nLab
symplectic dual pair

Contents

Contents

Idea

A kind of correspondence space between Poisson manifolds.

Definition

Definition

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 \,.

Examples

Example

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

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

Last revised on October 30, 2019 at 11:12:24. See the history of this page for a list of all contributions to it.