Showing papers on "Lie group published in 1970"
Book•
21 Feb 1970
TL;DR: In this article, an extension of the classical Littlewood-Paley theory in the context of symmetric diffusion semigroups is presented. But this work is restricted to the case of second order elliptic operators.
Abstract: This work deals with an extension of the classical Littlewood-Paley theory in the context of symmetric diffusion semigroups. In this general setting there are applications to a variety of problems, such as those arising in the study of the expansions coming from second order elliptic operators. A review of background material in Lie groups and martingale theory is included to make the monograph more accessible to the student.
1,077 citations
195 citations
88 citations
TL;DR: The group is extended via prolongation of its Lie derivatives to the complicated orbits characteristic of the contours recognizable in higher form vision and it is argued that perception consists of an exact sequence of prolonged Lie derivatives acting on the corresponding differential form.
Abstract: A Basic Lie group was derived in an earlier article that described the mathematical form of the visual constancies and primitive form memory. In the present article the group is extended via prolongation of its Lie derivatives to the complicated orbits characteristic of the contours recognizable in higher form vision. Arguments are given that the results of Hubel and Wiesel on areas 18 and 19 constitute electrohistological correlates of these prolongations. It is further argued that perception consists of an exact sequence of prolonged Lie derivatives acting on the corresponding differential form. Physiological arguments are juxtaposed with mathematical results to demonstrate that form memory resides in the proliferation of the neuronal arborescence, to which RNA generation during protein synthesis is incidental.
77 citations
TL;DR: In this paper, a factorization method was established for systems of second order linear q-difference equations, which correspond to irreducible representations of infinite-dimensional Lie algebras.
Abstract: A factorization method is established for systems of second order linear q-difference equations. The factorization types are shown to correspond to irreducible representations of infinite-dimensional Lie algebras. If the q-difference equations degenerate to differential equations (as q approaches 1) a Lie theory of hypergeometric and related functions is obtained in the limit. If the q-difference equations degenerate to ordinary difference equations a Lie theory of special functions of a discrete variable is obtained in the limit.
48 citations
TL;DR: In this article, the authors show how to construct invariant pseudo-Riemannian connections on suitable reductive homogeneous spaces M which make M into an Einstein manifold, where the corresponding manifold M = G/H is called a reductive pair.
Abstract: Let G be a connected Lie group and H a closed subgroup with Lie algebra such that in the Lie algebra g of G there exists a subspace m with (subspace direct sum) and In this case the corresponding manifold M = G/H is called a reductive homogeneous space and (g, ) (or ( G,H )) a reductive pair . In this paper we shall show how to construct invariant pseudo-Riemannian connections on suitable reductive homogeneous spaces M which make M into an Einstein manifold.
32 citations
Book•
01 Jan 1970
TL;DR: The theory of harmonic analysis for tempered distributions on real semisimple Lie groups has been studied in this paper, where the Fourier transform map is restricted to %(G).
Abstract: SUMMARY. Let G be a real semisimple Lie group. Harish-Chandra has defined the Schwartz space, V[G), on G. A tempered distribution on G is a continuous linear functional on RG). If the real rank of G equals one, Harish-Chandra has published a version of the Plancherel formula for I^(G) [3(k), 5241. We restrict the Fourier transform map to %(G), and we compute the image of the space V(G) [Theorem 31. This permits us to develop the theory of harmonic analysis for tempered distributions on G [Theorem 51. Summary
30 citations
TL;DR: In this article, the authors construct on the Lie group R × H3 two different families of left invariant metrics which satisfy the Einstein field equations with incoherent matter, called the Riemannian spaces M4, obtained this way, Class II and Class III universes.
Abstract: I construct on the Lie group R × H3 two different families of left‐invariant metrics which satisfy the Einstein field equations with incoherent matter, calling the Riemannian spaces M4, obtained this way, Class II and Class III universes. We discuss the geometry of these universes.
28 citations
25 citations
21 citations
TL;DR: In this article, the problem of determining arbitrary nonlinear representations of a given compact Lie group is studied with the object of constructing Lagrangians invariant under the group, and expressions for the covariant derivatives are obtained and it is shown how previous treatments based on global and on infinitesimal considerations are related.
Abstract: The problem of determining arbitrary nonlinear representations of a given compact Lie group is studied with the object of constructing Lagrangians invariant under the group. To achieve this, expressions for the covariant derivatives are obtained and it is shown how previous treatments based on global and on infinitesimal considerations are related. The noncompact case, the relationship between non‐linearity and zero‐mass particles, and the possibility of embedding the representation manifold in a higher‐dimensional space are all discussed.
Book Chapter•
01 Jan 1970
TL;DR: In this article, Arnold and Blancheton this article put it in the language of global analysis with perhaps some simplification, and the case of hydrodynamics was described.
