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Showing papers on "Lie group published in 1971"


Book
01 Jun 1971
TL;DR: Foundations of Differentiable Manifolds and Lie Groups as discussed by the authors provides a clear, detailed, and careful development of the basic facts on manifold theory and Lie groups, including differentiable manifolds, tensors and differentiable forms.
Abstract: Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. It includes differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provides a proof of the de Rham theorem via sheaf cohomology theory, and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find this beginning graduate-level text extremely useful.

1,992 citations


Journal ArticleDOI
TL;DR: In this paper, a combination of the methods of the formal calculus of variations with those of Lie's group theory is presented, which is not new; Hamel and Herglotz for special finite groups, Lorentz and his pupils (for instance Fokker, Weyl and Klein)1 for special infinite groups.
Abstract: The problems in variation here concerned are such as to admit a continuous group (in Lie's sense); the conclusions that emerge for the corresponding differential equations find their most general expression in the theorems formulated in Section I and proved in following sections. Concerning these differential equations that arise from problems of variation, far more precise statements can be made than about arbitrary differential equations admitting of a group, which are the subject of Lie's researches. What is to follow, therefore, represents a combination of the methods of the formal calculus of variations with those of Lie's group theory. For special groups and problems in variation, this combination of methods is not new; I may cite Hamel and Herglotz for special finite groups, Lorentz and his pupils (for instance Fokker, Weyl and Klein)1 for special infinite groups. Especially Klein's second Note and the present developments have been mutually influenced by each other, in which regard I may ...

880 citations


Journal ArticleDOI
TL;DR: In this paper, a constructive generalization to functions defined on spheres and projective spaces of the Jackson theorems on polynomial approxima- tion is presented. But this generalization is restricted to functions on the unit ball (which is not homogeneous) in a Euclidean space.
Abstract: This paper contains constructive generalizations to functions defined on spheres and projective spaces of the Jackson theorems on polynomial approxima- tion. These results, (3.3) and (4.6), give explicit methods of constructing uniform approximations to smooth functions on these spaces by polynomials, together with error estimates based on the smoothness of the function and the degree of the poly- nomial. The general method used exploits the fact that each space considered is the orbit of some compact subgroup, G, of an orthogonal group acting on a Euclidean space. For such homogeneous spaces a general result (2.1) is proved which shows that a G-invariant linear method of polynomial approximation to continuous functions can be modified to yield a linear method which produces better approximations to k-times differentiable functions. Jackson type theorems (3.4) are also proved for functions on the unit ball (which is not homogeneous) in a Euclidean space. Introduction. We previously extended the Jackson theorems to any smooth compact submanifold M of a Euclidean space E (see (12)). The proofs of these theorems were not really constructive and made use of a rather ad hoc extension of a function from M to some ball in E. In the present paper we show how, in case M is a sphere or projective space, constructive versions of these theorems can be proved which do not require us to extend functions of M. We begin in ?1 by defining differentiability and other smoothness properties for functions on a homogeneous manifold M in terms of the homogeneous structure of M. Then in ?2 we show how the homogeneity of M together with the Jackson theorems for C(M) can be combined to prove the Jackson estimates for Ck(M). These general results are applied to prove the Jackson estimates for spheres and balls in ?3 and for projective spaces over the reals, complexes and quaternions in ?4. A general reference for the differential geometry of homogeneous spaces of compact Lie groups which we use is Helgason (4). However we have tried in ??3 and 4 to be reasonably explicit so that the reader without a background in Lie

144 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present a criterion of zero homology for Holder functions with respect to a dynamical system, as well as some consequences of this criterion and a generalization for functions taking their values in a Lie group.
Abstract: Suppose that a Y-system Tt(Tk) acts on a manifold Mn. We present a criterion of zero homology for Holder functions with respect to this dynamical system, as well as some consequences of this criterion and a generalization for functions taking their values in a Lie group.

121 citations



Journal ArticleDOI
TL;DR: In this paper, it was shown that if α denotes an n × n antisymmetric matrix of operators αpq,p,q = 1, 2, …, n, which satisfy the commutation relations characteristic of the Lie algebra of SO(n), then α satisfies an nth degree polynomial identity.
Abstract: It is shown that if α denotes an n × n antisymmetric matrix of operators αpq,p,q = 1, 2, …, n, which satisfy the commutation relations characteristic of the Lie algebra of SO(n), then α satisfies an nth degree polynomial identity. A method is presented for determining the form of this polynomial for any value of n. An indication is given of the simple significance of this identity with regard to the problem of resolving an arbitrary n‐vector operator into n components, each of which is a vector shift operator for the invariants of the SO(n) Lie algebra.

