Showing papers on "Lie group published in 1972"

Coherent states for arbitrary Lie group

Abstract: The concept of coherent states originally closely related to the nilpotent group of Weyl is generalized to arbitrary Lie group. For the simplest Lie groups the system of coherent states is constructed and its features are investigated.

Harmonic Analysis on Semi-Simple Lie Groups II

Abstract: 6 Spherical Functions - The General Theory- 61 Fundamentals- 611 Spherical Functions - Functional Properties- 612 Spherical Functions - Differential Properties- 62 Examples- 621 Spherical Functions on Motion Groups- 622 Spherical Functions on Semi-Simple Lie Groups- 7 Topology on the Dual Plancherel Measure Introduction- 71 Topology on the Dual- 711 Generalities- 712 Applications to Semi-Simple Lie Groups- 72 Plancherel Measure- 721 Generalities- 722 The Plancherel Theorem for Complex Connected Semi-Simple Lie Groups- 8 Analysis on a Semi-Simple Lie Group- 81 Preliminaries- 811 Acceptable Groups- 812 Normalization of Invariant Measures- 813 Integration Formulas- 814 A Theorem of Compacity- 815 The Standard Semi-Norm on a Semi-Simple Lie Group- 816 Completely Invariant Sets- 82 Differential Operators on Reductive Lie Groups and Algebras- 821 Radial Components of Differential Operators on a Manifold- 822 Radial Components of Polynomial Differential Operators on a Reductive Lie Algebra- 823 Radial Components of Left Invariant Differential Operators on a Reductive Lie Group- 824 The Connection between Differential Operators in the Algebra and on the Group- 83 Central Eigendistributions on Reductive Lie Algebras and Groups- 831 The Main Theorem in the Algebra- 832 Properties of FT-I- 833 The Main Theorem on the Group- 834 Properties of FT- II- 835 Rapidly Decreasing Functions on a Euclidean Space- 836 Tempered Distributions on a Reductive Lie Algebra- 837 Rapidly Decreasing Functions on a Reductive Lie Group- 838 Tempered Distributions on a Reductive Lie Group- 839 Tools for Harmonic Analysis on G- 84 The Invariant Integral on a Reductive Lie Algebra- 841 The Invariant Integral - Definition and Properties- 842 Computations in sl(2, R)- 843 Continuity of the Map f ? ? f- 844 Extension Problems- 845 The Main Theorem- 85 The Invariant Integral on a Reductive Lie Group- 851 The Invariant Integral - Definition and Properties- 852 The Inequalities of Descent- 853 The Transformations of Descent- 854 The Invariant Integral and the Transformations of Descent- 855 Estimation of ?f and its Derivatives- 856 An Important Inequality- 857 Convergence of Certain Integrals- 858 Continuity of the Map f? ?f- 9 Spherical Functions on a Semi-Simple Lie Group- 91 Asymptotic Behavior of ?-Spherical Functions on a Semi-Simple Lie Group- 911 The Main Results- 912 Analysis in the Universal Enveloping Algebra- 913 The Space S(?,?)- 914 The Rational Functions ??- 915 The Expansion of ?-Spherical Functions- 916 Investigation of the c-Function- 917 Applications to Zonal Spherical Functions- 92 Zonal Spherical Functions on a Semi-Simple Lie Group- 921 Statement of Results - Immediate Applications- 922 The Plancherel Theorem for I2(G)- 923 The Paley-Wiener Theorem for I2(G)- 924 Harmonic Analysis in I1(G)- 93 Spherical Functions and Differential Equations- 931 The Weak Inequality and Some of its Implications- 932 Existence and Uniqueness of the Indices I- 933 Existence and Uniqueness of the Indices II- 10 The Discrete Series for a Semi-Simple Lie Group - Existence and Exhaustion- 101 The Role of the Distributions ?? in the Harmonic Analysis on G- 1011 Existence and Uniqueness of the ??- 1012 Expansion of Z-Finite Functions in C-(G)- 102 Theory of the Discrete Series- 1021 Existence of the Discrete Series- 1022 The Characters of the Discrete Series I - Implication of the Orthogonality Relations- 1023 The Characters of the Discrete Series II - Application of the Differential Equations- 1024 The Theorem of Harish-Chandra- Epilogue- Append- 3 Some Results on Differential Equations- 31 The Main Theorems- 32 Lemmas from Analysis- 33 Analytic Continuation of Solutions- 34 Decent Convergence- 35 Normal Sequences of is-Polynomials- General Notational Conventions- List of Notations- Guide to the Literature- Subject Index to Volumes I and II

System Theory on Group Manifolds and Coset Spaces

Abstract: The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a coset space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.

