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.

A Z_2-orbifold model of the symplectic fermionic vertex operator superalgebra

Abstract: We construct an irrational C_2-cofinite vertex operator algebra associatted to a finite dimensional vector space with a nondegenerate skew-symmetric bilinear form. We also classify its equivalence classes of irreducible modules and determine its automorphism group.

The symplectic ideal and a double centraliser theorem

Abstract: We interpret a result of Oehms as a statement about the symplectic ideal. We use this result to prove a double centraliser theorem for the symplectic group acting on , where V is the natural module for the symplectic group. This result was obtained in characteristic zero by Weyl. Furthermore, we use this to extend to arbitrary connected reductive groups G with simply connected derived group the earlier result of the author that the algebra K[G] of infinitesimal invariants in the algebra of regular functions on G is a unique factorisation domain.

On the structure of homogeneous symplectic varieties of complete intersection

Abstract: A normal complex algebraic variety X is called a symplectic variety if its regular locus Xreg admits a holomorphic symplectic 2-form ω such that it extends to a holomorphic 2-form on a resolution f : X → X . Affine symplectic varieties are constructed in various ways such as nilpotent orbit closures of a semisimple complex Lie algebra (cf. [CM]), Slodowy slices to nilpotent orbits (cf.[Sl]) or symplectic reductions of holomorphic symplectic manifolds with Hamiltonian actions. Usually these examples come up with C-actions. In this article we shall study a 2n-dimensional affine symplectic variety X ⊂ C defined as a complete intersection of r homogeneous polynomials fi(z1, ..., z2n+r) (1 ≤ i ≤ r). Here we assume that weights of all coordinates are 1: wt(z1) = ... = wt(z2n+r) = 1. The C -action on C induces a C-action on X . We also assume that the symplectic form ω is homogeneous with respect to thisC-action. Namely, for some integer l, we have tω = t ·ω where t ∈ C. The integer l is called the weight of ω and is denoted by wt(ω). A main result (Theorem 2) is that such an X is isomorphic (as a Cvariety) to a nilpotent orbit closure Ō of a semisimple complex Lie algebra and ω corresponds to the Kostant-Kirillov form on O. A nilpotent orbit closure does not generally have complete intersection singularities and the orbit that appears in Theorem 2 should be very restricted one. We conjecture that such a nilpotent orbit closure would be the nilpotent variety of a semisimple complex Lie algebra. As a particular case we consider a symplectic hypersurface, i.e. r = 1. As is studied in [LNSV] we have very few examples of quasihomogeneous symplectic hypersurfaces. In our homogeneous case it exists only when n = 1 and X must be isomorphic to an A1-surface singularity (Theorem 3).

Geometric Structures on Spaces of Weighted Submanifolds

Abstract: In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,!), we construct a weak symplectic structure on each leaf Iw of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangian submani- folds is equivalent to a heuristic weak symplectic structure of Weinstein (Adv. Math. 82 (1990), 133-159). When the weightings are positive, these symplectic spaces are symplec- tomorphic to reductions of a weak symplectic structure of Donaldson (Asian J. Math. 3 (1999), 1-15) on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplec- tomorphism of each leaf Iw consisting of positive weighted isotropic submanifolds onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space Iw can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.