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Showing papers on "Lie group published in 1985"


Journal ArticleDOI
TL;DR: A survey of the theory of Kats-Moody algebras is given in this paper, which contains a description of the connection between the infinite-dimensional Lie algebra of Kats and systems of differential equations generalizing the Korteweg-de Vries and sine-Gordon equations and integrable by the inverse scattering problem.
Abstract: The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats-Moody and systems of differential equations generalizing the Korteweg-de Vries and sine-Gordon equations and integrable by the method of the inverse scattering problem. A survey of the theory of Kats-Moody algebras is also given.

1,288 citations


Journal ArticleDOI
TL;DR: In this paper, le comportement asymptotique des solutions de □u=0 ou □=∂ t 2 −∂ 1 2...−∂ n 2 pour des conditions initiales u=0, u t =g(x) en t=0.
Abstract: On etudie le comportement asymptotique des solutions de □u=0 ou □=∂ t 2 −∂ 1 2 ...−∂ n 2 pour des conditions initiales u=0, u t =g(x) en t=0, avec g reguliere a support compact dans R n

592 citations


Journal ArticleDOI
TL;DR: It is shown that the conformal diffeomorphisms, which remain after imposing certain covariant gauge conditions for the general coordinate invariance, can be used to gauge away twice as many modes as there are gauge parameters.
Abstract: We discuss the spontaneous compactification of chiral N = 2 ten-dimensional supergravity from ten to five dimensions on S/sup 5/. Harmonic analysis on S/sup 5/ is used to compute the complete mass spectrum. Our results indicate that scalars and spinors in different SO(6) multiplets have different masses, even within the ''massless'' supermultiplet. We show that the conformal diffeomorphisms, which remain after imposing certain covariant gauge conditions for the general coordinate invariance, can be used to gauge away twice as many modes as there are gauge parameters. A doubleton multiplet of pure gauge modes is identified, and all modes in the massless supermultiplet lie at the beginning of infinite towers of modes.

590 citations


Journal ArticleDOI
TL;DR: In this article, decomposition formulas of general exponential operators in a Banach algebra and in a Lie algebra are presented that yield a basis of Monte Carlo simulation of quantum systems, and they are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena.
Abstract: Decomposition formulas of general exponential operators in a Banach algebra and in a Lie algebra are presented that yield a basis of Monte Carlo simulation of quantum systems. They are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena. A global approximation method of transient phenomena is also formulated on the basis of decomposition formulas. It is applied to the laser model as a simple example.

338 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the space of completely bounded multipliers of the Fourier algebra A(G) coincides with the space B(G), which is defined as a set of functions on G that are linear combinations of continuous positive definite functions.
Abstract: For any amenable locally compact group G, the space of multipliers MA(G) of the Fourier algebra A(G) coincides with the space B(G) of functions on G that are linear combinations of continuous positive definite functions. We prove that MA(G)\B(G) * 0 for many non-amenable connected groups. More specifically we prove that MOA(G)\B(G) * 0 for the classical complex Lie groups, and the general Lorentz groups SOO(n, 1), n > 2. MOA(G) is a certain subspace of MA(G), which we call the space of completely bounded multipliers of A(G). Unlike MA(G), the space MOA(G) has nice stability properties with respect to direct products of groups. It is known that the Fourier algebra of the free group on N generators (N 2 2) admits an unbounded approximate unit ((Pn), which is bounded in the multiplier norm. We extend this result to any closed subgroup of the general Lorentz group SOO(n, 1). Moreover we show that for these groups ((Pn) can be chosen to be bounded with respect to the MOA(G)-norm. By a duality argument we obtain that the reduced C*-algebra of every discrete subgroup of SOO(n, 1) has "the completely bounded approximation property. " In particular this property holds for C* (F2), the reduced C*-algebra of the free group on two generators. We also prove

311 citations


Journal ArticleDOI
TL;DR: In this paper, the effect of Wess-Zumino terms on nonlinear sigma models with torsion on the field manifold has been investigated in two dimensions, showing that the geometrical aspects of such models carry over completely.