The Crystals Associated to Barsotti-Tate Groups: With Applications to Abelian Schemes

Abstract: Conventions.- Definitions and examples.- The relation between barsotti-tate and formal lie groups.- Divided powers, exponentials and crystals.- The crystals associated to barsotti-tate groups.- The deformation theory and applications.

On the algebraic structure of bilinear systems.

Abstract: It is shown that a particular bilinear model is both quite general and easy to work with. A basic structure theory is developed with the aid of previous results. Some preliminary ideas are discussed together with the system interconnection, the canonical form, questions of controllability, aspects of observability, and equivalent realizations. It is pointed out that in actually determining equivalent realizations for systems and in the classification of systems, the results available in the study of Lie algebras are of fundamental importance.

Analysis on Lie Groups and Homogeneous Spaces

Abstract: Introduction Some geometric properties of differential operators Spherical functions on symmetric spaces Conical distributions on the space of horocycles Central eigendistributions and characters Bibliography.

Primitive actions and maximal subgroups of Lie groups

Abstract: The classification of the primitive transitive and effective actions of Lie groups on manifolds is a problem dating back to Lie. The classification of the infinite dimensional infinitesimal actions was originally done by Cartan [3] and was made rigorous by some joint work of Guillemin, Quillen and Sternberg [8], whose proof was further simplified by Guillemin [7] recently by using some results of Veisfieler [19]. The classification of the primitive actions of a given finite dimensional Lie group is equivalent to that of the Lie subgroups of that group, which satisfy a certain maximality condition (see Prop. 1.5). This correspondence although more or less known seems never to have been stated in the literature (under the assumption that the leaves of a foliation are connected) so in § 1 we state it. The rest of § 1 is devoted to showing that the isotropy subalgebras of primitive actions are an intrinsically well-defined class, namely, they are the Lie algebras which correspond to maximal Lie subgroups and contain no proper ideals. We call these subalgebras primitive and hasten to add that this terminology does not agree with the use of "primitive" in [7], [8], [11], [12], [13] and [16]. In these articles a "primitive subalgebra" is a maximal Lie subalgebra which contains no proper ideals. In light of Theorem 1.10 we do feel that this is a more reasonable terminology. Also we show that every subalgebra which is "primitive" in the old sense is primitive in the new sense. The main result of this paper is that there exist primitive subalgebras which are not maximal subalgebras, i.e., there exist maximal Lie subgroups whose Lie algebras are not maximal subalgebras. In § 3 we classify the primitive, maximal rank, reductive subalgebras of the (complex) classical algebras giving many examples of primitive subalgebras which are, in fact, not maximal. § 2 and § 4 combined show that non-maximal primitive algebras exist only when the containing algebra is simple and the primitive subalgebra is reductive. The proofs of this involves essentially classifying the primitive subalgebras. In doing so we duplicate results of Morozov [15] on the classification of the maximal primitive subalgebras of non-simple algebras and results of Karpelevich [10] and Ochiai [16] on the

Estimation for Rotational Processes with One Degree of Freedom

Abstract: : A class of bilinear estimation problems involving single-degree-of- freedom rotation is formulated and resolved. Both continuous and discrete time estimation problems are considered. Error criteria, probability distributions, and optimal estimates on the circle are studies. An effective synthesis procedure for continuous time estimation is provided, and a generalization to estimation on arbitrary abelian Lie groups is included. An intrinsic difference between the discrete and continuous problems is discussed, and the complexity of the equations in the discrete time case is analyzed in this setting. Applications of these results to a number of practical problems including FM demodulation and frequency stability are examined.

