Showing papers on "Symplectic representation published in 1983"

Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds

Abstract: We prove the existence of star-products and of formal deformations of the Poisson Lie algebra of an arbitrary symplectic manifold Moreover, all the obstructions encountered in the step-wise construction of formal deformations are vanishing

Addendum to “on the variation in the cohomology of the symplectic form of the reduced phase space”

Abstract: Here we have used the notation (~rlX) for the inner product of the vector field X with the form a, and ,Ix(m)= (X, J(m)) for the X-component of a. In an earlier paper [4] it was shown that the push forward J,(dm) of the Liouville measure dm on M under the momentum mapping J is a piecewise polynomial measure on !.*. Moreover, in case X has isolated isolated zeros on M an explicit formula for the integral

The weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights

Abstract: (1983). The weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights. Communications in Algebra: Vol. 11, No. 12, pp. 1309-1342.

Closed forms on symplectic fibre bundles

Abstract: A bundle of symplectic manifolds is a differentiable fibre bundle F--~ E-% B whose structure group (not necessarily a Lie group) preserves a symplectic structure on F. The vertical subbundle V=Ker (TTr)_ TE carries a field of bilinear forms which we call the symplectic structure along the fibres and denote by to. Any 2-form 12 on E has a restriction to F; if this restriction is to, we call O an extension of to. In this note, we discuss the problem of finding a closed extension of the symplectic structure along the fibres. This is the first step toward finding a symplectic extension- a problem already considered in special cases in [Th] and [Wn]. The first theorem shows that the existence of a closed extension is a purely topological problem.

Illustrative examples of the symplectic group

Abstract: Simple illustrative examples are given for understanding the symplectic group. Transformation properties of the Sp(2) group are discussed in terms of the two‐dimensional geometry of circles and ellipses, two‐by‐two matrices, and in terms of a group‐theoretical language similar to that of the Pauli spin matrices. The isomorphism between the Sp(2) and SO(2,1) groups is discussed in detail. It is noted that this isomorphism is similar to that between the SU(2) and rotation groups. Simple illustrative physcial examples are given also for using the symplectic group in nuclear physics, statistical mechanics, and in high‐energy physics.

Symplectic structure, Lagrangian, and involutiveness of first integrals of the principal chiral field equation

Abstract: We deal with a form of the chiral equation, for which first integrals can be written explicitly. For these equations, we find a symplectic structure, the Lagrangian and first integrals in involution.

Deformations of Algebras Associated with a Symplectic Manifold

Abstract: It is possible to give a complete description of Classical Mechanics in terms of symplectic geometry and Poisson brackets. It is the essential of the hamiltonian formalism. In a common program with Flato, D. Sternheimer and J. Vey and other scientists (Fronsdal, Arnal, M. Cahen, S. Gutt, M. de Wilde) we have studied properties and applications of the deformations of the trivial associative algebra and of the Poisson Lie algebra associated with a symplectic manifold. Such deformations give a new approach for Quantum Mechanics; this approach has been developped in other papers ((1),(2)). In this lecture, I will give recent results concerning the existence and the equivalence of associative deformations (or *υ-products).

Normal forms generalizing action-angle coordinates for Hamiltonian actions of Lie groups

Abstract: Let (M, Ω) be a symplectic manifold on which a Lie group G acts by a Hamiltonian action. Under some restrictive assumptions, we show that there exists a symplectic diffeomorphism ψ of a G-invariant open neighbourhood U of a given G-orbit in M, onto an open subset ψ(U) of a vector bundle F*, with base space G. Explicit expressions are given for the symplectic 2-form, for the momentum map and for a Hamiltonian vector field whose Hamiltonian function is G-invariant, on the model symplectic manifold ψ(U).

The symplectic character of the three-dimensional magnetic-binary problem

Abstract: The authors' purpose is to generalize in this paper the symplectic character of the particle motions in the three-dimensional magnetic-binary problem. By adopting the classical electrodynamical formulation they find out the general form of the symplectic matrix as well as some relations between the variations of an orbit, which way be used to check the accuracy of any numerical work related to the orbits investigation.

Special properties of the irreducible representations of the proper Lorentz group

Abstract: It is shown that the finite‐ and infinite‐dimensional irreducible representations ( j0, c) of the proper Lorentz group SO(3,1) may be classified into the two categories, namely, the complex‐orthogonal and the symplectic representations; while all the integral‐j0 representations are equivalent to complex‐orthogonal ones, the remaining representations for which j0 is a half‐odd integer are symplectic in nature. This implies in particular that all the representations belonging to the complementary series and the subclass of integral‐j0 representations belonging to the principal series are equivalent to real‐orthogonal representations. The rest of the principal series of representations for which j0 is a half‐odd integer are symplectic in addition to being unitary and this in turn implies that the D j representation of SO(3) with half‐odd integral j is a subgroup of the unitary symplectic group USp(2 j+1). The infinitesimal operators for the integral‐j0 representations are constructed in a suitable basis wherein these are seen to be complex skew‐symmetric in general and real skew‐symmetric in particular for the unitary representations, exhibiting explicitly the aforementioned properties of the integral‐j0 representations. Also, by introducing a suitable real basis, the finite‐dimensional ( j0=0, c=n) representations, where n is an integer, are shown to be real‐pseudo‐orthogonal with the signature (n(n+1)/2, n(n−1)/2). In any general complex basis, these representations (0, n) are also shown to be pseudo‐unitary with the same signature (n(n+1)/2, n(n−1)/2). Further it is shown that no other finite‐dimensional irreducible representation of SO(3,1) possesses either of these two special properties.