Showing papers on "Symplectic vector space published in 1985"
TL;DR: In this article, the authors define a parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J).
Abstract: Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the sphere S 2.
2,482 citations
TL;DR: In this article, a decomposition of the space of tensors which have the symmetries of the curvature of a torsionless symplectic connection into Sp (n)-irreducible components is established.
Abstract: 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 Kahler 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.
81 citations
TL;DR: In this article, it was shown that the characters of the complex irreducible characters of O(n, q) are real-valued in the split orthogonal groups of degree n over GF(q) provided that q z 1 (mod 4).
51 citations
TL;DR: In this paper, a smooth homotopy of diffeomorphisms between two symplectic manifolds is defined such that the timedependent vector field ξt defined byd/dt(φt-ξtoφT) is a globally hamiltonian vector field for allt, i.e. there exists a smooth real-valued timedependent Hamiltonian functionh(x, t) on X x [0, 1] such thatξt⌋ω=dht, whereht=h(ex,t).
Abstract: Two symplectic diffeomorphisms,φ0,φ1 of a symplectic manifold (X, ω) are said to be homologous if there exists a smooth homotopyφ1,t∋[0, 1] of symplectic diffeomorphisms between them such that the timedependent vector fieldξt defined byd/dt(φt-ξtoφt is a globally hamiltonian vector field for allt, i.e. there exists a smooth real-valued timedependent hamiltonian functionh(x, t) onX x [0, 1] such thatξt⌋ω=dht, whereht=h(x,t).
49 citations
TL;DR: Deformations admitting a unit element of a local associative algebra defined on the space of functions on a manifold as discussed by the authors have been studied in statistical mechanics (KMS conditions) and conformal symplectic geometry.
Abstract: Deformations admitting a unit element of a local associative algebra defined on the space of functions on a manifold Definition and properties of the *
f
-products and conformal symplectic geometry Deformations of a *
f
-products A theorem of rigidity Application to statistical mechanics (KMS conditions)
41 citations
26 citations
TL;DR: In this paper, the authors show that K2 can be determined in explicit and closed form as the overlap of coherent states of the symplectic group, which can then be obtained by expanding this overlap in terms of appropriate eigenstates.
Abstract: The discussion of boson realisations of symplectic algebras requires as an essential ingredient an operator K needed for the passage from the Dyson to the Holstein-Primakoff realisation. In previous papers the matrix form of K2 was derived through appropriate recursion relations. In the present analysis the authors show that K2 can be determined in explicit and closed form as the overlap of coherent states of the symplectic group. The matrix elements of K2 can then be obtained by expanding this overlap in terms of appropriate eigenstates.
14 citations
9 citations
6 citations
TL;DR: In this paper, it was shown that every one-differentiable deformation of order k of the Poisson Lie algebra of a manifold which is invariant with respect to a Lie subalgebra G of its manifold is also an invariant star-product.
4 citations
01 Jan 1985
TL;DR: In this article, a variation of the lemma of Jacobson-Morozov on the imbedding of a nonzero nilpotent element of the real symplectic algebra into the split simple three-dimensional Lie algebra is proved.
Abstract: A variation of the lemma of Jacobson–Morozov on the imbedding of a nonzero nilpotent element of the real symplectic algebra into the split simple three‐dimensional Lie algebra is proved. The proof is algorithmic and relies on our earlier work on the theory of normal forms for the real symplectic algebra.