辛几何引论（英文版）

Contents

1 Some Algebra Basics 1

1.1 Skew-Symmetric Forms 1

1.2 0rthogonality Defined by a Skew-Symmetric 2-Form 3

1.3 Symplectic Vector Spaces， Symplectic Bases 6

1.4 The Canonical Linear Representation of s/(2， k) in the Algebra of the Skew-Symmetric Forms on a Symplectic Vector Space 8

1.5 Symplectic Groups 11

1.6 Symplectic Complex Structures 16

2 Symplectic Manifolds 21

2.1 Symplectic Structures on Manifolds 21

2.2 0perators of the Algebra of Differential Forms on a Symplectic

2.3 Symplectic Coordinates 30

2.4 Hamiltonian Vector Fields and Symplectic Vector Fields 35

2.5 Poisson Brackets Under Symplectic Coordinates 44

2.6 Submanifolds of Symplectic Manifolds 48

3 Cotangent Bundles 57

3.1 Liouville Forms and Canonical Symplectic Structures on Cotangent Bundles 57

3.2 Symplectic Vector Fields on a Cotangent Bundle 61

3.3 Lagrangian Submanifolds of a Cotangent Bundle 68

4 Symplectic G-Spaces 75

4.1 Definitions and Examples 76

4.2 Hamiltonian q-Spaces and Moment Maps 79

4.3 Equivariance of Moment Maps 87

5 Poisson Marufolds 91

5.1 The Structure of a Poisson Manifold 91

5.1.1 The Schouten-Nijenhuis Bracket 91

5.2 The Leaves of a Poisson Manifold 95

5.3 Poisson Structures on the Dual of a Lie Algebra 98

6 A Graded Case 109

6.1 (0， n)-Dimensional Supermanifolds 109

6.2 (0， n)-Dimensional Symplectic Supermanifolds 114

6.3 The Canonical Symplectic Structure on TP 115

Bibliography 117