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