INTEGRABILITY AND SYMMETRIC SPACES II: THE COSET SPACES

1989 ◽  
Vol 04 (03) ◽  
pp. 675-699 ◽  
Author(s):  
L. A. FERREIRA
Keyword(s):  
Symmetric Space ◽  
Poisson Bracket ◽  
Algebraic Structure ◽  
Symmetric Spaces ◽  
Sufficient Condition ◽  
Conserved Quantities ◽  
Coset Space ◽  
Canonical Variables ◽  
Coset Spaces

It is shown that a sufficient condition for a model describing the motion of a particle on a coset space to possess a Fundamental Poisson bracket Relation, and consequently charges in involution, is that it must be a symmetric space. The conditions, a Hamiltonian, or any functions of the canonical variables, has to satisfy in order to commute with these charges, are studied. It is shown that, for the case of the noncompact symmetric spaces, these conditions lead to an algebraic structure which plays an important role in the construction of conserved quantities.

TERNARY ALGEBRAIC CONSTRUCTION OF EXTENDED SUPERCONFORMAL ALGEBRAS

1991 ◽  
Vol 06 (19) ◽  
pp. 1733-1743 ◽  
Author(s):  
MURAT GÜNAYDIN ◽  
SEUNGJOON HYUN
Keyword(s):  
Lie Groups ◽  
Symmetric Spaces ◽  
Coset Space ◽  
Algebraic Construction ◽  
Hermitian Symmetric Spaces ◽  
Magic Square ◽  
Triple Systems ◽  
Coset Spaces ◽  
Jordan Triple Systems ◽  
Ternary Algebras

We give a construction of extended (N = 2 and N = 4) superconformal algebras over a very general class of ternary algebras (triple systems). For N = 2 this construction leads to superconformal algebras corresponding to certain coset spaces of Lie groups with non-vanishing torsion and generalizes a previous construction over Jordan triple systems which are associated with Hermitian symmetric spaces. In general, a given Lie group admits more than one coset space of this type. We give examples for all simple Lie groups. In particular, the division algebras and their tensor products lead to N = 2 superconformal algebras associated with the groups of the Magic Square. For a very special class of ternary algebras, namely the Freudenthal triple (FT) systems, the N = 2 superconformal algebras can be extended to N = 4 superconformal algebras with the gauge group SU (2) × SU (2) × U (1). We give a complete list of the FT systems and the corresponding N = 4 models. They are associated with the unique quaternionic symmetric spaces of Lie groups.

INTEGRABILITY AND SYMMETRIC SPACES I: THE GROUP MANIFOLD

1989 ◽  
Vol 04 (03) ◽  
pp. 649-674 ◽  
Author(s):  
L. A. FERREIRA
Keyword(s):  
Poisson Bracket ◽  
Lie Group ◽  
Symmetric Spaces ◽  
Algebraic Structures ◽  
Semisimple Lie Group ◽  
Conserved Charges ◽  
Group Manifold ◽  
Coset Spaces

It is shown that any nonsingular Lagrangian describing the motion of a particle on a semisimple Lie group possesses a Fundamental Poisson bracket Relation (FPR) and consequently charges in involution. This property is independent of the dynamics of the model and can be derived in a quite simple and general way from the geometric and algebraic structures of the group manifold. The conditions a Hamiltonian has to satisfy in order those charges are to be conserved are discussed. These conditions lead to an algebra which plays an important role in the construction of conserved charges. In the second paper of the series, this work is extended to the coset spaces which are symmetric spaces.

Minimum and Conjugate Points in Symmetric Spaces

1962 ◽  
Vol 14 ◽  
pp. 320-328 ◽  
Author(s):  
Richard Crittenden
Keyword(s):  
Symmetric Space ◽  
Algebraic Structure ◽  
Tangent Bundle ◽  
Tangent Space ◽  
Symmetric Spaces ◽  
Conjugate Points ◽  
Minimum Cut ◽  
Cut Locus ◽  
Connected Case ◽  
Simply Connected

The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

Generalized non-linear σ-models in two dimensions

2021 ◽  
pp. 692-720
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Two Dimensions ◽  
Asymptotic Freedom ◽  
Quantum Field Theories ◽  
Field Equations ◽  
Local Conservation ◽  
Non Linear ◽  
Coset Spaces ◽  
Formal Properties