Asymptotic behaviour of eigen functions on a semisimple Lie group: The discrete spectrum

Abstract: Let G be a connected noncompact real form of a simply connected complex semisimple Lie group. For many questions of Fourier Analysis on G it is useful to have a good knowledge of the behaviour, at infinity on G, of the matrix coefficients of the irreducible unitary representations of G. In this paper we restrict ourselves to the discrete series of representations of G, and study the rapidity with which the corresponding matrix coefficients decay at infinity on the group. Let K be a maximal compact subgroup of G. Given any p, with 1 ~0 , 7 > 0 , q~>0 (depending on ]) such that

Lie algebras and linear differential equations

Abstract: Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

Lie Theory and the Lauricella Functions FD

Abstract: It is shown that the Lauricella functions FD in n variables transform as basis vectors corresponding to irreducible representations of the Lie algebra sl(n+3,C). Group representation theory can then be applied to derive addition theorems, transformation formulas, and generating functions for the FD. It is clear from this analysis that the use of SL(m,C) symmetry in atomic and elementary particle physics will lead inevitably to the remarkable functions FD.

Decision surface estimate of nonlinear system stability domain by Lie series method

Abstract: The Lie series recursive algorithm for Zubov's partial differential equation is used to generate two sets of points, where one represents the exact asymptotic stability boundary of an equilibrium state of the nonlinear system under consideration and the other is interior to it. Based on these two sets of data as training samples of two classes, a decision hypersurface can be determined such that it is a close approximation of the asymptotic stability boundary.

On the algebraic structure of a class of solvable quantum problems

Abstract: A new algebraic approach to the treatment of quantum problems is introduced, which allows us to understand completely the algebraic structure of all those systems whose eigenvalue equations can be reduced, by means of proper functional transformations, to the hypergeometric equation. The results obtained constitute therefore the natural extension of the program we started in previous papers. In particular, it is proved that the eigenfunctions of the simple but physically interesting differential equation (3.8) are associated to the states of the irreducible representations Open image in new window (l=0,1,2,…) of the Lie algebraA1≈B1≈C1, characterized by the fact that the eigenvalue of the generatorJ3 of the algebra takes the constant value 1/2[1+2g]1/2 for the considered states. When consideration is given to theS-wave Schrodinger equation for two particles interacting through the Hulthen potential, it is proved that the energy levels are associated with a properly selected submanifold of the space carrying the irreducible representations Open image in new window or Open image in new window of the same algebra. Finally, a general procedure is introduced which allows an analogous treatment of all problems whose eigenvalue equations can be reduced to the hypergeometric equation. An interesting three-body problem is also exhaustively investigated from the algebraic point of view.

Uniform Finite Generation of SU(2) and SL(2, R)

Abstract: A connected Lie group H is generated by a pair of one-parameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups, i.e., if and only if the subalgebra generated by the corresponding pair of infinitesimal transformations is equal to the whole Lie algebra h of H (observe that the subgroup of all finite products is arcwise connected and hence, by Yamabe's theorem [5], is a sub-Lie group). If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n ; otherwise define it as infinity.

The Structure of Real Semi-Simple Lie Groups

Abstract: In order to formulate and prove the deeper results in the representation theory of semi-simple Lie groups, it is necessary to have at hand detailed information on the structure of these groups. The purpose of this chapter is to set down the salient facts.

Certain Controllability Properties of Analytic Control Systems

Abstract: In this article we consider analytic control systems defined on an analytic manifold M. In particular, we study the relationship between the accessibility property, i.e., attainable sets having nonempty interior, and controllability. This study is restricted to analytic control systems because they admit a very simple algebraic criterion for determining the accessibility property; this criterion is a generalization of the well-known rank condition for linear systems.The main difficulty in extending the accessibility property to yield controllability lies in the fact that the time coordinate is nondecreasing. Beside systems of the form $F(x,u) = - F(x, - u)$, right invariant control systems defined on a compact Lie group are controllable whenever they have the accessibility property. It seems reasonable that a similar result should hold for arbitrary compact manifolds.

Invariant pseudo-differential operators on a Lie group

Abstract: The invariant peendo-differential operators on a Lie gronp G with Lie algebra d3 are characterized in terms of a function on (~* called the Lie symbol. A calculus of Lie symbols is developed in terms of the algebraic structure of o. We also define global Sobolev spaces for non-compact groups.

Geodesics and Classical Mechanics on Lie Groups

Abstract: Hamilton‐Jacobi Theory on Lie groups is discussed, and an error in a previous paper is corrected. A method for ``analytically continuing'' geodesics from compact to noncompact Lie groups is presented.