Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

Resolution of symplectic cyclic orbifold singularities

Abstract: In this paper we present a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S^1-manifold at a regular value. As an application, we show that all isolated cyclic singularities of a symplectic orbifold admit a resolution and that pre-quantisations of symplectic orbifolds are symplectically fillable by a smooth manifold.

Complex symplectic geometry with applications to ordinary differential operators

Abstract: Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems. 1. Fundamental definitions for complex symplectic spaces, and three motivating illustrations Complex symplectic spaces, as defined below, are non-trivial generalizations of the real symplectic spaces of Lagrangian classical dynamics [AM], [MA]. Further, these complex spaces provide important algebraic structures clarifying the theory of boundary value problems of linear ordinary differential equations, and the theory of the associated self-adjoint linear operators on Hilbert spaces [AG], [DS], [NA]. These fundamental concepts are introduced in connection with three examples or motivating discussions in this first introductory section, with further technical details and applications presented in the Appendix at the end of this paper. The new algebraic results are given in the second and main section of this paper, which developes the principal theorems of the algebra of finite dimensional complex symplectic spaces and their Lagrangian subspaces. A preliminary treatment of these subjects, with full attention to the theory of self-adjoint operators, can be found in the earlier monograph of these authors [EM]. Definition 1. A complex symplectic space S is a complex linear space, with a prescribed symplectic form [:], namely a sesquilinear form (i) u, v → [u : v], S × S → C, so [c1u + c2v : w] = c1[u : w] + c2[v : w], (1.1) which is skew-Hermitian, (ii) [u : v] = −[v : u], so [u : c1v + c2w] = c̄1[u : v] + c̄2[u : w] Received by the editors August 19, 1997. 1991 Mathematics Subject Classification. Primary 34B05, 34L05; Secondary 47B25, 58F05.

Eigenvalues of the Casimir operators of the orthogonal and symplectic groups

Abstract: Eigenvalues of the Casimir operators of the orthogonal and the symplectic groups are obtained in closed and simple form by diagonalizing directly the matrices introduced by Perelomov and Popov This method unifies the treatment of the problem for the semisimple Lie groups

Formal Symplectic Groupoid of a Deformation Quantization

Abstract: We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique formal symplectic groupoid ‘with separation of variables’ over an arbitrary Kahler-Poisson manifold.