Normal forms for Poisson maps and symplectic groupoids around Poisson transversals

Area of Science:

• Differential Geometry
• Mathematical Physics
• Symplectic Geometry

Background:

• Poisson manifolds are fundamental in mathematical physics.
• Poisson transversals are key submanifolds for studying these structures.
• Understanding maps and integrations related to these transversals is crucial.

Purpose of the Study:

• To establish a normal form theorem for Poisson maps around Poisson transversals.
• To investigate the conditions for integrability of neighborhoods of Poisson transversals.
• To develop a normal form theorem for symplectic groupoids related to integrable Poisson transversals.

Main Methods:

• Proving a normal form theorem for Poisson maps.
• Analyzing the integrability of Poisson transversals and their neighborhoods.
• Developing a normal form theorem for symplectic groupoids.

Main Results:

• Poisson maps can be simplified to be transversally linear around Poisson transversals.
• A neighborhood of a Poisson transversal is integrable if and only if the transversal itself is integrable.
• Normal forms are established for symplectic groupoids over integrable Poisson transversals.

Conclusions:

• The study provides powerful tools for simplifying and analyzing Poisson manifolds.
• Results demonstrate a deep connection between the integrability of a Poisson transversal and its neighborhood.
• Applications are illustrated using examples from Lie algebra theory.