Abstract: Introduction. Some of the most classical and important examples in mechanics are systems whose configuration space is a Lie group. The particular examples we have in mind are the rigid body (on the Lie group SO(3» and the perfect fluid (on the Lie group of volume preserving diffeomorphisms). Most of what we have to say is classical and well known. What we do is to put it in the language of global analysis with perhaps some simplification. Our sources are mainly the papers of Arnold and Blancheton [3], [4]. The paper is divided into two parts. In the first we present the general theory. In the second we describe the case of hydrodynamics. Some connections will be made with the calculus of variations in the future. In addition, a more complete exposition of the presen t work will appear in lecture note forms shortly [11].
Journal Article•
TL;DR: In this article, a modification weyl(E, cr) of the universal enveloping algebra of the Heisenberg Lie algebra is defined and the Lie algebras E E9 R in this algebra are realized.
Abstract: Some facts concerning symplectic vector spaces, their automorphism groups Spl(2n, R, E), and derivation Lie algebras spl(2n, R, E) are given. For every element R of these Lie algebras a solvable Lie group exp(RR) x E x R is constructed, which is nilpotent iff R is nilpotent. We calculate the Lie algebras f~R (B E of these groups, all of which contain the Heisenberg Lie algebra. Automorphism groups and derivation Lie algebras of RR fl3 E fl3 R, and faithful finite dimensional representations of them together with the corresponding representations of exp(RR) x E x R are given. In Part II a modification weyl(E, cr) of the universal enveloping algebra of the Heisenberg Lie algebra is defined. We realize the Lie algebras E E9 R in this algebra. Finally some automorphisms and derivations of are constructed by means of the adjoint representation of weyl(E, ~). Attention is given to the case of the harmonic oscillator and especially to the free nonrelativistic particle whose group is nilpotent. RESUME. Quelques qualites concernant des espaces vectoriels symplectiques, leurs groupes d’automorphismes Spl(2n, R, E) et leurs algèbres de Lie des derivations sont discutes. Pour chaque element R d’une telle algèbre de Lie, on construit un groupe de Lie solvable, exp(RR) x E x R, qui est nilpotent si et seulement si R est nilpotent. On calcule les algèbres de Lie RR © E p R de ces groupes qui contiennent tous 1’algebre de Lie d’Heisenberg. On donne les groupes d’automorphismes et les algèbres de ANN. POINCARÉ, w-XIl1-2 8
TL;DR: In this article, a potential theory canonically equivalent to the Lee model in all sectors is deduced with algebraic techniques, for the Vθ sector this potential is free of disconnected graph difficulties and so is soluble in closed form.
Abstract: A potential theory canonically equivalent to the Lee model in all sectors is deduced with algebraic techniques. For the Vθ sector this potential is free of disconnected graph difficulties and so is soluble in closed form. For higher sectors the problem can be reduced to Fredholm equations.
TL;DR: In this article, the authors give an account of some of the techniques which lie at the foundation of solvable Lie group theory, including integer semisimple splitting, and show how their apparent differences are superficial.
TL;DR: The Lie algebra defined by the currents in the Sugawara model is defined in a way that is natural from the point of view of Lie transformation theory and differential geometry as discussed by the authors.
Abstract: The Lie algebra defined by the currents in the Sugawara model is defined in a way that is natural from the point of view of Lie transformation theory and differential geometry. Previous remarks that the Sugawara model is associated with a field‐theoretical dynamical system on a Lie group manifold are made more precise and presented in a differential geometric setting.
Dissertation•
01 Jan 1970
TL;DR: In this paper, Segal et al. studied intertwining sesquilinear forms on the spaces of C -vectors of two representations of a Lie group in a Banach space and a Hilbert space respectively.
Abstract: First we study intertwining sesquilinear forms on the spaces of C -vectors of two representations of a Lie group in a Banach space and a Hilbert space respectively. Using regularity methods such forms are identified with certain closed densely defined intertwining operators for the representations. This gives new criteria for irreducibility and equivalence of unitary representations. The results are applied to study families of representations having a common space of Cw-vectors, and the theory is illustrated by some examples. Secondly we introduce C" -systems of imprimitivity for a unitary representation of a Lie group. The usual projection valued measure is replaced by a measure whose values are (possibly unbounded) positive operators. A 00 C -system of imprimitivity gives rise to an induced representation, and we give a complete classification of such systems. The result is a generalization of Mackey's imprimitivity theorem, and the proof is based on the regularity of a certain sesquilinear form on the space of 00 C -vectors for the representation. Thesis supervisor: Irving E. Segal Title: Professor of Mathematics
TL;DR: In this paper, it was shown that the problem of determining relativistic pion transition amplitudes may be reduced to the study of unitary representations of the non-compact dynamical group (SO(3, 1), where matrix elements of physical observables form a closed algebra which is identical with the Lie algebra of this group.
Abstract: Making use of rather general dynamical assumptions, it is proven that the problem of determining relativistic pion transition amplitudes may be completely reduced to the study of unitary representations of the noncompact dynamical group $SO(3, 1)\ensuremath{\bigotimes}SO(4, 3)$, since matrix elements of physical observables are shown to form a closed algebra which is identical with the Lie algebra of this group. These assumptions are Lorentz and isospin invariance of strong interactions, the Lehmann-Symanzik-Zimmermann reduction technique, an effective-interaction Lagrangian or partial conservation of axial-vector current, the usual equal-time commutator algebra between axial charges, and the absence of exotic states. The connection with the dynamical group approach previously proposed by Barut is discussed.