96 citations



Journal ArticleDOI
TL;DR: In this paper, it was shown that for any analytic mapping from a connected analytic manifold M to an analytic manifold V, the inverse image of a point in IV is either the whole of M or has measure zero in V. This is easily proved by taking iVl to be an open subset of EP' and using induction on m, and Fubini's theorem.

72 citations


Journal ArticleDOI
TL;DR: In this paper, Canonical sets of cyclic vectors for principal series representations of semisimple Lie groups having faithful representations are found, which are used to obtain estimates for the number of irreducible subrepresentations of a principal series representation.
Abstract: Canonical sets of cyclic vectors for principal series representations of semisimple Lie groups having faithful representations are found. These cyclic vectors are used to obtain estimates for the number of irreducible subrepresentations of a principal series representations. The results are used to prove irreducibility for the full principal series of complex semisimple Lie groups and for SL(2n + 1, R), n > 1.

48 citations




Journal ArticleDOI
TL;DR: In this paper, the Fourier inversion formula is used to define analytic functions for the left regular representation of a simply-connected nilpotent Lie group, with complexification.
Abstract: . Let G be a simply-connected nilpotent Lie group, with complexificationGc. The functions on G which are analytic vectors for the left regular representation ofG on L2(G) are determined in this paper, via a dual characterization in terms of theiranalytic continuation to Gc, and by properties of their L2 Fourier transforms. Theanalytic continuation of these functions is shown to be given by the Fourier inversionformula. An explicit construction is given for a dense space of entire vectors for theleft regular representation. In the case G = R this furnishes a group-theoretic settingfor results of Paley and Wiener concerning functions holomorphic in a strip. Introduction. Let G be a simply-connected nilpotent Lie group. Since G is aseparable, type I, unimodular group [1], there exists by the general Planchereltheorem [3] a unique Borel measure p. on the space G of equivalence classes ofirreducible unitary representations of G with the following properties:(i) (Harmonic analysis). Fix a p. measurable cross-section C -*■ n( from G toconcrete irreducible unitary representations (ir( e £). Then a function/e L2(G) has aFourier transform/defined ii-a.e. on G, such that/(£) is a Hilbert-Schmidt operatoron the space ^(-n*-) of tt(. The map f ->/(i) is a p-measurable field of operators,

Journal ArticleDOI
TL;DR: In this article, the authors propose a representation theory for complex semi-simple Lie groups, where the dual object has not yet been completely determined, except in the case of SL(2, C) and possibly SL(n, C), for n> 2.
Abstract: Introduction. Although the work of Harish-Chandra yields, the Plancherel formula for complex semi-simple Lie groups, the (unitary) representation theory of such groups remains incomplete in the sense that the dual object has not yet been completely determined, except in the case of SL(2, C)and possibly SL (n, C) for n> 2. The suggested constituents of the dual object fall into various series of representations which one refers to as pritncipal series or complementary series.


Journal ArticleDOI
TL;DR: In this article, a method for determining the Hamiltonians of quantum-mechanical systems having a given Lie algebra as their spectrum-generating algebra (SGA) is introduced.
Abstract: A method is introduced to determine the HamiltoniansH of the quantum-mechanical systems having a given Lie algebra as their spectrum-generating algebra (SGA). Application of this method to the particular case in which the SGA is assumed to be the Lie algebraSO2,1 allows under several assumptions the determination of a general solutionH, for a system of two spinless particles, having this algebra as its SGA, and possessing both a discrete and a continuous spectrum. Velocity-dependent potentials have been included, the only restrictions imposed onH being that it be at most quadratic in the momentum, and rotationally and time-reversal invariant. The potential obtained as the general solution contains as an essential part a term which asymptotically is Coulomb-like and is more general than those which have been obtained previously in the same context. The requirement that the potential be angular-momentum independent reduces the possible Hamiltonian to that of the hydrogen atom, with an extra cubic force, which has already been solved. In the case in which a purely discrete spectrum is allowed, the general answer is not given, but still a sufficiently general Hamiltonian is obtained withSO2,1 as its SGA, that includes as a particular case the known harmonic oscillator with an extra cubic force.


