Skip to main content

Reduction

  • Chapter
Poisson Structures

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 347))

  • 3154 Accesses

Abstract

We explain in this chapter a few more advanced constructions, which are related to the general concept of reduction. Roughly speaking, reduction means that the object under study (here a Poisson structure), is replaced by an object of the same type, but on a manifold of smaller dimension, in classical terms “with less degrees of freedom”. Poisson reduction deals with Poisson structures on quotients of Poisson manifolds or of coisotropic submanifolds of Poisson manifolds. Poisson–Dirac reduction is concerned with Poisson structures on submanifolds, which are not necessarily Poisson submanifolds. As examples and applications, we consider the transverse Poisson structure to an arbitrary symplectic leaf of a Poisson manifold, fixed point sets and quotient spaces of groups acting on Poisson varieties/manifolds and the reduced space of the momentum map, associated to the action. We also describe each of the constructions in the corresponding algebraic context.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P. (2013). Reduction. In: Poisson Structures. Grundlehren der mathematischen Wissenschaften, vol 347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31090-4_5

Download citation

Publish with us

Policies and ethics