Classifying Linear Canonical Relations

Preprint English OPEN
Lorand, Jonathan;
(2015)
  • Subject: Mathematics - Symplectic Geometry | 15A21, 53D99
    arxiv: Mathematics::Symplectic Geometry

In this Master's thesis, we consider the problem of classifying, up to conjugation by linear symplectomorphisms, linear canonical relations (lagrangian correspondences) from a finite-dimensional symplectic vector space to itself. We give an elementary introduction to th... View more
  • References (12)
    12 references, page 1 of 2

    1 Introduction 3 1.1 Summary and Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Conventions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2 Symplectic Linear Algebra 6 2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Lagrangian splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Reduction, Witt-Artin decomposition . . . . . . . . . . . . . . . . . . . . . 11

    3 Linear Relations 13 3.1 Definitions, Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    4 Linear Canonical Relations 18 4.1 Definitions, Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    5 Coisotropic Pairs 23 5.1 General classification of coisotropic pairs . . . . . . . . . . . . . . . . . . . . 25 5.2 Elementary types and normal forms . . . . . . . . . . . . . . . . . . . . . . 30

    6 Classification of Linear Canonical Relations 39 6.1 Reduced classification problem . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    [1] Artin, E., Geometric Algebra, Interscience, New York, 1957.

    [2] Bates, S., Weinstein, A., Lectures on the geometry of quantization, Berkeley Mathematics Lecture Notes, American Mathematical Society, Providence, 1997.

    [3] Benenti, S., Tulczyjew, W., Relazioni lineari simplettiche, Memorie dell'Accademia delle Scienze di Torino 5 (1981), 71-140.

    [15] Weinstein, A., Symplectic Categories, (2009) arXiv:0911.4133.

  • Metrics
    No metrics available
Share - Bookmark