298 citations


Journal ArticleDOI
TL;DR: Theorem III of this introduction is a character formula for any special unipotent representation as discussed by the authors, which can be deduced from the Kazhdan-Lusztig conjecture.
Abstract: algebraic group over R or C. In this paper, we restrict attention to C. We generalize Arthur's definition slightly (or perhaps simply make it more precise). All of the resulting representations, except for a finite set, are then unitarily induced from representations of the same kind on proper parabolic subgroups. We call the finite set remaining special unipotent representations; a precise definition will be given later (Definition 1.17). Our main result (Theorem III of this introduction) is a character formula for any special unipotent representation. Of course such a formula can be deduced from the Kazhdan-Lusztig conjecture (cf. [V3]). The advantages of Theorem III are that it is in closed form, and that it lends itself to verification of some conjectures of Arthur in [A]. So let G be a connected complex semisimple Lie group, and q its Lie algebra. Choose

218 citations


Book
23 Dec 1985
TL;DR: The Noether Theorem REDUCTION, ACTIONS OF GROUPS and ALGEBRAS: Reduction of Dynamical Systems through Regular Foliations Foliation of Symplectic Manifolds and Reduction of Hamiltonian Systems Algebra and Group Actions as discussed by the authors.
Abstract: Preface FOUNDATIONS OF MECHANICS: Introduction to Part I A Digression on Manifolds and Diffeomorphisms Construction of Q: From Observables to the Configuration Space Manifold Time and Transformations on Time A Digression on Calculus on Manifolds From Trajectories to the Linear Field Lifting to a Carrier Space: Canonical Lifting A Digression on Sub-manifolds and Smooth Maps Transformations on TQ Integrating the Dynamics on TQ: Hamiltonian and Lagrangian Formalisms From the Tangent Bundle to the Cotangent Bundle The Canonical Hamiltonian Formalism on TQ Equivalent Lagrangians and Hamiltonians Other Carrier Spaces: Action-angle Variables and the Hamilton-Jacobi Method The Noether Theorem REDUCTION, ACTIONS OF GROUPS AND ALGEBRAS: Reduction Introduction to Part II Linear Dynamical Systems: A Prelude to Reduction A Digression on Foliations and Distributions Reduction of Dynamical Systems Through Regular Foliations Foliation of Symplectic Manifolds and Reduction of Hamiltonian Systems Algebra and Group Actions A Digression on Lie Algebras and their Actions on Manifolds Actions of Lie Algebras on Symplectic Manifolds A Digression on Lie Groups and their Action on Manifolds Actions of Lie Groups on Symplectic Manifolds Parallelizable Manifolds, Dynamics on Lie Groups Examples and Applications Conclusion References Further Reading Index. .pa

206 citations


Book
01 Jan 1985

197 citations


Journal ArticleDOI
TL;DR: In the quasi-classical limit, these solutions reproduce the known ones of the classical triangle equations for the classical Lie algebras SL( N), SP( N ), O( N ) as mentioned in this paper.

195 citations


Journal ArticleDOI
TL;DR: It is shown that the three-cocycle arises when a representation of a transformation group is nonassociative, so that the Jacobi identity fails.
Abstract: It is shown that the three-cocycle arises when a representation of a transformation group is nonassociative, so that the Jacobi identity fails. A physical setting is given: When the translation group in the presence of a magnetic monopole is represented by gauge-invariant operators, a (trivial) three-cocycle occurs. Insisting that finite translations be associative leads to Dirac's monopole quantization condition. Attention is called to the possible relevance of three-cocycles in the quark model's U(6) \ensuremath{\bigotimes} U(6) algebra.

Journal ArticleDOI
TL;DR: The orbit space of a representation of a compact Lie group has a natural semialgebraic structure as mentioned in this paper, and explicit ways of finding the inequalities defining this structure can be found.
Abstract: The orbit space of a representation of a compact Lie group has a natural semialgebraic structure. We describe explicit ways of finding the inequalities defining this structure, and we give some applications.

Journal ArticleDOI
TL;DR: In this article, all possible anomalous terms in quantum gauge theory in the natural class of polynomials of differential forms were computed using the appropriate cohomological and algebraic methods, for all dimensions of spacetime and all structure groups with reductive Lie algebras.
Abstract: We compute all possible anomalous terms in quantum gauge theory in the natural class of polynomials of differential forms. By using the appropriate cohomological and algebraic methods, we do it for all dimensions of spacetime and all structure groups with reductive Lie algebras.

