Showing papers on "Lie group published in 1968"
TL;DR: The generalized Borel-Weil theorem and Theorem 1 are proved, which show that the structure of a holomorphic line bundle such that the action of G on D lifts to the sheaf of germs of holomorphic sections of Lx, O(L,) is an infinite-dimensional Frechet space on which G acts continuously.
Abstract: Let G be a connected semisimple Lie group, K a maximal compact subgroup of G. Assume that rank K = rank G. We keep fixed a Cartan subgroup H of K; H is then also a Cartan subgroup of G. Denote the Lie algebras of G, K, H by go, fo, to, and their complexifications by g, E, b. The adjoint action of b determines a rootspace decomposition g = t) ®( ga, where A is the set of nonzero roots of the pair (g, t). A root a C A is called compact if ga C d, otherwise noncompact. The complexified tangent space of the manifold G/H at eH is naturally isomorphic to 2avEA ha. If A+ C A is a particular system of positive roots, there exists a unique G-invariant complex structure on G/H such that the space of (1,0)-tangent vectors at eH corresponds to Zany+ g¶ The manifold G/H, endowed with this complex structure, will be denoted by D. Let s = 1/2 dim fo/ o THEOREM 1. The complex manifold D is (s + l)-complete.1' 2 In the obvious manner, we identify the character group of H with a lattice A in fR*, the dual space of bR = V/-1 bo. Every X C A associates a complex line bundle LX -* D to the principal bundle H -G -. D; £, can be given the structure of a holomorphic line bundle such that the action of G on D lifts to the sheaf of germs of holomorphic sections of Lx, O(L,). According to a conjecture announced by R. P. Langlands,3 the representations of the discrete series4 of G should turn up on the \"L2-cohomology groups\" of such line bundles Lx, in analogy to Bott's generalized Borel-Weil theorem for compact Lie groups. There is some hope that an understanding of the ordinary sheaf cohomology groups of the line bundles L), may lead to information about the \"L2-cohomology groups\" and an explicit realization of the discrete series representations of G. Let Wt be the Weyl group of the pair (f, i), and pr one half of the sum of the positive compact roots. The Killing form of g0 induces an inner product (, ) on lR*. We introduce an integer-valued function Q on A: Q(,u) is the number of distinct ways in which 1A can be expressed as a-possibly empty-sum of positive noncompact roots. Observe that the action of G on O(2,) determines a Gmodule structure on the cohomology groups HI(D, 0(Lx)). By means of the generalized Borel-Weil theorem and Theorem 1 one can prove: THEOREM 2. Suppose that X C A is such that (X,a) < -b for every a C A+, where b is a suitably chosen positive constant. Then Hi(D, 0(Lx)) = 0 for i 5 s, and Hs(D, 0(Lx)) is an infinite-dimensional Frechet space on which G acts continuously. If the action of G on H3(D, 0(3x)) is restricted to K, the irreducible K-module with highest weight5' 1, u C A, occurs with multiplicity
106 citations
TL;DR: In this article, the Schrodinger equation for the motion of an electric charge in the field of a magnetic monopole is examined to see how the quantization of the interaction constant follows from the requirement of rotational invariance.
74 citations
TL;DR: In this article, it was shown that Borel measurable cross-sections exist if G is any locally compact group and M any closed subgroup which is metrizable (first countable).
Abstract: If M is a closed subgroup of a locally compact group G, we consider the problem of finding a measurable transversal for the cosets G\M — {gM: qeG}—a. measurable subset T c G which meets each coset just once. To each transversal T corresponds a unique cross-section map τ: G/M —> T c G such that π o τ — id, where π: G —> G/M is the canonical mapping. For many purposes it is important to produce reasonably well behaved cross sections for the cosets G/M, and the generality of results obtained is often limited by one's ability to prove that such cross-sections exist. It is well known that, even if G is a connected Lie group, smooth (continuous) cross-sections need not exist; however Mackey ([3], pp. 101-139) showed, using the theory of standard Borel spaces, that a Borel measurable cross section exists if G is a separable (second countable) locally compact group. In this paper topological methods, independent of the theory of standard Borel spaces, are applied to show that Borel measurable cross-sections exist if G is any locally compact group and M any closed subgroup which is metrizable (first countable). The constructions become very simple if G is separable, and give a direct proof that Borel cross-sections exist in this familiar situation.
74 citations
01 Jan 1968
TL;DR: In this paper, the authors show that the list given by Mostert (loc. cit.) for the compact case with n = 2 has some omissions and give a completed list for this case, and a brief indication of how it may be obtained when the simplifying assumption of differentiability is made.
