Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 072, 10 pages      arXiv:1503.00169
Contribution to the Special Issue on Poisson Geometry in Mathematics and Physics

(Co)isotropic Pairs in Poisson and Presymplectic Vector Spaces

Jonathan Lorand a and Alan Weinstein b
a) Department of Mathematics, ETH Zurich, Zurich, Switzerland
b) Department of Mathematics, University of California, Berkeley, CA 94720 USA

Received March 01, 2015, in final form September 03, 2015; Published online September 10, 2015

We give two equivalent sets of invariants which classify pairs of coisotropic subspaces of finite-dimensional Poisson vector spaces. For this it is convenient to dualize; we work with pairs of isotropic subspaces of presymplectic vector spaces. We identify ten elementary types which are the building blocks of such pairs, and we write down a matrix, invertible over $\mathbb{Z}$, which takes one 10-tuple of invariants to the other.

Key words: coisotropic subspace; direct sum decomposition; Poisson vector space; presymplectic vector space.

pdf (353 kb)   tex (17 kb)


  1. Atiyah M.F., On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France 84 (1956), 307-317.
  2. Benenti S., Tulczyjew W., Symplectic linear relations, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (5) 5 (1981), 71-140.
  3. Brenner S., Endomorphism algebras of vector spaces with distinguished sets of subspaces, J. Algebra 6 (1967), 100-114.
  4. Etingof P., Golberg O., Hensel S., Liu T., Schwendner A., Vaintrob D., Yudovina E., Introduction to representation theory, Lecture notes, available at \url etingof/replect.pdf.
  5. Gel'fand I.M., Ponomarev V.A., Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space, in Hilbert Space Operators and Operator Algebras (Proc. Internat. Conf., Tihany, 1970), Colloq. Math. Soc. János Bolyai, Vol. 5, North-Holland, Amsterdam, 1972, 163-237.
  6. Gutt J., Normal forms for symplectic matrices, Port. Math. 71 (2014), 109-139, arXiv:1307.2403.
  7. Li-Bland D., Weinstein A., Selective categories and linear canonical relations, SIGMA 10 (2014), 100, 31 pages, arXiv:1401.7302.
  8. Lorand J., Classifying linear canonical relations, arXiv:1508.04568.
  9. Lorand J., Weinstein A., Coisotropic pairs, arXiv:1408.5620.
  10. Lorand J., Weinstein A., Decomposition of (co)isotropic relations, in preparation.
  11. Roman S., Advanced linear algebra, Graduate Texts in Mathematics, Vol. 135, 3rd ed., Springer, New York, 2008.
  12. Sergeichuk V.V., Classification of pairs of subspaces in spaces with scalar product, Ukrainian Math. J. 42 (1990), 487-491.
  13. Towber J., Linear relations, J. Algebra 19 (1971), 1-20.
  14. Weinstein A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), 705-727.
  15. Weinstein A., A note on the Wehrheim-Woodward category, J. Geom. Mech. 3 (2011), 507-515, arXiv:1012.0105.
  16. Weinstein A., Categories of (co)isotropic linear relations, arXiv:1503.06240.
  17. Williamson J., On the normal forms of linear canonical transformations in dynamics, Amer. J. Math. 59 (1937), 599-617.

Previous article  Next article   Contents of Volume 11 (2015)