Showing papers on "Lie group published in 1989"
TL;DR: In this paper, some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented.
Abstract: Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painleve equations, cannot be obtained using the standard Lie group method for finding group‐invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.
922 citations
TL;DR: In this article, for a smooth manifold equipped with a Poisson bracket, the authors formulate a C*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the bracket.
Abstract: ForM a smooth manifold equipped with a Poisson bracket, we formulate aC*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the Poisson bracket. We then show that the much-studied non-commutative tori give examples of such deformation quantizations, invariant under the usual action of ordinary tori. Going beyond this, the main results of the paper provide a construction of invariant deformation quantizations for those Poisson brackets on Heisenberg manifolds which are invariant under the action of the Heisenberg Lie group, and for various generalizations suggested by this class of examples. Interesting examples are obtained of simpleC*-algebras on which the Heisenberg group acts ergodically.
371 citations
TL;DR: In this paper, a new form of equations of motion is proposed for d = 4 massless fields of all spins interacting with gravity, which are described in terms of a free differential algebra constructed from 1-forms and 0-forms belonging both to the adjoint representation of the superalgebra of higher-spin and auxiliary fields.
320 citations
Book•
01 Jan 1989
TL;DR: The geometry of cones wedges in Lie algebras invariant to cones is described in this article, where the local Lie theory of semigroups subsemigroups of Lie groups positivity embedding semiggroups into Lie groups.
Abstract: The geometry of cones wedges in Lie algebras invariant cones the local Lie theory of semigroups subsemigroups of Lie groups positivity embedding semigroups into Lie groups.
217 citations
Book•
01 Jan 1989
TL;DR: In this article, the authors present an introductory course on modern, coordinate-free differential geometry which is taken by first-year theoretical physics PhD students, or by students attending the one-year MSc course “Fundamental Fields and Forces” at Imperial College.
Abstract: These notes are the content of an introductory course on modern, coordinate-free differential geometry which is taken by the first-year theoretical physics PhD students, or by students attending the one-year MSc course “Fundamental Fields and Forces” at Imperial College. The book is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen with an eye to the way in which differential geometry is applied these days to modern theoretical physics. This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, non-linear sigma-models and other types of non-linear field systems that feature in modern quantum field theory. This volume is in three parts dealing with, respectively, (i) introductory coordinate-free differential geometry, (ii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds, (iii) introduction to the theory of fibre bundles. In the first part of the book the author has laid considerable stress on the basic ideas of “tangent space structure” which he develops from several different points of view: some geometrical, and others more algebraic. This is done with the awareness of the difficulty which physics graduate students often experience when being exposed for the first time to the rather abstract ideas of differential geometry.
206 citations
01 Jan 1989
TL;DR: In this paper, the Laplace Spherical Functions are used to represent the representation of Lie groups and the representations of SU2 and S03, and the groups are decomposed into a regular representation.
Abstract: Basic Notions.- General Properties of Representations.- Invariant Subspaces.- Complete Reducibility of Representations of Compact Groups.- Basic Operations on Representations.- Properties of Irreducible Complex Representations.- Representations of Finite Groups.- Decomposition of the Regular Representation.- Orthogonality Relations.- Representations of Finite Groups.- The Groups SU2 and S03.- Matrix Elements of Compact Groups.- The Laplace Spherical Functions.- Representations of Lie Groups.- General Properties of Homomorphisms and Representations of Lie Groups.- Representations of SU2 and S03.- Appendices.- References.
189 citations
TL;DR: In this paper, the descending chain condition for continuous automorphisms of a compact, metrizable group X was introduced, and it was shown that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X and if X is a compact group and (X, Γ) satisfies the descending-chain condition, then every ergodic element of G has a dense set of periodic points.
Abstract: We study finitely generated, abelian groups Γ of continuous automorphisms of a compact, metrizable group X and introduce the descending chain condition for such pairs (X, Γ). If Γ acts expansively on X then (X, Γ) satisfies the descending chain condition, and (X, Γ) satisfies the descending chain condition if and only if it is algebraically and topologically isomorphic to a closed, shift-invariant subgroup of GΓ, where G is a compact Lie group. Furthermore every such subgroup of GΓ is a (higher dimensional) Markov shift whose alphabet is a compact Lie group. By using the descending chain condition we prove, for example, that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X. Furthermore, if X is a compact group and (X, Γ) satisfies the descending chain condition, then every ergodic element of Γ has a dense set of periodic points. Finally we give an algebraic description of pairs (X, Γ) satisfying the descending chain condition under the assumption that X is abelian.
156 citations
TL;DR: In this article, the authors study the expansion of the solution of a stochastic differential equation as an (infinite) sum of iterated Stratonovitch integrals, which enables them to give a universal and explicit formula for any invariant diffusion on a Lie group in terms of Lie brackets.
