Decomposition of (Co)isotropic Relations


We identify 13 isomorphism classes of indecomposable coisotropic relations between Poisson vector spaces and show that every coisotropic relation between finite-dimensional Poisson vector spaces may be decomposed as a direct sum of multiples of these indecomposables. We also find a list of 13 invariants, each of which is the dimension of a space constructed from the relation, such that the 13-vector of multiplicities and the 13-vector of invariants are related by an invertible matrix over \({\mathbb{Z}}\). It turns out to be simpler to do the analysis above for isotropic relations between presymplectic vector spaces. The coisotropic/Poisson case then follows by a simple duality argument.

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Correspondence to Alan Weinstein.

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Dedicated to the memory of Louis Boutet de Monvel

Jonathan Lorand: Research partially supported by a scholarship of the Anna and Hans Kägi Foundation.

Alan Weinstein: Research partially supported by UC Berkeley Committee on Research.

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Lorand, J., Weinstein, A. Decomposition of (Co)isotropic Relations. Lett Math Phys 106, 1837–1847 (2016).

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Mathematics Subject Classification

  • Primary 18B10
  • Secondary 53D99


  • linear relation
  • duality
  • symplectic vector space
  • coisotropic relation