1 Some Algebra Basics 1.1 Skew-Symmetric Forms 1.2 0rthogonality Defined by a Skew-Symmetric 2-Form 1.3 Symplectic Vector Spaces, Symplectic Bases 1.4 The Canonical Linear Representation of s/(2, k) in the Algebra of the Skew-Symmetric Forms on a Symplectic Vector Space 1.5 Symplectic Groups 1.6 Symplectic Complex Structures 2 Symplectic Manifolds 2.1 Symplectic Structures on Manifolds 2.2 0perators of the Algebra of Differential Forms on a Symplectic 2.3 Symplectic Coordinates 2.4 Hamiltonian Vector Fields and Symplectic Vector Fields 2.5 Poisson Brackets Under Symplectic Coordinates 2.6 Submanifolds of Symplectic Manifolds 3 Cotangent Bundles 3.1 Liouville Forms and Canonical Symplectic Structures on Cotangent Bundles 3.2 Symplectic Vector Fields on a Cotangent Bundle 3.3 Lagrangian Submanifolds of a Cotangent Bundle 4 Symplectic G-Spaces 4.1 Definitions and Examples 4.2 Hamiltonian q-Spaces and Moment Maps 4.3 Equivariance of Moment Maps 5 Poisson Marufolds 5.1 The Structure of a Poisson Manifold 5.1.1 The Schouten-Nijenhuis Bracket 5.2 The Leaves of a Poisson Manifold 5.3 Poisson Structures on the Dual of a Lie Algebra 6 A Graded Case 6.1 (0, n)-Dimensional Supermanifolds 6.2 (0, n)-Dimensional Symplectic Supermanifolds 6.3 The Canonical Symplectic Structure on T*P Bibliography Index