Journal ArticleDOI
TL;DR: In this article, it was shown that an abstract CR manifold whose structure is invariant under a transversal Lie group action can always be locally embedded in complex space as a generic submanifold.
Abstract: Part I presents results on local embedding of CR structures. We consider an abstract CR manifold whose structure is invariant under a transversal Lie group action. We show that such a manifold can always be locally embedded in complex space as a generic submanifold. The proof is based on selection of canonical coordinates and repeated use of the Newlander-Nirenberg theorem (13). When the Lie group is abelian the embedding can be given a particularly simple form. Let l~ 1 be the codimension of our submanifold (called M throughout the paper); it is then convenient to denote by n + I the dimension of the ambient complex space and by zl,..., z,, w 1 .... , w t the complex coordinates; we shall systematically write

Journal ArticleDOI
TL;DR: A REDUCE package for determining the group of Lie symmetries of an arbitrary system of partial differential equations is described in this paper, which can be used both interactively and in a batch mode.
Abstract: A REDUCE package for determining the group of Lie symmetries of an arbitrary system of partial differential equations is described. It may be used both interactively and in a batch mode. In many cases the system finds the full group completely automatically. In some other cases there are a few linear differential equations of the determining system left the solution of which cannot be found automatically at present. If it is provided by the user, the infinitesimal generators of the symmetry group are returned.

Journal ArticleDOI
TL;DR: On considere un espace symetrique de Riemann simplement connexe M, la connexion de Levi-Civita ⊇ and le fibre torseur metrique #7B-F(M)⊆J~(M), don't les fibres #7b-F x (M) pour x∈M est constitue de structures complexes orthogonales as discussed by the authors.
Abstract: On considere un espace symetrique de Riemann simplement connexe M, la connexion de Levi-Civita ⊇ et le fibre torseur metrique #7B-F(M)⊆J~(M) dont les fibres #7B-F x (M) pour x∈M est constitue de structures complexes orthogonales

Journal ArticleDOI
TL;DR: In this paper, the authors studied all naturally reductive homogeneous spaces of G when G is either semisimple of non-compact type or nilpotent and gave necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be natural reductive with respect to some transitive subgroup of G.
Abstract: The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are (i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and (ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

Journal ArticleDOI
TL;DR: In this paper, it was shown how to obtain an explicit boson realization of a sp(2d) Lie algebra for an arbitrary irrep of the SU(2) group, which is a problem of considerable physical interest.
Abstract: Holstein and Primakoff derived long ago the boson realization of a su(2) Lie algebra for an arbitrary irreducible representation (irrep) of the SU(2) group. The corresponding result for su(1,1)≅sp(2) is also well known. This raises the question of whether it is possible to obtain in an explicit, analytic, and closed form, and for any integer d, the boson realization of a sp(2d) Lie algebra for an arbitrary irrep of the Sp(2d) group, which is a problem of considerable physical interest. The case d=2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp(4). The Dyson realization is well known, and the passage to bosons satisfying the appropriate Hermiticity conditions can be done by a similarity transformation through an operator K. What we want, though, is an explicit boson realization for sp(2d) similar to the one that exists for sp(2). In Sec. VI we show how we can get it for sp(4) if the operator K is known. Unfortunately while the matrix form of K2 can be ex...


Journal ArticleDOI
TL;DR: In this article, the Grassmannian formalism of the Kyoto school and the group of dressing transformations have been compared in the context of Kortewegde Vries equations.
Abstract: We study several methods of describing ‘explicit’ solutions to equations of Korteweg-de Vries type: (i) the method of algebraic geometry (Krichever, I.M. Usp. mat. Nauk 32, 183-208 (1977)); (ii) the Grassmannian formalism of the Kyoto school (iii) acting on the trivial solution by the ‘group of dressing transformations’ (Zakharov, V. E. & Shabat, A. B. Funct. Anal. Appl. 13 (3), 13-22 (1979)). I show that the three methods are more or less equivalent, and in particular that the ‘ r -functions’ of method (ii) arise very naturally in the context of method (iii).