This chapter describes the formal properties, and discusses the renormalization, of quantum field theories (QFT) based on homogeneous spaces: coset spaces of the form G/H, where G is a compact Lie group and H a Lie subgroup. In physics, they appear naturally in the case of spontaneous symmetry breaking, and describe the interaction between Goldstone modes. Homogeneous spaces are associated with non-linear realizations of group representations. There exist natural ways to embed these manifolds in flat Euclidean spaces, spaces in which the symmetry group acts linearly. As in the example of the non-linear σ-model, this embedding is first used, because the renormalization properties are simpler, and the physical interpretation of the more direct correlation functions. Then, in a generic parametrization, the renormalization problem is solved by the introduction of a Becchi–Rouet–Stora–Tyutin (BRST)-like symmetry with anticommuting (Grassmann) parameters, which also plays an essential role in quantized gauge theories. The more specific properties of models corresponding to a special class of homogeneous spaces, symmetric spaces (like the non-linear σ-model), are studied. These models are characterized by the uniqueness of the metric and thus, of the classical action. In two dimensions, from the classical field equations an infinite number of non-local conservation laws can be derived. The field and the unique coupling renormalization group (RG) functions are calculated at one-loop order, in two dimensions, and shown to imply asymptotic freedom.

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

2015 ◽  
Vol 26 (05) ◽  
pp. 1550039
Author(s):  
Salma Nasrin
Keyword(s):  
Symmetric Space ◽  
Lie Algebras ◽  
Lie Group ◽  
Symmetric Spaces ◽  
Coadjoint Orbit ◽  
Natural Projection ◽  
Real Form ◽  
Single Orbit ◽  
Complex Semisimple ◽  
Complex Symmetric

Let Gℂ be a complex simple Lie group, GU a compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂ is either an empty set or a single orbit of GU if [Formula: see text] is isomorphic to a complex symmetric space.

DECAY RATES FOR THE SHIFTED WAVE EQUATION ON A SYMMETRIC SPACE OF NONCOMPACT TYPE

2013 ◽  
Vol 10 (04) ◽  
pp. 677-701
Author(s):  
CARLOS ALMADA
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Symmetric Space ◽  
Decay Rate ◽  
Wave Operator ◽  
Symmetric Spaces ◽  
Decay Rates ◽  
Noncompact Type ◽  
Killing Fields ◽  
The Cauchy Problem

We derive L∞–L1 decay rate estimates for solutions of the shifted wave equation on certain symmetric spaces (M, g). The Cauchy problem for the shifted wave operator on these spaces was studied by Helgason, who obtained a closed form for its solution. Our results extend to this new context the classical estimates for the wave equation in ℝn. Then, following an idea from Klainerman, we introduce a new norm based on Lie derivatives with respect to Killing fields on M and we derive an estimate for the case that n = dim M is odd.

scholarly journals On projective-symmetric spaces

1964 ◽  
Vol 4 (1) ◽  
pp. 113-121 ◽  
Author(s):  
Bandana Gupta
Keyword(s):  
Symmetric Space ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Covariant Differentiation ◽  
Metric Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor ◽  
Image Position ◽  
Riemannian Spaces

This paper deals with a type of Remannian space Vn (n ≧ 2) for which the first covariant dervative of Weyl's projective curvature tensor is everywhere zero, that is where comma denotes covariant differentiation with respect to the metric tensor gij of Vn. Such a space has been called a projective-symmetric space by Gy. Soós [1]. We shall denote such an n-space by ψn. It will be proved in this paper that decomposable Projective-Symmetric spaces are symmetric in the sense of Cartan. In sections 3, 4 and 5 non-decomposable spaces of this kind will be considered in relation to other well-known classes of Riemannian spaces defined by curvature restrictions. In the last section the question of the existence of fields of concurrent directions in a ψ will be discussed.

CLASSES MINIMALES DE RÉSEAUX ET RÉTRACTIONS GÉOMÉTRIQUES ÉQUIVARIANTES DANS LES ESPACES SYMÉTRIQUES

2001 ◽  
Vol 64 (2) ◽  
pp. 275-286 ◽  
Author(s):  
CHRISTOPHE BAVARD
Keyword(s):  
Finite Group ◽  
Euclidean Space ◽  
Symmetric Space ◽  
Group Action ◽  
Symmetric Spaces ◽  
Classification Problem ◽  
Finite Group Action ◽  
Natural Geometry ◽  
A Finite Group

Equivariant and cocompact retractions of certain symmetric spaces are constructed. These retractions are defined using the natural geometry of symmetric spaces and in relation to the theory of lattices of euclidean space. The following cases are considered: the symmetric space corresponding to lattices endowed with a finite group action, from which is obtained some information relating to the classification problem of these lattices, and the Siegel space Sp2g(R)/Ug, for which a natural Sp2g(Z)-equivariant cocompact retract of codimension 1 is obtained.

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

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