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Showing papers on "Lie group published in 1976"


Journal ArticleDOI
TL;DR: In this paper, the author outlines what is known to the author about the Riemannian geometry of a Lie group which has been provided with a metric invariant under left translation.

1,403 citations


Book
01 Jan 1976
TL;DR: In this article, the assumptions are: Cusp forms, Eisenstein series, lemmas, and some functional equations, and the main theorem of main theorem 1.1.
Abstract: The assumptions.- Cusp forms.- Eisenstein series.- Miscellaneous lemmas.- Some functional equations.- The main theorem.

657 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that under very general conditions an analytic symplectic map can be written as a product of Lie transformations, which can be combined to form a single Lie transformation by means of the Campbell-Baker-Hausdorff theorem.
Abstract: Symplectic maps (canonical transformations) are treated from the Lie algebraic point of view using Lie series and Lie algebraic techniques. It is shown that under very general conditions an analytic symplectic map can be written as a product of Lie transformations. Under certain conditions this product of Lie transformations can be combined to form a single Lie transformation by means of the Campbell–Baker–Hausdorff theorem. This result leads to invariant functions and generalizes to several variables a classic result of Birkhoff for the case of two variables. It also provides a new approach since the connection between symplectic maps, Lie algebras, invariant functions, and Birkhoff’s work has not been previously recognized and exploited. It is expected that the results obtained will be applicable to the normal form problem in Hamiltonian mechanics, the use of the Poincare section map in stability analysis, and the behavior of magnetic field lines in a toroidal plasma device.

352 citations


Book
01 Jan 1976
TL;DR: The structure of nilpotent Lie algebras and Lie groups is described in this article, where Singular integrals on spaces of homogeneous type are used to represent Lie groups.
Abstract: Structure of nilpotent Lie algebras and Lie groups.- Nilpotent Lie algebras as tangent spaces.- Singular integrals on spaces of homogeneous type.- Applications.

138 citations


Journal ArticleDOI
TL;DR: In this article, the authors make the assumption that G contains a compact Cartan subgroup and show that the discrete series representations of a connected, semisimple Lie group can all be built up from discrete-series representations of subgroups of G. The main technique in the representation theory of compact Lie groups is to investigate the restriction of any given representation to a maximal torus.
Abstract: Let G be a connected, semisimple Lie group. In Harish-Chandra's work on the Plancherel formula, the discrete series of irreducible unitary representations [5] plays a crucial role. Roughly speaking, besides the discrete series itself, the representations which enter the Plancherel decomposition of L2(G) can all be built up from discrete series representations of subgroups of G. Among the various irreducible unitary representations, the discrete series representations are therefore of particular importance. The main technique in the representation theory of compact Lie groups is to investigate the restriction of any given representation to a maximal torus. Similarl~r to understand the structure of an irreducible representation of a noncompact semisimple Lie group, one may try to determine its restriction to a maximal compact subgroup. Blattner's conjecture predicts how the discrete series representations should break up under the action of a maximal compact subgroup. It formally resembles the formula for the multiplicity of a weight for finite-dimensional representations. A companion statement to Blattner's conjecture, analogous to the theorem of the highest weight, characterizes discrete series representations, up to infinitesimal equivalence, in terms of their restriction to a maximal compact subgroup [13]. Once and for all, we make the assumption that G contains a compact Cartan subgroup. Exactly in this situation G has a non-empty discrete series [5]. Let K be a maximal compact subgroup, with HcK. The Lie algebras of G, K, H will be written as go, ~0, bo, and their complexifications as g, [, b. A non-zero root of (g, b) is called compact or noncompact, depending on whether or not its root space lies in L We denote the sets of compact and noncompact roots by, respectively, q~ and 4"; ~ = ~c w 4" is thus the setof all non-zero roots of (g, b). The Killing form of g induces an inner product ( , ) on ib*, the space of linear functions on b which assume imaginary values on b0An element ~teib* is said to be singular if it is perpendicular to at least one ct~q~, and nonsingular otherwise. The differentials of the characters of H form a lattice A cib*, which contains ~b. Since

