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
鄂公網(wǎng)安備 42010302001612號 