Other

- 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

- Klaas Landsman,
*Functorial quantization and the Guillemin-Sternberg conjecture*, Proc. Bialowieza 2002 (arXiv:math-ph/0307059)

Last revised on June 26, 2020 at 10:39:55. See the history of this page for a list of all contributions to it.