Showing papers on "Lie group published in 1979"
TL;DR: In this article, a generalized method of dimensional reduction, applicable to theories in curved space, is described, where the extra dimensions are related to the manifold of a Lie group and the resulting theory has no cosmological constant, a well-behaved potential, and a number of arbitrary mass parameters.
1,262 citations
TL;DR: The representation theory of a group G over the field of complex numbers involves two problems: first, the construction of the irreducible representations of G; and second, the problem of expressing each suitably restricted complex valued function on G, as a linear combination (or a limit of linear combinations), of the coefficients of the IRQs as discussed by the authors.
Abstract: The representation theory of a group G over the field of complex numbers involves two problems: first, the construction of the irreducible representations of G; and second, the problem of expressing each suitably restricted complex valued function on G, as a linear combination (or a limit of linear combinations), of the coefficients of the irreducible representations. For example, if G is the additive group of real numbers mod 1 (the one-dimensional torus), one considers integrable functions on G, or what is the same thing, integrable periodic functions of period 1 on the additive group of real numbers. In this case the irreducible representations of G are given by the exponential functions x -» e, where k is an integer, and are the continuous homomorphisms from G into the multiplicative group of complex numbers. The expression of an integrable function in terms of the irreducible representations {e} is the Fourier expansion of/,
256 citations
201 citations
TL;DR: In this article, tensor products of representations of the holomorphic discrete series of a Lie group were computed for the conformal group O(4, 2) and a detailed study was done for the case of O(1, 2).
110 citations
TL;DR: In this article, it was shown that a non-commutative Lie group G belongs to the class of solvable Lie groups if and only if there exists a commutative ideal u of codimension 1 and an element b^n such that [b, X]=Λ: for every xEΞU.
Abstract: With J. Milnor [2] we consider a special class @ of solvable Lie groups. A non-commutative Lie group G belongs to @ if its Lie algebra g has the property that [x, y] is a linear combination of x and y for any elements x and y in Q. It is shown that g has this property if and only if there exist a commutative ideal u of codimension 1 and an element b^n such that [b, X]=Λ: for every xEΞU. Milnor has shown that if G G @ , then every left-invariant (positive-definite) Riemannian metric on G has negative constant sectional curvature. The simplest example is given by
85 citations
TL;DR: In this article, a theory in which the Linear Group in four dimensions GL (4, R) and its affine extension GA ( 4, R ) bear a direct relationship to the Physics of hadrons, and indirectly to that of the leptons is outlined.
83 citations
TL;DR: In this article, the accuracy of the semiclassical approximation for Green's functions is discussed and the special features of these representations due to the compactness and curvature are analysed.
Abstract: Classical and quantal dynamics on the compact simple Lie group and on a sphere of arbitrary dimensionality are considered. The accuracy of the semiclassical approximation for Green's functions is discussed. Various path integral representations of Green's functions are presented. The special features of these representations due to the compactness and curvature are analysed. Basic results of the theory of Lie algebras and Lie groups used in the main text are presented in the Appendix.
73 citations
70 citations
69 citations
01 Jan 1979
TL;DR: In this paper, the authors generalize Milnor's theorem in a different direction by finding bounds for the 2-dimensional real characteristic numbers of flat G-bundles where G is any connected semi-simple Lie group with finite center.
Abstract: I. A well-known theorem of J. Milnor [8] states that on an oriented surface of genus h any flat Sl(2,~)-bundle has Euler number of numerical value at most h-]. Here "flat" means that there exists a system of local trivializations for the bundle such that all the transition functions are constant. D. Sullivan [9] has generalized this result by finding bounds for the Euler number of a flat Sl(2n, ~)-bundle on a 2n-dimensional manifold. In this note we shall generalize Milnor's theorem in a different direction by finding bounds for the 2-dimensional real characteristic numbers of flat G-bundles where G is any connected semi-simple Lie group with finite center. By a real characteristic number we simply mean the evaluation of a real characteristic class (i.e. the pull-back under the classifying map of a class in H2(BG,~)) on a given homology class in the base and we want to estimate this number independently of the flat bundle. Actually it suffices to consider the characteristic numbers of flat bundles over surfaces (see Remark 2 following Proposition 2.2 below) and in this case our results are given by Proposition 2.2 and Theorem 4.1 below. The results depend on the particular simple description due to Guichardet and Wigner [4] which one has for 2-dimensional continuous cochains on Lie groups and I am indebted to Professor A. Guichardet for informing me about his work.