Journal ArticleDOI
TL;DR: In this article, the question of whether a homomorphism can be lifted to a lattice for the case that the lattice is in a Lie group of type T is investigated.
Abstract: Let and be simply connected Lie groups, and let be a lattice in . In the present article we investigate the question whether the homomorphism can be lifted to a homomorphism for the case that or is a Lie group of type . Incidentally we prove some of the properties of lattices in such groups.Bibliography: 13 items.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every Riemannian space admitting a multiply transitive Lie algebra of Killing fields is locally isometric to a homogeneous Riemanian space.
Abstract: The following new results are proved: (1) Every Riemannian space admitting a multiply transitive Lie algebra of Killing fields is locally isometric to a homogeneous Riemannian space. (2) For every closed connceted subgroupH 0 of the invariance group of a non degenerate quadratic form a homogeneous Riemannian space exists whose isotropy group containsH 0. (3) Necessary and sufficient conditions are derived for a Lie algebra $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{G} $$ to have a realization as multiply transitive Killing fields. These conditions are constructire in the sense that, for a given linear connected isotropy group, Lie algebras $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{G} $$ can be calculated algebraically, (4) It is shown how the Riemann tensor of a bomogencous space and its covariant derivatives can be expressed in terms of the metric at one point and the structure constants of the Lie algebra of Killing fields.

Journal ArticleDOI
TL;DR: The definition and classification of classical relativistic particles requires the classification of certain invariant tensor fields on the inhomogeneous Lorentz group as mentioned in this paper, and the entire 10-parameter set is exhibited.
Abstract: The definition and classification of classical relativistic particles requires the classification of certain invariant tensor fields on the inhomogeneous Lorentz group. The entire 10-parameter set is exhibited. At the same time, a much larger class of Lie groups is treated. The connection with particles will be presented in the succeeding article.


Journal ArticleDOI
TL;DR: In this article, the transitive actions of compact connected Lie groups on certain spaces X which are not spheres, whose dimension is not too small and whose rational cohomology algebra is an exterior algebra on homogeneous generators of odd degree are investigated.
Abstract: In this paper we investigate transitive actions of compact connected Lie groups on certain spaces X which are not spheres, whose dimension is not too small and whose rational cohomology algebra is an exterior algebra on homogeneous generators of odd degree. In case X is a simply connected classical group, a 3-connected real or a 5-connected complex or a quaternionic Stiefel manifold, we obtain (in principle) the classification of the transitive actions on X up to equivariant homeomorphism.

Journal ArticleDOI
TL;DR: Sufficient conditions are found for a biinvariant operator on a compact Lie group to be bounded on L(p), 1 < p < infinity, and an analog to the familiar relationship between differentiation and multiplication under the Fourier transform is used.
Abstract: Sufficient conditions are found for a biinvariant operator on a compact Lie group to be bounded on Lp, 1 < p < ∞. The proof uses properties of g-functions on such a group, and an analog to the familiar relationship between differentiation and multiplication under the Fourier transform.


Journal ArticleDOI
TL;DR: The formal moduli for one-parameter formal Lie groups were constructed by Lubin and Tate (1) by using Lazard's methods as mentioned in this paper, and they were proved to be the same for higher dimensional formal groups.
Abstract: The formal moduli for one-parameter formal Lie groups was constructed by Lubin and Tate (1) by using Lazard’s methods. The aime of this paper is to prove the existence of the formal moduli for higher dimensional formal groups.

Journal ArticleDOI
01 Nov 1971-Topology
TL;DR: In this paper, the authors describe methods which may be applied to the study of arbitrary topological torus actions, and apply these methods to actions on a product of two odd spheres.





Journal ArticleDOI
TL;DR: The reducibility of a complete integrable linear Pfaff system to a system with constant coefficients with respect to a natural parallelism on a manifold was studied in this article.
Abstract: Completely integrable linear Pfaff systems are investigated, and some of their generalizations to manifolds M=G/Γ, where G is a Lie group and Γ is a discrete subgroup of G, are studied The reducibility of such a system to a system with constant coefficients with respect to a natural parallelism on M is considered