Abstract: We study the expansion of the solution of a stochastic differential equation as an (infinite) sum of iterated stochastic (Stratonovitch) integrals. This enables us to give a universal and explicit formula for any invariant diffusion on a Lie group in terms of Lie brackets, as well as a universal and explicit formula for the brownian motion on a Riemannian manifold in terms of derivatives of the curvature tensor. The first of these formulae contains, and extends to the non nilpotent case, the results of Doss [6], Sussmann [17], Yamato [18], Fliess and Normand-Cyrot [7], Krener and Lobry [19] and Kunita [11] on the representation of solutions of stochastic differential equations.
142 citations
TL;DR: In this article, the authors consider the problem of finding a permutation group on n letters in the orthogonal group O ( n ) with the latter acting on R n (n + 1)/2 via a symmetricized tensor product.
TL;DR: In this paper, the authors quantize the Chern-Simons theory in the functional Schrodinger representation and show how to solve the Gauss law constraint in the non-Abelian theory.
TL;DR: In this article, the authors presented some new similarity solutions of the modified Boussinesq equation, which is a completely integrable soliton equation, using a direct method which involves no group theoretical techniques.
Abstract: In this paper the author presents some new similarity solutions of the modified Boussinesq equation, which is a completely integrable soliton equation. These new similarity solutions include reductions to the second and fourth Painleve equations which are not obtainable using the standard Lie group method for finding group-invariant solutions of partial differential equations; they are determined using a new and direct method which involves no group theoretical techniques.
TL;DR: In this paper, an equivalence is exhibited between a classical group of monopoles over ℝ3, with maximal symmetry breaking at infinity, and a family of (rank (G)) algebraic curves in Tℙ1, along with divisors on those curves, satisfying certain constraints.
Abstract: ForG a classical group, an equivalence is exhibited between:
A)
G monopoles over ℝ3, with maximal symmetry breaking at infinity,
B)
families of (rank (G)) algebraic curves inTℙ1, along with divisors on those curves, satisfying certain constraints,
C)
solutions of Nahm's equations over (rank(G)) intervals, satisfying the appropriate boundary conditions.
A) and B) are linked by twistor techniques, B) and C) via the Krichever method for solving non-linear differential equations, and A) and C) via the ADHMN construction, providing a unified picture of techniques for solution. Amongst other things, an asymptotic formula for the Higgs field of the monopole is computed.
TL;DR: In this article, the authors discuss the supersymmetry and the semiclassical quantum-mechanical interpretation of the Weyl character formula for compact, semisimple Lie groups.
TL;DR: In this article, quantum integrable models associated with non-degenerate solutions of classical Yang-Baxter equations related to simple Lie algebras are investigated, and the analogy with the quantum inverse scattering method is demonstrated.
Abstract: Quantum integrable models associated with nondegenerate solutions of classical Yang–Baxter equations related to the simple Lie algebras are investigated. These models are diagonalized for rational and trigonometric solutions in the cases of sl(N)/gl(N)/, o(N) and sp(N) algebras. The analogy with the quantum inverse scattering method is demonstrated.
TL;DR: In this article, the authors investigated the Scissors Congruence Problem in 3-dimensional hyperbolic and spherical spaces and showed that it has a close relation with the Quillen algebraic Ks-group of iF where [F denotes one of the three classical real division algebras R, C or IH.
TL;DR: In this paper, the authors introduce quadratic Poisson structures on Lie groups associated with a class of solutions of the modified Yang-Baxter equation and apply them to the Hamiltonian description of Lax systems.
Abstract: We introduce quadratic Poisson structures on Lie groups associated with a class of solutions of the modified Yang-Baxter equation and apply them to the Hamiltonian description of Lax systems. The formal analog of these brackets on associative algebras provides second structures for certain integrable equations. In particular, the integrals of the Toda flow on generic orbits are shown to satisfy recursion relations. Finally, we exhibit a third order Poisson bracket for which ther-matrix approach is feasible.
TL;DR: In this article, the YM2-measure space is analyzed in both the continuum and the lattice, and the expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group.
Abstract: The two dimensional Yang-Mills theory (YM2) is analyzed in both the continuum and the lattice. In the complete axial gauge the continuum theory may be defined in terms of a Lie algebra valued white noise, and parallel translation may be defined by stochastic differential equations. This machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesC. The expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group. The time parameters of the heat kernels are determined by the areas enclosed by the collectionC, and the arguments are determined by the crossing topologies of the curves inC. The expectations for the Wilson lattice models have a similar structure, and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations. It is also shown that the lasso variables advocated by L. Gross [36] exist and are sufficient to generate all the measurable functions on the YM2-measure space.
TL;DR: In this article, the authors classified second-order ordinary differential equations according to their Lie algebra of point symmetries, and established the Canonical forms of generators for equations with three-point symmetry.