Abstract: In [3] Paul S. Mostert classifies all topological actions of compact connected Lie groups on connected (n + l)-dimensional manifolds which have n-dimensional orbits. If one starts from the assumption of differentiability a classification is very much easier, but our results show that the list given by Mostert (loc. cit.) for the compact case with n = 2 has some omissions. In this note we therefore give a completed list for this case, and a brief indication of how it may be obtained when the simplifying assumption of differentiability is made.
72 citations
65 citations
TL;DR: In this paper, the authors studied the structure of [C]-groups and showed that these groups are far away from being semidirect products of connected locally compact groups and totally disconnected compact groups.
Abstract: Let G be a topological group and H a closed normal subgroup of G. !f G contains a closed subgroup K isomorphic to the quotient group G/H such that Hc~K={1}, the identity of G and that the map (h, k ) ~ h k is a homeomorphism of H x K onto G, then G is said to be a semidirect product of H and K, and we say that G splits over H. In recent years, a great deal has been learned about the structure of connected locally compact groups on the one hand and of compact totally disconnected groups on the other. The natural synthesis of these two seems to be locally compact groups which are compact modulo component of the identity. Let [C] denote the class of all such groups. In this paper, we are primarily concerned with study of [C]-groups. In studying the structure of [C]-groups, the chief purpose we have in mind is to see how far these groups are away from being semidirect products of connected locally compact groups and totally disconnected compact groups. Even though one can hope for such a splitting in rare instances, we are able to find a totally disconnected compact 'supplement ' to the identity component of a [C]-group. The proof of existence of such a supplement given in this paper uses the method which HOFMANN and MOSTERT [6] have developed in their investigation of splittings of topological groups over vector subgroups. As is well known, a study of [C]-groups sometimes requires aconsiderable knowledge on Lie algebras, since these groups may be approximated by Lie groups. Thus we devote the first chapter to an investigation of automorphisms of Lie algebras. Although this chapter is not an integral part of our topological investigations, it is closely related to the following chapters. The following are outlines of results in the order in which the chapters containing them occur.
43 citations
TL;DR: In this article, it is proposed that particle interactions can be described by means of simple group operations in non-compact Lie groups, and the structure of these group operations are formulated which are motivated by the study of simple models.
Abstract: It is proposed that particle interactions can be described by means of simple group operations in non-compact Lie groups. Prescriptions for the structure of these group operations are formulated which are motivated by the study of simple models, in particular of the dynamics of the hydrogen atom. If they are fulfilled we call the structure “group dynamics”.
Neglecting at first internal symmetries, a few simple models are investigated in which group dynamics is possible. The group O(3,1) turns out to describe the coupling of pseudoscalar mesons, O(4,2) that of photons to baryons quite well: The pionic decay rates of baryon resonances up to spin 19/2 and the electromagnetic form factors of the nucleons are predicted in good agreement with experiment.
The internal symmetry SU(3) is included in the O(3.1) model of the pionic coupling in the simplest way, by assuming O(3,1) × SU(3) to be the dynamical group. This gives rise to a minimal symmetry breaking of the amplitudes and relates it to the mass differences in SU(3) multiplets: The amplitude consists of a product of a SU(3) Clebsch Gordan coefficient and a universal function of the velocity of the final baryon. The way the particle masses enter the decay rates is uniquely prescribed in this approach. The agreement with experiment is excellent.
Finally, the connection of this purely algebraic approach with related ideas, like the use of infinite component wave equations, is discussed.
42 citations
TL;DR: The general theory of realizations of finite Lie groups by means of canonical transformations in classical mechanics, which has been developed in a preceding paper and already applied to the rotation group, is now applied for the Galilei group as mentioned in this paper.
Abstract: The general theory of the realizations of finite Lie groups by means of canonical transformations in classical mechanics, which has been developed in a preceding paper and already applied to the rotation group, is now applied to the Galilei group. Some complements to the general theory are introduced; in particular, a new kind of possible canonical realizations connected with the singularity surfaces of the functions Q(y),P(y),J(y) are discussed (singular realizations). In agreement with the situation encountered in quantum mechanics, the constants dρσ appearing in the fundamental Poisson bracket relations among the infinitesimal generators ({yρ,yσ}=cρστyτ+dρσ) cannot all be reduced to zero. There remains a single independent constant m, which, in the physically significant cases (m > 0), represents the mass of the system. No physical interpretation seems to be attachable to the realizations corresponding to m = 0. For m ≠ 0, two different kinds of irreducible realizations exist: one of a singular type wh...
33 citations
TL;DR: In this article, a Lie group derived earlier that describes the constants of visual perception is extended to one more dimension to take into account the nonlinear flow of an afferent volley of nerve impulses through the layers of the visual cortex.
