Showing papers on "Symplectic vector space published in 1974"

Symplectic homogeneous spaces

Abstract: It is proved in this paper that for a given simply connected Lie group G with Lie algebra g, every left-invariant closed 2-form induces a symplectic homogeneous space. This fact generalizes the results in [7] and [12] that if H1l(g)= H2(g) =0, then the most general symplectic homogeneous space covers an orbit in the dual of the Lie algebra S. A one-to-one correspondence can be established between the orbit space of equivalent classes of 2-cocycles of a given Lie algebra and the set of equivalent classes of simply connected symplectic homogeneous spaces of the Lie group. Lie groups with left-invariant symplectic structure cannot be semisimple; hence such groups of dimension four have to be solvable, and connected unimodular groups with left-invariant symplectic structure are solvable [4]. 1. Symplectic manifolds. Let M be a 2n-dimensional connected differentiable manifold. A symplectic structure on M is defined by a closed differential 2form c) which is everywhere of maximal rank. Such a form is called a symplectic form of the symplectic structure defined on M. On a symplectic manifold M, a one-to-one map from the space of vector fields X(M) onto the space of linear differential forms D1(M) can be defined as follows. If X is a vector field, the map x "i(X)w (where i(X)w denotes the interior product of X with wo) is a bijective map from X(M) onto DI(M). In fact, at each point x c M, this map from TX(M) onto 7* (M) is given by the nonsingular bilinear form o) A classical theorem attributed to Darboux [11] states that for an n-dimensional manifold M with a closed 2-form w) of rank exactly p everywhere there can be introduced about every point a system of coordinates x1,*. *, xn-P, y * yP, in terms of which the local representation of cw becomes Received by the editors April 23, 1973 and, in revised form, July 10, 1973. AMS (MOS) subject classifications (1970). Primary 53C30; Secondary 57F15, 22E25.

Generating elements and defining relations for symplectic groups over certain rings

Abstract: We determine the generating elements and defining relations for symplectic groups over local rings in which each finitely generated right ideal is principal.

The development of the boson calculus for the orthogonal and symplectic groups

Abstract: introduce \"modified boson operators\" a. which satisfy in particular the Is ft o traceless condition p a a = 0 (summation), where p is the metric for P<7 P Q Spin) or 0(n) . With these operators, which behave as vectors under Spin) or 0{n) , we are able to construct manifestly traceless tensors (multivectors) of arbitrary symmetry. Furthermore, all objects to be studied may be defined in terms of these operators. In general, we find that the modified boson calculus which we develop has, for Sp{n) and 0(n) , a domain of application which not only includes that of ordinary boson operators, but is considerably larger.

On the branching theorem of the symplectic groups

Abstract: In [1], Zhelobenko introduced the concept of a Gauss decomposition ZtDZ of a topological group and gave characterizations of irreducible representations of the classical groups. In this setting, vectors of representation spaces are polynomial solutions of a system of differential equations and the problem of obtaining branching theorem with respect to a subgroup G0 is to find all polynomial solutions that are invariant under Z ∩ G0 and have dominant weight with respect to D ∩ G0

The symplectic group and parity

Abstract: It is shown that the discrete operator of space inversion is an element of the cortinuous sympletic group. (LBS)