TL;DR: In this paper, the class [S] of locally compact groups G is considered, for which the algebra L to the power of 1(G) is symmetric (=Hermitian).
TL;DR: In this article, the SU(2) Yang-Mills equations are studied in compactified Minkowski space and the manifold is identified with that of the Lie group U(1) ×SU(2), and a classification is made of all principal bundles over this base space in terms of homotopy classes of mappings f:S3→S3.
Abstract: In this work the SU(2) Yang–Mills equations are studied in compactified Minkowski space. The manifold is identified with that of the Lie group U(1) ×SU(2) and a classification is made of all SU(2) principal bundles over this base space in terms of homotopy classes of mappings f:S3→S3. Invariance of gauge fields under transformation groups is defined in terms of bundle mappings and the case of invariance under SU(2) translations is shown to imply a trivial bundle structure. All solutions to the field equations invariant under U(1) ×SU(2) translations are obtained as well as all (anti‐) self‐dual solutions invariant under SU(2) translations.
TL;DR: A derivation of the group of automorphisms of the Lie group of isometries characteristic of each of the nine Bianchi types of cosmologies is presented in this article.
Abstract: A derivation of the group of automorphisms of the Lie group of isometries characteristic of each of the nine Bianchi types of cosmologies is presented.
TL;DR: In this paper, a simple form for the Plancherel measure for rank one, linear simple groups, including the normalizing constant, is given, which is a form that we use in this paper.
Abstract: In this paper we find a very explicit, simple form for the Plancherel measure for rank one, linear simple groups, including the normalizing constant.
TL;DR: The Lie superalgebraic properties of the ordinary quantum statistics are discussed in this article, where it is indicated that the algebra generated byn pairs of Fermi operator is isomorphic to the classical simple Lie algebraB n, whereas n pairs of Bose operators generate the simple Lie super algebraB(0,n).
Abstract: The Lie superalgebraical properties of the ordinary quantum statistics are discussed. It is indicated that the algebra generated byn pairs of Fermi operator is isomorphic to the classical simple Lie algebraB n , whereasn pairs of Bose operators generate the simple Lie superalgebraB(0,n). An idea of how one can introduce new classes of creation and annihilation operators that satisfy the second quantization postulates and generate other simple Lie superalgebras is given. The statistics corresponding to the Lie algebraA n is considered in more details.
TL;DR: In this article, it was proved that G admits only the "obvious" weakly almost periodic functions, i.e., the functions of a semisimple analytic group with finite center.
Abstract: IfG is a semisimple analytic group with finite center, it is proved thatG admits only the “obvious” weakly almost periodic functions. The analysis yields also an intrinsic proof of Moore's ergodicity theorem [7].
TL;DR: In this article, it is shown that a function on Rn which can be well approximated by polynomials, in the mean over Euclidean balls, is Lipschitz smooth in the usual sense.
01 Jan 1979
TL;DR: In this article, the authors give a rapid review of recent developments in nonlinear filtering and suggest a method of classification of estimation problems based on the Lie algebra generated by the operators which appear in the conditional density equation.
Abstract: We give a rapid review of some recent developments in nonlinear filtering and suggest a method of classification of estimation problems based on the Lie algebra generated by the operators which appear in the conditional density equation. This is, to some extent implicit in [1] and [2]. We then go on to study a natural class of automorphisms of this algebra thus giving a systematic method of generating problems of equivalent difficulty. Finally, we give here a new class of nonlinear filtering problems which have filtering equations which are themselves nonlinear in an essential way.
TL;DR: In this article, the authors review and develop geometrical gauging involving the sequence: Lie group/Principal Bundle, for an Internal symmetry group/Soft Group Manifold, for Non-Internal groups.
TL;DR: In this paper, the authors study the extensions of Banach space representations of a Lie group G. They introduce different spaces of 1-cohomology on G, or on its Lie algebra G, and make the connection between these spaces and the equivalence (or weak equivalence) classes of extensions.
