Showing papers on "Symplectic vector space published in 1975"

Symplectic homogeneous spaces

Abstract: In this paper we make various remarks, mostly of a computational nature, concerning a symplectic manifold X on which a Lie group G acts as a transitive group of symplectic automorphisms. The study of such manifolds was initiated by Kostant [41 and Souriau [5] and was recently developed from a more general point of view by Chu [2]. The first part of this paper is devoted to reviewing the Kostant, Souriau, Chu results and deriving from them a generalization of the Cartan conjugacy theorem. In the second part of this paper we apply these results to Lie algebras admitting a generalized (k, p) decomposition. In this paper we make various remarks, mostly of a computational nature, concerning a symplectic manifold X on which a Lie group G acts as a transitive group of symplectic automorphisms. The study of such manifolds was initiated by Kostant [4] and Souriau [5] and was recently developed from a more general point of view by Chu [2]. For the convenience of the reader we will begin by summarizing the basic facts. 1. General facts. Let G be a Lie group and X = G/H a homogeneous space for G where H is a closed subgroup, and let ir: G G/H = X be the projection. If Q is an invariant form on X then it is clear that a = 7T*Q is a left invariant form on G which satisfies (i) tI a = 0 for all t E h where h is the Lie algebra of H; (ii) a is invariant under right multiplication by elements of H, and hence under Ad for elements of H. Conversely, it is clear that any left invariant form a on G satisfying (i) and (ii) arises from G/H. If Q2 is a symplectic form then it is clear that a left invariant vector field will satisfy tla = 0 if and only if t E h. Furthermore, since do = 0, the set of all vector fields satisfying tla = 0 forms an integrable subbundle of TG, and in particular, the left invariant ones form a subalgebra of the Lie algebra of G; let us call it ha. We have thus recovered h. Let Ha be the group generated by ha. Notice that for any t E ha we have Dto = Ida + d(QJ1) = 0 so that a is invariant under Ha. The only problem is that Ha need not be closed. Let Received by the editors May 23, 1974. AMS (MOS) subject classifications (1970). Primary 53C15, 53C30. (1)This research was partially supported by NSF Grant GP43613X. Copyriglht

Eigenvalues of the invariant operators of the orthogonal and symplectic groups

Abstract: Eigenvalues of the invariant operators of the orthogonal and symplectic groups have been obtained in closed form. All semisimple Lie groups, the unitary, orthogonal, and symplectic groups, are treated in a systematic way by modifying Perelomov and Popov’s method. The eigenvalues of the invariant operators for the orthogonal and symplectic groups are then calculated with reference to the unitary group.

A canonical form for symmetric and skew-symmetric extended symplectic modular matrices with applications to Riemann surface theory

Abstract: The (extended) symplectic modular group (Ad) i,n is the set of all 2n x 2n integer matrices M such that (MJtM = t J), MJtM = J, j = [ ], I being the n x n identity matrix. Let Sn ={M E An lI nM M} and Tn = {M E An --_I"lM tM}. We say M N if there exists K E rn such that M = KN tK. This defines an equivalence relation on each of these sets separately and we obtain a canonical form for this equivalence. We use this canoinical form to study two types of Riemann surfaces which are conformally equivalent to their conjugates and obtain characterizations of their period matrices. We also obtain characterizations of the symplectic matrices which the conformal equivalence induces on the first homology group. One type of surface dealt with is the symmetric Riemann surfaces, i.e. those surfaces which have a conjugate holomorpliic self-map of order 2. The other type of surface studied we we call pseudo-symmetric surfaces. These are the hyperelliptic surfaces with the property that the sheet interchange is the square of a conjugate holomorphic automorphism.

On holomorphic representations of symplectic groups

Abstract: In this paper we shall give a simple and concrete realization of a set of representatives of all irreducible holomorphic representations of G. This realization, which involves the G-module structure of a symmetric algebra of polynomial functions is inspired by the work of B. Kostant [1] and follows the general scheme formulated in [2]. Detailed proofs will appear elsewhere. 1. The symmetric algebra S(E *). Set E = C X2k with k>n>2\\ then G acts linearly on E by right multiplication. Let ( • , • ) denote the skew-symmetric bilinear form on E given by

The symplectic property of matrizants in Störmer's problem

Abstract: The matrizants of periodic solutions in Stormer's problem, because of their symplectic character, can be transformed by means of multiplication with constant matrices into symmetric ones. As a result the six bilinear relations between their elements, existing on account of the symplectic property, are replaced by 14 linear and simple forms. This fact is very useful in numerical integrations where these relations are used as criteria of accuracy.

Some results on symplectic matrices

Abstract: The principal results are that if A is an integral matrix such that AAT is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.