129 citations


BookDOI
01 Jan 1976

109 citations




Journal ArticleDOI
Jiro Sekiguchi1
TL;DR: In this paper, the radial components of the invariant differential operators and the zonal spherical functions were investigated for real, complex or quanternion unimodular groups, and it was shown that the system of differential equations on A satisfied by the Zonal spherical function has as many parameters as dim a.
Abstract: Let G be a real semisimple Lie group with finite center, and K a maximal compact subgroup of G. A zonal spherical function on the symmetric space X=G/K is an simulatneous eigeiifunction (p(x) of all the invariant differential operators on X satisfying (p(kx) = cp(x) for any x^X, k^K, and (p(eK) =1, where e is the identity element in G. By the Cartan decomposition G = KAK, (p(x) is considered as a function on A. And by the separation of variables, we obtain differential operators on A from the invariant differential operators, which are called their radial components. In this paper, we investigate the radial components of the invariant differential operators and the zonal spherical functions when G is a real, complex or quanternion unimodular group. The eigenvalues of the zonal spherical functions is parametrized by the element in a*. Therefore, the system of differential equations on A satisfied by the zonal spherical function has as many parameters as dim a. However, we can construct a new system of differential equations which admits the other parameter V. It is shown that the zonal spherical function on the real, complex or quaternion unimodular group corresponds to the case in which V = -o-, 1, 2, respectively.

90 citations


Journal ArticleDOI
01 Jan 1976
TL;DR: In this paper, it was shown that if a semisimple Lie group G is a closed subgroup such that the quotient space G/H carries finite measure, then for any finite-dimensional representation of G, each H-inuariant subspace is G-inariant.
Abstract: We prove the following theorem of Borel: If G is a semisimple Lie group, H a closed subgroup such that the quotient space G / H carriesfinite measure, then for any finite-dimensional representation of G , each H-inuariant subspace is G-inuariant. The proof depends on a consideration of measures on projective spaces. The following is a relatively elementary proof of A. Borel's "density" theorem [ l ] (cf. also [5, Chapter V]). This theorem implies, among other things, that if r is a lattice subgroup of a connected semisimple real algebraic Lie group G with no compact factors, then r is Zariski dense in G. The main idea of the following proof can be found in [2, p. 3471, but the connection with Borel's theorem escaped our notice. Following von Neumann we call a locally compact topological group G minimally almost periodic (m.a.p.) if any continuous homomorphism of G into a compact group (equivalently, compact Lie group) is trivial [3]. The outstanding example of a m.a.p. group is a connected semisimple Lie group with no compact factors, but there are also discrete m.a.p. groups. (Cf. [3] and [4]. The semisimple case follows from the fact that the image of a semisimple group in a Lie group is closed.) Note that a m.a.p. group has no proper subgroups of finite index. Also a m.a.p. group is unimodular. In fact, any homomorphism of a m.a.p. group to the reals (indeed, to any abelian group) is trivial, since the reals have enough homomorphisms into compact groups to separate points. Let V be a finite dimensional linear space. P (V) will denote the corresponding projective space. If v E V, 5 will denote the corresponding point of P(V); if W is a subspace of V, /, will designate the corresponding linear subvariety of P(V). Finite unions of linear subvarieties will be called quasilinear subvarieties. As for all algebraic subvarieties, these satisfy the descending chain condition. This leads to LEMMA1. If A is a subset of P ( V), there exists a unique minimal quasi-linear subvariety q(A) c P ( V) containing A. A set of projective transformations is either relatively compact, or it is possible to extract a sequence of transformations that converges pointwise to Received by the editors February 9, 1975. AMS (MOS) subject classifications (1970). Primary 22E40, 22D40; Secondary 28A65.

