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.

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 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 December 13, 2017 at 05:47:37. See the history of this page for a list of all contributions to it.