TL;DR: In this article, a bilinear signal process driven by a Gauss-Markov process was considered in additive, white, Gaussian noise and an exact stochastic differential equation for the least squares filter was derived when the Lie algebra associated with the signal process is nilpotent.
Abstract: We consider a bilinear signal process driven by a Gauss-Markov process which is observed in additive, white, Gaussian noise. An exact stochastic differential equation for the least squares filter is derived when the Lie algebra associated with the signal process is nilpotent. It is shown that the filter is also bilinear and moreover that it satisfies an analogous nilpotency condition. Finally, some special cases and an example are discussed, indicating ways of reducing the filter dimensionality.
TL;DR: In this paper, the authors generalize the Figa-Talamanca theorem for unimodular groups to the case of connected Lie groups and show that every positive definite function p on G which vanishes at infinity is associated with the regular representation R, i.e., p(g) = (Rgϑ, ϑ) for some L2 function ϑ.
01 Mar 1979
TL;DR: In this article, the divergence of Fourier series on compact Lie groups has been shown to be polynomial in the length of a compact connected semisimple Lie group.
Abstract: ABSTRACr. The following theorem is proved: let G denote a compact connected semisimple Lie group. There exists 9 O(G) (3 9, IIIZ tIc1jdj1.lIIIP > constp NI" where III * Illp denotes the LP(G) convolutor norm and constp and aap(G) are positive constants. Results on divergence of Fourier series on compact Lie groups are deduced.
TL;DR: In this paper, it was shown that a Hilbert space over the real Clifford algebra C7 provides a mathematical framework, consistent with the structure of the usual quantum mechanical formalism, for models for unification of weak, electromagnetic and strong interactions utilizing the exceptional Lie groups.
Abstract: It is shown that a Hilbert space over the real Clifford algebra C7 provides a mathematical framework, consistent with the structure of the usual quantum mechanical formalism, for models for the unification of weak, electromagnetic and strong interactions utilizing the exceptional Lie groups. In particular, in case no further structure is assumed beyond that of C7, the group of automorphisms leaving invariant a minimal subspace acts, in the ideal generated by that subspace, as G2, and the subgroup of this group leaving one generating element (e7) fixed acts, in this ideal, as the color gauge group SU(3). A generalized phase algebra A⊇C7 is defined by the requirement that quantum mechanical states can be consistently constructed for a theory in which the smallest linear manifolds are closed over the subalgebra C(1,e7) (isomorphic to the complex field) of C7. Eight solutions are found for the generalized phase algebra, corresponding (up to an overall sign), in effect, to the use of ±e7 as imaginary unit in e...
TL;DR: In this paper, the authors studied semi-free (free off the fixed-point set) smooth actions of a compact Lie group on disks and spheres with fixed point set a disk or sphere, respectively.
Abstract: We study semi-free (=free off the fixed-point set) smooth actions of a compact Lie group G on disks and spheres with fixed-point set a disk or sphere, respectively. In dimensions ^6 and codimension Φ2 we obtain a complete classification for such actions on disks and a partial classification for spheres, together with partial results in dimension 5 or codimension 2. We show that semi-free smooth actions of G on the %-disk Dn, %^64- dim G, with fixed-point set an (n-k)-άisk, kφ2, are classified by two invariants: (1) a free orthogonal action of G on the (λ -l)-sphere Sk~λ (the representation at the fixed points) and (2) an element of the Whitehead group Wh(τro(G)).
TL;DR: In this paper, the authors established the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation and showed that consideration of only classical Lie point groups is insufficient.
TL;DR: In this paper, the authors studied the contractions of Lie groups and their representations and proved that they can be defined by deformations in Poisson algebras of symplectic manifolds on which the groups act.
Abstract: The contractions of Lie groups and Lie algebras and their representations are studied geometrically. We prove they can be defined by deformations in Poisson algebras of symplectic manifolds on which the groups act. These deformations are given by Dirac constraints which induce on C∞ functions on the deformed manifold an associative twisted product, characterizing the contracted group or its representations. We treat the contractions of SO(n) to E(n) and apply this theory to thermodynamical limits in spin systems.