77 citations


Journal ArticleDOI
TL;DR: In this article, the basic facts required to discuss c~-functions of compact symmetric spaces from the representation-theoretic viewpoint are discussed, in principle, everything here in w 1 is contained in Garth Warner's book [6], and refer to Warner [6] and Helgason [4] for the original sources.
Abstract: We assemble the basic facts required to discuss c~-functions of compact symmetric spaces from the representation-theoretic viewpoint, in principle, everything here in w 1 is contained in Garth Warner's book [6], and we refer to Warner [6] and Helgason [4] for the original sources (of which Caftan [3] is the principal one). Fix a compact riemannian symmetric space M and let G be the largest connected group of isometries. Thus G is a compact connected Lie group with an involutive automorphism a, and M = G/K where K is an open subgroup of G~= {geG :a(g) =g}, and the riemannian metric on M derives from a positive definite invariant bilinear form on the Lie algebra of G. (~ denotes the set of all equivalence classes In] of irreducible unitary representations n of G. Given In], V~ denotes the (finite dimensional complex Hilbert) space on which n represents G. A class [n]e(~ is of class 1 relative to K if there exists

Journal ArticleDOI
TL;DR: In this article, the authors applied the Lie theory of differential equations to the equation of motion of the classical one-dimensional harmonic oscillator and found that the equation is invariant under a global Lie group of point transformations.
Abstract: Lie's theory of differential equations is applied to the equation of motion of the classical one-dimensional harmonic oscillator. The equation is found to be invariant under a global Lie group of point transformations that is shown to be SL(3, R). The physical significance of the analysis and the results is considered. It is shown that the periodicity of the motion is a local topological property of the equation, while the length of the periods depends upon global properties.

Journal ArticleDOI
TL;DR: In this article, the authors consider two variants of the Cramer-Rao inequality for estimating the parameters of canonical states, in particular, the canonical parameters of a Lie group, and show that these bounds are globally attainable only for canonical states for which there exist e¢ cient measurements or quasimeasurements.
Abstract: We consider two variants of a quantum-statistical generalization of the Cramer-Rao inequality that establishes an invariant lower bound on the mean square error of a generalized quantum measurement. The proposed complex variant of this inequality leads to a precise formulation of a generalized uncertainty principle for arbitrary states, in contrast to Helstrom’s variant [1] in which these relations are obtained only for pure states. A notion of canonical states is introduced and the lower mean square error bound is found for estimating of the parameters of canonical states, in particular, the canonical parameters of a Lie group. It is shown that these bounds are globally attainable only for canonical states for which there exist e¢ cient measurements or quasimeasurements.




Book
01 Jan 1976
TL;DR: In this paper, the authors proved a double coset theorem for finite coverings, expressing p*(K, G) o T(H, G): BH -BG as a sum of other compositions.
Abstract: AnsTRAcT. Let G be a compact Lie group with H and K arbitrary closed subgroups. Let BG, BH, BK be i-universal classifying spaces, with p(H, G): BH -BG the natural projection. Then transfer homomorphisms 7(H, G): h(BH) -+ h(BG) are defined for h an arbitrary cohomology theory. One of the basic properties of the transfer for finite coverings is a double coset formula. This paper proves a double coset theorem in the above more general context, expressing p*(K, G) o T(H, G) as a sum of other compositions. The main theorems were announced in the Bulletin of the American Mathematical Society in May 1977.

Journal ArticleDOI
TL;DR: When the bases for irreducible representations of a semisimple group are reduced according to a semi-simple subgroup, the number of functionally independent missing label operators available is just twice the total number of missing labels as mentioned in this paper.
Abstract: When the bases for irreducible representations of a semisimple group are reduced according to a semisimple subgroup the number of functionally independent missing label operators available is just twice the number of missing labels. The argument presented suggests that the result holds for any Lie group.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the parafermi and parabose algebras for f degrees of freedom correspond to the Lie algesas of SO (2+1) and of a graded version of Sp (2/R) respectively.
Abstract: The usual theory of the Lie algebras and the Lie groups rs shown to be formally extended to the cases in which the group parameters commute and/or anticommute in the most general manner. It is then proved that the parafermi and parabose algebras for f degrees of freedom correspond to the Lie algebras of SO (2/+ 1) and of a graded version of Sp (2/, R), respectively. Further the trilinear and bilinear commutation relations for a general system comprising parafermi and parabose fields are shown to coincide with the Lie commutation relations of a certain group in the above-mentioned sense. Throughout our argumentation parafermi and parabose fields can formally be treated in an analogous man­ ner. It is thus concluded that the fermi-bose similarity that is known to hold in the ordi­ nary field theory persists also in parafielcl theory.