Abstract: A basic Lie group derived earlier that describes the \"constancies\" of visual perception is extended to one more dimension to take into account the nonlinear flow of an afferent volley of nerve impulses through the layers of the visual cortex. One is then led, via the usual determination of the solutions of a Lagrange partial differential equation in terms of an associated Pfaffian system of ordinary differential equations, to a correspondence between neuron cell body, Lie group germ, and critical point of the system of ordinary differential equations governing the orbits. The local phase portraits of the latter bear a marked resemblance to one or the other of the neuron types defined by Sholl. Since \"brains are as different as faces\", the concept of structural stability plays an important role in analyzing the connectivity of the neural network. Finally, Lukasiewicz's theory of parentheses is used to obtain a graph-theoretic representation of the Jacobi identity, which then serves to explain the branching of neuronal processes (dendrites).
01 Jan 1968
TL;DR: A compendium of examples of exotic differentiable actions of compact Lie groups on homotopy spheres is given in this paper, mainly for the benefit of the participants in the Tulane Conference on Compact Transformation Groups, May 8-June 2, 1967.
Abstract: In this review article I attempt to give a compendium of examples of exotic differentiable actions of compact Lie groups on homotopy spheres. It was written mainly for the benefit of the participants in the Tulane Conference on Compact Transformation Groups, May 8-June 2, 1967.
TL;DR: In this paper, it was shown that any finite, local group scheme can be embedded into a formal Lie group with coefficients in the same field, on the same number of parameters.
Abstract: The answers to (A) and to the weaker question (B) are negative in general. However if in (B) moreover is given that N o is a commutative finite group scheme, the answer is affirmative; it is the aim of this paper to give a proof of this fact via deformation theory of finite group schemes in characteristic p > 0. As a byproduct we obtain a proof for the fact that any finite, local group scheme can be embedded into a formal Lie group with coefficients in the same field, on the same number of parameters.
TL;DR: In this paper, the transitive transitive action on the Stiefel manifold of orthonormal (n-k)-frames in the euclidean n-space has been studied.
Abstract: Introduction. Let G be a compact connected Lie group and H a closed subgroup of G. Then the coset space G/H is a smooth manifold with G acting transitively as translations. It is easy to see that the natural action of G on G/H is effective if and only if H contains no nontrivial normal subgroup of G. For a given compact homogeneous space M= G/H, one might ask whether there are any other (differentiably nonequivalent) transitive effective actions on M? And furthermore, what are all the possible nonequivalent transitive effective actions on M? In the special case that M is a sphere, the above classification problem has been completely solved by the successive efforts of Montgomery and Samelson [10], Borel [1], and Poncet [11]. The purpose of this paper is to continue, along this direction, to classify the transitive effective actions on the Stiefel manifolds. Let Vfl,k be the Stiefel manifold of orthonormal (n-k)-frames in the euclidean n-space. If we consider Vf,k as a subset of the space of n x (n k) matrices, then SO(n) acts on Vfl,k by matrix multiplication from the left and SO(n-k) acts on Vfl,k by matrix multiplication from the right. Suppose G is any compact Lie group such that SO(n) c G c SO(n) x SO(n k), then it is easy to see that G acts on V ,k transitively (and effectively in many cases). Our main result is that: "For many values of n and k, every transitive effective action on Vfl,k is differentiably equivalent to one of the above examples." Parallel results are also proved for the complex and symplectic Stiefel manifolds. Technically, the most difficult part of the proof is to show that any transitive group G on Vn k contains a simple normal subgroup G1 that already acts transitively! In the case of transitive groups on spheres, the above fact is an easy consequence of the particularly simple structure of the cohomology group of spheres [10], which is no longer available in our case. For this purpose we introduce the concept of irreducible transitive action, namely, G is said to be an irreducible transitive group on M if there is no proper normal subgroup of G that is already transitive on M. With the help of a cohomological criterion of irreducible transitivity, the uniqueness of irreducible transitive effective action on a Stiefel manifold is established
TL;DR: In this paper, the class of "elementary" representations for a complex semisimple Lie group, obtained by analytic continuation from the Gel'fand-Naĭmark "fundamental series," were studied and sufficient conditions for the irreducibility of these representations were established.
Abstract: We study the class of "elementary" representations for a complex semisimple Lie group, obtained by analytic continuation from the Gel'fand-Naĭmark "fundamental series." We establish necessary and sufficient conditions for the irreducibility of these representations. Here the term "irreducibility" is to be understood to mean both topological irreducibility and complete irreducibility in the sense of R. Godement.
TL;DR: In this paper, the possibility of constructing a relativistic spectrum-generating algebraic scheme for hadrons is re-examined, interpreting the symmetry operators in terms of local weak currents.
30 Sep 1968
TL;DR: In this paper, the real and complex representation rings RO(F4) and R(F 4) of F4, which is a simply connected compact Lie group of exceptional type F, were determined.
