Symplectic modules

A symplectic module is a finite group with a regular antisymmetric form. The paper determines sufficient conditions for the invariants of the maximal isotropic subgroups (Lagrangians), and asymptotic values for a lower bound of a group which contains Lagrangians of all symplectic modules of a fixed orderp ⁿ. These results have application to the splitting fields of universal division algebras.     

Symplectic Spreads

We construct an infinite family of symplectic spreads in spaces of odd rank and characteristic.     

Symplectic Implosion

Let K be a compact Lie group. We introduce the process of symplectic implosion, which associates to every Hamiltonian K-manifold a stratified space called the imploded cross-section. It bears a resemblance to symplectic reduction, but instead of quotienting by the entire group, it cuts the symmetries down to a maximal torus of K. We examine the nature of the singularities and describe in detail the imploded cross-section of the cotangent bundle of K, which turns out to be identical to an affine variety studied by Gelfand, Popov, Vinberg, and others. Finally we show that "quantization commutes with implosion".     

Symplectic Matroids

A symplectic matroid is a collection B of k-element subsets of J = {1, 2, ..., n, 1*, 2*, ...; n*}, each of which contains not both of i and i* for every i n, and which has the additional property that for any linear ordering of J such that i j implies j* i* and i j* implies j i* for all i, j n, B has a member which dominates element-wise every other member of B. Symplectic matroids are a special case of Coxeter matroids, namely the case where the Coxeter group is the hyperoctahedral group, the group of symmetries of the n-cube. In this paper we develop the basic properties of symplectic matroids in a largely self-contained and elementary fashion. Many of these results are analogous to results for ordinary matroids (which are Coxeter matroids for the symmetric group), yet most are not generalizable to arbitrary Coxeter matroids. For example, representable symplectic matroids arise from totally isotropic subspaces of a symplectic space very similarly to the way in which representable ordinary matroids arise from a subspace of a vector space. We also examine Lagrangian matroids, which are the special case of symplectic matroids where k = n, and which are equivalent to Bouchet's symmetric matroids or 2-matroids.     

Symplectic Geometry

Book Review

Symplectic GeometryA. T. Fomenko: Advanced Studies in Contemporary Mathematics, Volume 5, Gordon and Breach Science Publishers, 1988, 387 pp     

Symplectic orthogonality spaces

Let E be a finite dimensional vector space over the Galois field GF(2). Let lin(E) denote the set of one-dimensional subspaces of E. Let Φ be the symplectic inner product on E. Consider the elements of lin(E) as vertices of a graph, two vertices being connected exactly when they are distinct and orthogonal with respect to Φ. This graph is characterized abstractly.     

Symplectic curvature tensors

In the paper, one establishes the decomposition of the space of tensors which have the symmetries of the curvature of a torsionless symplectic connection into Sp (n)-irreducible components. This leads to three interesting classes of symplectic connections: flat, Ricci flat, and similar to the Levi-Civita connections of Kähler manifolds with constant holomorphic sectional curvature (we call them connections with reducible curvature). A symplectic manifold with two transversal polarizations has a canonical symplectic connection, and we study properties that are encountered if this canonical connection belongs to the classes mentioned above. For instance, in the reducible case we can compute the Pontrjagin classes, and these will be zero if the polarizations are real, etc. If the polarizations are real and there exist points where they are either singular or nontransversal, one has residues in the sense ofLehmann [L], which should be expected to play an interesting role in symplectic geometry.     

Polar Symplectic Representations

We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case.     

Symplectic half-flat solvmanifolds

We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.     

Symplectic Difference Schemes for Hamiltonian Systems in General Symplectic Structure

We consider the construction of phase flow generating functions and symplectic difference schemes for Hamiltonian systems in general symplectic structure with variable coefficients.     

Error Analysis of the Symplectic Lanczos Method for the Symplectic Eigenvalue Problem

A rounding error analysis of the symplectic Lanczos algorithm for the symplectic eigenvalue problem is given. It is applicable when no break down occurs and shows that the restriction of preserving the symplectic structure does not destroy the characteristic feature of nonsymmetric Lanczos processes. An analog of Paige's theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. As to be expected, it follows that (under certain assumptions) the computed J-orthogonal Lanczos vectors loose J-orthogonality when some Ritz values begin to converge.     

Symplectic embeddings of polydisks

If P and P ^′ are symplectic polydisks of radii R ₁≤...≤R _n and R ₁ ^′≤...≤R _n ^′, respectively, then we prove that P symplectically embeds in P ^′ provided that C(n)R ₁≤R ₁ ^′ and C(n)R ₁...R _n ≤R ₁ ^′...R _n ^′. Up to a constant factor, these conditions are optimal.