Journal ArticleDOI
TL;DR: In this paper, the authors apply the Lie group of the heat equation to explain the phenomenon of separation of variables and show that it can be used to find particular solutions for particular solutions.

Journal ArticleDOI
TL;DR: In this article, a model of the mammalian visual system was built on the basis of the representation theory of the rotation group SO(3) and primary visual analysis was considered as an expansion of visual images in basic function series related to the group.
Abstract: A model of the mammalian visual system has been built on the basis of the representation theory of the rotation group SO(3). Primary visual analysis is considered as an expansion of visual images in basic function series related to the group. The receptive field concept is realized as some system of fragments of spherical harmonics Y m l (ϕ, θ). It is assumed that lower layers of the model visual system (retina, lateral geniculate body and layer IY of the striate cortex) are used for the pointlike description of visual images and deal with the infinitesimal representations of the rotation group. Higher layers, it is postulated, deal with the integral representations. Many physiological data supporting the model are given at the end of the paper.

Journal ArticleDOI
TL;DR: In this article, the authors introduce a class of factorisable unitary representations of Ce∞(Rn,G) with the property that the unitary operator corresponding to a function f in Ce ∞(rn,g) depends not only on f but also on the derivatives off up to a certain order.
Abstract: LetCe∞(Rn,G) denote the group of infinitely differentiable maps fromn-dimensional Euclidean space into a simply connected and connected Lie group, which have compact support. This paper introduces a class of factorisable unitary representations ofCe∞(Rn,G) with the property that the unitary operatorUf corresponding to a functionf inCe∞(Rn,G) depends not only onf, but also on the derivatives off up to a certain order. In particular these representations can not be extended to the group of all continuous functions fromRn toG with compact support.

Journal ArticleDOI
TL;DR: In this paper, it was shown that higher derivatives of Debever-Penrose spinors can be expressed in terms of lower order derivatives, and that conformal motion in vacuum with zero cosmological constant must be homothetic, unless the conformal tensor vanishes or is of type N.
Abstract: The concept of Killing spinor is analyzed in a general way by using the spinorial formalism. It is shown, among other things, that higher derivatives of Killing spinors can be expressed in terms of lower order derivatives. Conformal Killing vectors are studied in some detail in the light of spinorial analysis: Classical results are formulated in terms of spinors. A theorem on Lie derivatives of Debever–Penrose vectors is proved, and it is shown that conformal motion in vacuum with zero cosmological constant must be homothetic, unless the conformal tensor vanishes or is of type N. Our results are valid for either real or complex space–time manifolds.






Journal ArticleDOI
TL;DR: In this paper, the relativistic canonical formalism of Bakamjian and Thomas describing direct particle interactions is defined in terms of the total momentum, the center-of-mass position, and a complete set of additional intrinsic canonical variables.
Abstract: In the relativistic canonical formalism of Bakamjian and Thomas describing direct particle interactions the generators are defined in terms of the total momentum, the center-of-mass position, and a complete set of additional intrinsic canonical variables. In the interaction region of phase space the transformation linking these variables to individual particle coordinates and momenta is not determined by basic principles. In this paper canonical transformations to single-particle variables valid to order c/sup -2/ and the corresponding approximate Hamiltonians are constructed for a two-particle system; approximate many-body Hamiltonians are then constructed from the two-body ones, maintaining the Lie algebra of the Poincare group to the same order. If, and only if, the nonrelativistic limit of the potential is velocity independent (except for a possible spin-orbit interaction) it is possible to require, to order c/sup -2/, transformation properties of the position operators corresponding to the classical world-line conditions. This requirement implies restrictions on admissible canonical transformations to single-particle variables. The cluster separability condition is then automatically satisfied. In the classical limit the class of approximately relativistic Hamiltonians for spinless particles is identical with that obtained by Woodcock and Havas from expansion of an exact Poincare-invariant Fokker-type variational principle automatically satisfying the world-line conditions.more » Conversely, direct quantization of their classical Hamiltonians is shown to lead to the approximate quantum-mechanical ones resulting from the Bakamjian-Thomas theory. The relation of these results to various approximately relativistic Hamiltonians built up by several authors starting from the nonrelativistic theory is discussed, as well as their implications for phenomenological nucleon-nucleon potentials. (AIP)« less