nLab symplectic dual pair

Idea

A kind of correspondence space between Poisson manifolds.

Definition

Definition

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:

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

Examples

Example

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

Morita Equivalence

One may also impose additional conditions on $i_1$ 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.

Related concepts

• symplectic category

References

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)