Abstract: Second‐order ordinary differential equations are classified according to their Lie algebra of point symmetries. The existence of these symmetries provides a way to solve the equations or to transform them to simpler forms. Canonical forms of generators for equations with three‐point symmetries are established. It is further shown that an equation cannot have exactly r ∈{4,5,6,7} point symmetries. Representative(s) of equivalence class(es) of equations possessing s ∈{1,2,3,8} point symmetry generator(s) are then obtained.
TL;DR: It is found that, prior to spontaneous breaking of the electroweak subgroup, the minimal Weyl representations and their charges are uniquely determined by insisting on all three known chiral gauge anomaly-free conditions in four dimensions.
Abstract: The uniqueness of the Weyl representations of the standard gauge group is reexamined. We find that, prior to spontaneous breaking of the electroweak subgroup, the minimal Weyl representations and their charges are uniquely determined by insisting on all three known chiral gauge anomaly-free conditions in four dimensions: (1) cancellation of triangular anomalies; (2) absence of the global SU(2) anomaly; and (3) cancellation of the mixed-gauge-gravitational anomaly. The uniqueness question for the left-right-symmetric group and the simple (grand-unified-theory) group are discussed from the anomalies viewpoint.
TL;DR: In this article, the analysis of integral operators on nilpotent Lie groups with kernels supported on submanifolds and/or containing an oscillatory factor of polynomial type was studied.
TL;DR: In this article, the potential group SO(2, 2) was used to realize a class of solvable potentials and the scattering matrices can be obtained by purely algebraic techniques.
TL;DR: In this article, the moduli spaces of monopoles with maximal symmetry breaking at infinity for SU(N), SO(N) and SP(N)) were derived and shown to be equivalent to holomorphic maps from ℙ1 into flag manifolds.
Abstract: By studying a construction of Nahm, we compute the moduli spaces of monopoles with maximal symmetry breaking at infinity forSU(N),SO(N) andSp(N); these are found to be equivalent to spaces of holomorphic maps from ℙ1 into flag manifolds.
TL;DR: In this article, the classical theory of Lyapunov characteristic exponents is reformulated in invariant geometric terms and carried over to arbitrary noncompact semisimple Lie groups with finite center.
Abstract: The classical theory of Lyapunov characteristic exponents is reformulated in invariant geometric terms and carried over to arbitrary noncompact semisimple Lie groups with finite center. A multiplicative ergodic theorem (a generalization of a theorem of Oseledets) and the global law of large numbers are proved for semisimple Lie groups, as well as a criterion for Lyapunov regularity of linear systems of ordinary differential equations with subexponential growth of coefficients.
Book•
11 Dec 1989
TL;DR: In this article, the authors present a dimensional reduction of pure Yang-Mills theories and a general procedure for solving the equations of spontaneous compactification within EYM systems, including gravity and theories with fermions included.
Abstract: This monograph presents in detail the reduction method for studying the unification of fundamental actions. The mathematical (differential geometrical) methods make extensive use of Lie Groups and the concept of homogeneous spaces. The main topic of the book is the dimensional reduction of pure Yang-Mills theories. A rather complete analysis of the structure of the scalar field potential is given and a general procedure for solving the equations of spontaneous compactification within Einstein-Yang-Mills systems is presented. The authors also discuss gravity and theories with fermions included and they review attempts to construct realistic models. The book presents the basic ideas and the calculations in detail and should be of interest to researchers and graduate students in mathematical physics.
TL;DR: In this paper, the authors studied the notion of minimax invariance of a G-space and developed a purely topological construction, based on the original approach of Lusternik and Schnirelman.
TL;DR: In this paper, Cartan's method of equivalence was used to give a complete classification, in terms of differential invariants, of second-order ordinary differential equations admitting Lie groups of fiber preserving point symmetries.
Abstract: We use Elie Cartan's method of equivalence to give a complete classification, in terms of differential invariants, of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries. We then apply our results to the determination of all second-order equations which are equivalent, under fibre-preserving transformations, to the free particle equation. In addition we present those equations of Painleve' type which admit a transitive symmetry group. Finally we determine the symmetry group of some equations of physical interest, such as the Duffing and Holmes-Rand equations, which arise as models of non-linear oscillators.
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TL;DR: An elementary introduction to the notions of the quantum Lie Groups and quantum Lie algebras is given in this article, based on the fundamental commutation relations which appeared first in the quantum inverse scattering method.
Abstract: An elementary introduction to the notions of the quantum Lie Groups and quantum Lie algebras is given. The approach is based on the fundamental commutation relations which appeared first in the quantum inverse scattering method.
TL;DR: In this paper, it was shown that local CR automorphisms of non-degenerate standard CR manifolds are rational and form a finite-dimensional Lie group, and that proper holomorphic mappings of Siegel domains are biholomorphic and rational.
Abstract: It is shown that local CR automorphisms of nondegenerate standard CR manifolds are rational and form a finite-dimensional Lie group. It is established that proper holomorphic mappings of nondegenerate Siegel domains are biholomorphic and rational. Bibliography: 14 titles.