TL;DR: The non-linear realizations of compact connected Lie groups are considered mainly from the point of view of algebraic topology in this article, where all homogeneous spaces of the groupS U(2) are listed, the construction of a few nonlinear realisations of S U (2) is given, and the orbit structure of linear and non linear realizations are discussed.
Abstract: The non-linear realizations of compact connected Lie groups are considered mainly from the point of view of algebraic topology. In particular, all homogeneous spaces of the groupS U(2) are listed, the construction of a few non-linear realizations ofS U(2) is given and the orbit structure of linear and non-linear realizations are discussed.
TL;DR: In this article, it was shown that the existence of primitive roots of unity in an H-manifold is independent of the choice of the map m in the sense that m is a map such that mr(x, e) rm(e,x) ==x for all xC M.
Abstract: Let (M be a group with unit element e. An element g C (M is a k-th root of unity, for some integer ki ? 2, if gk e. If, in addition, gi j e for all j 1,2, 2 , Ic-i, then g is a primitive k-th root. The purpose of this paper is to show that, when (M is a compact connected Lie group, the structure of primitive roots of unity is, to a considerable extent, determined by the topology of (M and hence is independent of the algebraic structure on (M. Our approach will be, instead of considering only Lie groups, to study a larger class of objects. By an H-manifold we shall mean a triple (M. m, e) where M is a compact connected topological manifold without boundary, e C M, and m: M X M -> M is a map such that mr(x, e) rm(e,x) ==x for all xC M. Define in1: M --VI by m, (x) ==x and, for 7A>2, define Mk(X) m (X, mk-l (X) ). A primitive 1-th root of unity in the H-manifold (M, m, e) is a point x C M such that mk(x) = e but mj (x) 7& e for all j < k. We will studyprimitive roots of unity in this setting. We remark that the subject is of interest, independent of its application to Lie groups. Recent results by Hilton, Belfi [1], Morgan [6], and Zabrodsky produce many new classes of H-manifolds which are not homeomorphic to any Lie group or even, in some cases, of the same homotopy type as any Lie group. We shall show that the existence of primitive roots of unity in an Hmanifold (M, m, e) is independent of the choice of the map m in the sense that:
TL;DR: In this paper, the spectra of the Casimir invariants for the classical compact simple Lie groups are presented and it is proved that these invariants are irreducible and functionally independent.
Abstract: The spectra of the Casimir invariants for the classical compact simple Lie groups are presented. It is proved that these invariants are irreducible and functionally independent. The highest weight of a representation is determined in terms of its invariant spectrum.
TL;DR: In the space-time picture for dual-resonance models proposed by Susskind, the authors considered the groupSL2R suggested as the group of gauge transformations for this theory and analyzed disc-preserving mappings in a Lagrangian scheme.
Abstract: In the space-time picture for dual-resonance models proposed by Susskind we consider the groupSL2R suggested as the group of gauge transformations for this theory The analysis of disc-preserving mappings in a Lagrangian scheme provides the expression for the operator that generates the linear relations for the first mode and, by an extension, those found by Virasoro for all modes
TL;DR: In this article, the characters of the Poincare group P as solutions of linear differential equations on an analytic manifold in a space of distributions were derived for a large class of Lie groups.
Abstract: We calculate the characters of the Poincare group P as solutions of differential equations in a way which is valid for a large class of Lie groups. We discuss the solutions of linear differential equations on an analytic manifold in a space of distributions. Then, we calculate the characters of SL(2, R) and the central distributions on P. Finally, we give the characters of all unitary irreducible representations of P, including mass 0, and some expressions which may be ``characters'' of nonunitary representations.
TL;DR: The notion of principal orbit type is due to D. Montogomery as mentioned in this paper, who has shown that those points which lie on principal orbits form an open dense submanifold.
TL;DR: In this article, it was shown that for no mappings ψ is not necessarily true that the sum ψ(ξ1) + ψ (ξ2) + ξn+1 converges almost everywhere.
Abstract: We consider products of independent random variables ξ1ξ2...ξn, 1≤n < ∞, taking values in an arbitrary compact Lie group. In some neighborhood of the identity let the coordinates of the group be given by a mapping of G into a neighborhood of the zero of Rs, where s is the dimension of the group. It is shown that for no mappings ψ is it necessarily true that the sum ψ(ξ1) + ψ(ξ2) + ... converges almost everywhere if the product ξ1ξ2...ξn converges almost everywhere. Nevertheless it is established that there exist elements αn of G such that for ξn' = αn-1ξnαn+1 the sum ψ(ξ1') + ... + ψ(ξn') + ... and the product ξ1ξ2...ξn are both convergent almost everywhere or else neither of them has this property. Bibliography: 3 items.