BookDOI
01 Jan 1985
TL;DR: The Lie Group Structure of Diffeomorphism Groups and Invertible Fourier Integral Operators with Applications as mentioned in this paper have been studied in the context of Kac-Moody groups.
Abstract: The Lie Group Structure of Diffeomorphism Groups and Invertible Fourier Integral Operators with Applications.- On Landau-Lifshitz Equation and Infinite Dimensional Groups.- Flat Manifolds and Infinite Dimensional Kahler Geometry.- Positive-Energy Representations of the Group of Diffeomorphisms of the Circle.- Instantons and Harmonic Maps.- A Coxeter Group Approach to Schubert Varieties.- Constructing Groups Associated to Infinite-Dimensional Lie Algebras.- Harish-Chandra Modules Over the Virasoro Algebra.- Rational Homotopy Theory of Flag Varieties Associated to Kac-Moody Groups.- The Two-Sided Cells of the Affine Weyl Group of Type An.- Loop Groups, Grassmannians and KdV Equations.- An Adjoint Quotient for Certain Groups Attached to Kac-Moody Algebras.- Analytic and Algebraic Aspects of the Kadomtsev-Petviashvili Hierarchy from the Viewpoint of the Universal Grassmann Manifold.- Comments on Differential Invariants.- The Virasoro Algebra and the KP Hierarchy.

Journal ArticleDOI
TL;DR: The long history of the theory of complex atomic spectra is reviewed in this article from the period of the 1930s, when quantum mechanics was rapidly applied to solve a variety of problems, to the present day, when, at a single stroke, elaborate computer programs are used to fit many hundreds of atomic energy levels to theoretical models.
Abstract: The long history of the theory of complex atomic spectra, as distinct from series spectra, is reviewed from the period of the 1930s, when quantum mechanics was rapidly applied to solve a variety of problems, to the present day, when, at a single stroke, elaborate computer programs are used to fit many hundreds of atomic energy levels to theoretical models. Emphasis is placed on the use of of annihilation and creation operators. With their help, the role that Lie groups play in atomic spectra can be described in analogy to SO(3), the special orthogonal group corresponding to rotations in ordinary three-dimensional space. Configuration interaction is represented by effective operators that act within the states of the unperturbed configuration under study. These effective operators are also usefully constructed from annihilation and creation operators. A table is given in which the least-squares fits to the levels of atomic configurations comprising at least three electrons (or electron holes and electrons) are listed.

Journal ArticleDOI
TL;DR: In this article, generalised characters of the infinite dimensional, holomorphic, discrete series, unitary, irreducible representations of the non-compact groups U(p, q), Sp(2n, R) and SO*(n) are explicitly expressed in terms of characters of finite dimensional unitary group representations.
Abstract: Generalised characters of the infinite dimensional, holomorphic, discrete series, unitary, irreducible representations of the non-compact groups U(p, q), Sp(2n, R) and SO*(2n) are explicitly expressed in terms of characters of finite dimensional unitary group representations. These formulae are remarkably succinct despite involving certain infinite series of Schur functions. Similar formulae are derived for harmonic series unitary representation of both U(p, q) and Sp(2n, R). Consideration of the branching rules from U(p, q) to U(q)*(p) and from Sp(2n, R) to U(n) enables holomorphic representations to be identified as a subset of the harmonic representations. The branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations. In the case of the former a known result is recast in terms of closed formulae involving Schur functions and for the latter various generalisations of these formulae are given. A conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of Sp(2n, R) and U(p, q). Illustrative examples are presented.

Journal ArticleDOI
Abstract: In this paper we study the geometry of oscillator groups: they are the only non commutative simply connected solvable Lie groups which have a biinvariant Lorentzian metric. We first study curvature and geodesics, and then give a full analysis of lattices - i.e. discrete co-compact subgroups - getting examples of compact Lorentzian homogeneous varieties.

Journal ArticleDOI
TL;DR: In this paper, it was shown that for discrete groups of matrices this boundary can be identified with the boundary of the corresponding Lie group, which is the Poisson boundary for the μ-harmonic functions on the group.
Abstract: If μ is a probability measure on a countable group there is defined a notion of the Poisson boundary for μ which enables one to represent all bounded μ-harmonic functions on the group. It is shown that for discrete groups of matrices this boundary can be identified with the boundary of the corresponding Lie group.