Abstract: Introduction The aim of this paper is to determine the real and complex representation rings RO(F4) and R(F4) of F4, which is a simply connected compact Lie group of exceptional type F. Let 3 denote the Jordan algebra of all 3-hermitian matrices over the division ring of Cayley numbers, We know that the grottp F4 is obtainedi as the automorphism group of 5. In Chapter I, we shall arrange some properties of F4 : the sttbgroups Spin(8), Spin(9), maximal torus T, the Weyl group VV and the Lie algebra &, The origin of the results of Chapter I are found in H, Freudenthal [1], however we rewrite them with some modifications. In Chapter II,
TL;DR: In this article, a tensor product of representations of Lie groups over a continuous variable such as a space coordinate is constructed, which gives realizations of the equal-time commutation relations of a quantized current density, which generates local transformations of any Lie group.
Abstract: We propose a method for constructing tensor products of representations of Lie groups over a continuous variable such as a space co-ordinate. This gives realizations of the equal-time commutation relations of a quantized current density, which generates local transformations of any Lie groupG. The operators act in a linear space furnished with a «scalar product», which however is not always nonnegative. IfG=SU2 we obtain an indefinite «metric» in the ferromagnetic representation, and ifG=Heisenberg-group we obtain the usual Fock representation, which of course has got a positive metric. Finally by a change of localization the usual free relativistic quantized boson field is obtained.
TL;DR: In this article, the symmetry group of a simple material is a Lie subgroup of the unimodular group and the classification of symmetry groups and the determination of representation formulae for objective response functions can be established by the application of the theory of Lie groups.
TL;DR: In this article, the homogeneous space M = G/H is called reductive if there exists a decomposition (subspace direct sum) with where g is the Lie algebra of G (resp. H ).
Abstract: Let G be a connected Lie group and H a closed subgroup, then the homogeneous space M = G/H is called reductive if there exists a decomposition (subspace direct sum) with where g (resp. ) is the Lie algebra of G (resp. H ); in this case the pair (g, ) is called a reductive pair .
TL;DR: In this paper, the authors define a homogeneous space of physically admissable real simple Lie groups for the symmetric pseudo-Riemannian space SOo(4,2)/SOo( 4,1).
01 Jan 1968
TL;DR: In this article, the authors compare the behavior of differentiable transformation groups with natural actions on smooth manifolds, such as euclidean spaces, spheres, homogeneous spaces, etc.
Abstract: In differentiable transformation groups, one mainly studies the geometric behavior of differentiable actions of compact Lie groups on certain given types of manifolds. Since the existence of some interesting differentiable action on a smooth manifold M is itself a non-trivial fact which is not universally enjoyed by general manifolds (i.e., not every manifold has interesting symmetries), it seems rather logical to consider firstly smooth actions on those manifolds with sufficiently rich natural symmetries, such as euclidean spaces, spheres, homogeneous spaces, etc. For manifolds with “natural actions”, a simple-minded but rather fruitful approach is to compare the behavoir of general differentiable actions with the behavior of “natural actions”. From the above viewpoint, it is quite fair to say that euclidean spaces, spheres and discs are among the best testing spaces for the study of differentiable transformation groups. For these testing spaces, linear actions are clearly the “natural actions” and the comparisons between general differentiable actions and linear actions consist of the following two complementary efforts. Namely, one tries to prove more and more resemblances between differentiable actions and linear actions on the one hand, and on the other hand one tries to construct more and more varieties of differentiable examples to see how differentiable actions may differ from linear actions.
01 Jun 1968
TL;DR: The Lie-Series can be used to solve systems of differential equations by applying differential operators to known functions or to invert systems of analytical functions as discussed by the authors, and has been applied to Kepler's problem of an undisturbed planet round the sun (two-body problem), to the study of perturbations, and to the process of obtaining the characteristics for general dynamical problems.
Abstract: Since Newton, the stumbling block in celestial mechanics has been the three-body problem Only restricted cases have yielded solutions This paper describes a device, the “Lie-Series,“that first appeared in Lie’s work on analytical transformations; Grobner has shown that they can be used to solve systems of differential equations by applying differential operators to known functions or to invert systems of analytical functions The series are applied to Kepler’s problem of an undisturbed planet round the sun (two-body problem), to the study of perturbations, and to the process of obtaining the characteristics for any general dynamical problem CONTENTS Abstract , ii DEFINITION AND PROPERTIES OF LIE-SERIES 1 APPLICATION TO KEPLER’S PROBLEM ‘7 APPLICATION TO THE PROBLEM OF PERTURBATIONS 16 APPLICATION TO THE CONSTRUCTION OF THE CHARACTERISTICS OF DYNAMICAL PROCESSES 23 ACKNOWLEDGMENT 28 References 28
TL;DR: In this article, a method was proposed to obtain a set of commutation relations for creation and annihilation operators for the orthogonal group of even dimensions, and it was shown that the irreducible representations with unique vacuum state lead to the well-known para-Fermi commutation rules.