publication . Preprint . 2015

# Classifying Linear Canonical Relations

Lorand, Jonathan;
Open Access English
• Published: 19 Aug 2015
Abstract
Comment: 46 pages, 4 figures
Subjects
arXiv: Mathematics::Symplectic Geometry
free text keywords: Mathematics - Symplectic Geometry, 15A21, 53D99
Related Organizations

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.

[16] Weinstein, A., A note on the Wehrheim-Woodward category, Journal of Geometric Mechanics 3 (2011), 507-515.

[17] Williamson, J., On the Normal Forms of Linear Canonical Transformations in Dynamics, American Journal of Mathematics 59 (1937), 599-617.

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