Journal ArticleDOI
TL;DR: In this paper, the general linear group GL(n,R) is decomposed into a Markov-type Lie group and an abelian scale group and the Markov type Lie group basis is shown to generate all singly stochastic matrices which are continuously connected to the identity when non-negative parameters are used.
Abstract: The general linear group GL(n,R) is decomposed into a Markov‐type Lie group and an abelian scale group. The Markov‐type Lie group basis is shown to generate all singly stochastic matrices which are continuously connected to the identity when non‐negative parameters are used. A basis is found which shows that it in turn contains a Lie subgroup which corresponds to doubly stochastic matrices, which basis, over the complex field, is shown to give the symmetric group for certain discrete values of the complex parameters. The basis of the Markov algebra is shown to give the negative of the corresponding M‐matrices with property ‘‘C’’ (for non‐negative combinations). These stochastic Lie groups are shown to be isomorphic to the affine group and the general linear group in one less dimension. The basis generates transformations with a natural interpretation for physical applications.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the geometrical structure given by an S-invariant metric and an S invariant Yang Mills field on E with gauge group R. They showed that there is a one to one correspondence between such structures and quadruplets of fields defined solely on M.
Abstract: LetE be a manifold on which a compact Lie groupS acts simply (all orbits of the same type);E can be written locally asM×S/I,M being the manifold of orbits (space-time) andI a typical isotropy group for theS action. We study the geometrical structure given by anS-invariant metric and anS-invariant Yang Mills field onE with gauge groupR. We show that there is a one to one correspondence between such structures and quadruplets of fields defined solely onM; γμv is a metric onM,h αβ are scalar fields characterizing the geometry of the orbits (internal spaces), Φ α i are other scalar fields (Higgs fields) characterizing theS invariance of the Lie(R)-valued Yang Mills field and is a Yang Mills field for the gauge groupN(I)|I×Z(λ(I)),N(I) being the normalizer ofI inS, λ is a homomorphism ofI intoR associated to theS action, andZ(λ(I)) is the centralizer ofλ(I) inR. We express the Einstein-Yang-Mills Lagrangian ofE in terms of the component fields onM. Examples and model building recipes are given.

01 Jan 1985
TL;DR: In this article, the equivalence relations on Lie groups that are generated by the action of a countable subgroup are studied and necessary conditions for the amenability of such a relation are described.
Abstract: We study the equivalence relations on Lie groups that are generated by the action of a countable subgroup. In particular, we describe some necessary conditions for the amenability of such a relation On etudie les relations d'equivalence sur un groupe de Lie engendrees par l'action d'un sous groupe denombrable. En particulier, on donne des conditions pour qu'une telle relation soit moyennable

Journal ArticleDOI
TL;DR: In this article, it was shown that the Holstein-Primakoff representation of the Sp(2d,R) algebra cannot be written in such a compact form for a generic irreducible representation.
Abstract: Both non‐Hermitian Dyson and Hermitian Holstein–Primakoff representations of the Sp(2d,R) algebra are obtained when the latter is restricted to a positive discrete series irreducible representation 〈λd +n/2,...,λ1+n/2〉. For such purposes, some results for boson representations, recently deduced from a study of the Sp(2d,R) partially coherent states, are combined with some standard techniques of boson expansion theories. The introduction of Usui operators enables the establishment of useful relations between the various boson representations. Two Dyson representations of the Sp(2d,R) algebra are obtained in compact form in terms of ν=d(d+1)/2 pairs of boson creation and annihilation operators, and of an extra U(d) spin, characterized by the irreducible representation [λ1⋅⋅⋅λd]. In contrast to what happens when λ1=⋅⋅⋅=λd=λ, it is shown that the Holstein–Primakoff representation of the Sp(2d,R) algebra cannot be written in such a compact form for a generic irreducible representation. Explicit expansions are,...

Journal ArticleDOI
TL;DR: The general Lie point transformation group and the associated reduced differential equations and similarity forms for the solutions are derived in this paper for the coupled (nonlinear) Vlasov-Maxwell equations in one spatial dimension.
Abstract: The general Lie point transformation group and the associated reduced differential equations and similarity forms for the solutions are derived here for the coupled (nonlinear) Vlasov-Maxwell equations in one spatial dimension. The case of one species in a background is shown to admit a larger group than the multispecies case. Previous exact solutions are shown to be special cases of the above solutions, and many of the new solutions are found to constrain the form of the distribution function much more than, for example, the BGK solutions do. The individual generators of the Lie group are used to find the possible subgroups. Finally, a simple physical argument is given to show that the asymptotic solution for a one-species, one-dimensional plasma is one of the general similarity solutions.