Showing papers on "Lie group published in 1973"
TL;DR: In this article, it is shown that in constructing a theory for the most elementary class of control problems defined on spheres, some results from the Lie theory play a natural role, and that to understand controllability, optimal control, and certain properties of stochastic equations, Lie theoretic ideas are needed.
Abstract: It is shown that in constructing a theory for the most elementary class of control problems defined on spheres, some results from the Lie theory play a natural role. To understand controllability, optimal control, and certain properties of stochastic equations, Lie theoretic ideas are needed. The framework considered here is the most natural departure from the usual linear system/vector space problems which have dominated control systems literature. For this reason results are compared with those previously available for the finite dimensional vector space case.
247 citations
01 Jan 1973
TL;DR: The theory of differential equations and control have been linked very closely because most of the early applications of control theory were to engineering problems of the type which are most naturally described by ordinary differential equations as discussed by the authors.
Abstract: The theory of differential equations and control have been linked very closely because most of the early applications of control theory were to engineering problems of the type which are most naturally described by ordinary differential equations. The questions of importance in control have helped to revitalize certain problem areas in differential equations and methods and tools from control have been useful in obtaining new results in differential equation theory. On the other hand, going back to the era of Lie himself, there has been close ties between Lie theory and differential equations. Thus it is not surprising that one finds that Lie theory and control are also closely connected. This „triangle“ is the subject of this set of notes.
229 citations
Book•
01 Jan 1973
TL;DR: In this article, the Spectral Theorem for Compact Operators in Hilbert Space has been proposed, which is based on the idea of the spectrum of an element and the spectrum-of-an-element.
Abstract: I. Algebras and Banach Algebras.- 1. Algebras and Norms.- 2. The Group of Units and the Quasigroup.- 3. The Maximal Ideal Space.- 4. The Spectrum of an Element.- 5. The Spectral Norm Formula.- 6. Commutative Banach Algebras and their Ideals.- 7. Radical and Semisimplicity.- 8. Involutive Algebras.- 9. H* Algebras.- Remarks.- II. Operators and Operator Algebras.- 1. Topologies on Vector Spaces and on Operator Algebras.- 2. Compact Operators.- 3. The Spectral Theorem for Compact Operators.- 4. Hilbert-Schmidt Operators.- 5. Trace Class Operators.- 6. Vector Valued Line Integrals.- 7. Homomorphisms into A. The Spectral Mapping Theorem.- 8. Unbounded Operators.- Remarks.- III. The Spectral Theorem, Stable Subspaces and v. Neumann Algebras.- 1. Linear Functionals on Vector Lattices and their Extensions.- 2. Linear Functionals on Lattices of Functions.- 3. The Spectral Theorem for SelfAdjoint Operators in Hilbert Space.- 4. Normal Elements and Normal Operators.- 5. Stable Subspaces and Commutants.- 6. von Neumann Algebras.- 7. Measures on Locally Compact Spaces.- Remarks.- IV. Elementary Representation Theory in Hilbert Space.- 1. Representations and Morphisms.- 2. Irreducible Components, Equivalence.- 3. Intertwining Operators.- 4. Schur's Lemma.- 5. Multiplicity of Irreducible Components.- 6. The General Trace Formula.- 7. Primary Representations and Factorial v. Neumann Algebras.- 8. Algebras and Representations of Type I.- 9. Type II and III v. Neumann Algebras.- Remarks.- Preliminary Remarks to Chapter V.- V. Topological Groups, Invariant Measures, Convolutions and Representations.- 1. Topological Groups and Homogeneous Spaces.- 2. Haar Measure.- 3. Quasi-Invariant and Relatively Invariant Measures.- 4. Convolutions of Functions and Measures.- 5. The Algebra Representation Associated with ?:S??(?).- 6. The Regular Representations of Locally Compact Groups.- 7. Continuity of Group Representations and the Gelfand-Raikov Theorem.- Remarks.- VI. Induced Representations.- 1. The Riesz-Fischer Theorem.- 2. Induced Representations when G/H has an Invariant Measure.- 3. Tensor Products.- 4. Induced Representations for Arbitrary G and H.- 5. The Existence ofa Kernel for L1(G)??(K).- 6. The Direct Sum Decomposition of the Induced Representation ?:G?u(K).- 7. The Isometric Isomorphism between ?2 and HS(K2, K1). The Computation of the Trace in Terms of the Associated Kernel.- 8. The Tensor Product of Induced Representations.- 9. The Theorem on Induction in Stages.- 10. Representations Induced by Representations of Conjugate Subgroups.- 11. Mackey's Theorem on Strong Intertwining Numbers and Some of its Consequences.- 12. Isomorphism Theorems Implying the Frobenius Reciprocity Relation.- Remarks.- VII. Square Integrable Representations, Spherical Functions and Trace Formulas.- 1. Square Integrable Representations and the Representation Theory of Compact Groups.- 2. Zonal Spherical Functions.- 3. Spherical Functions of Arbitrary Type and Height.- 4. Godement's Theorem on the Characterization of Spherical Functions.- 5. Representations of Groups with an Iwasawa Decomposition.- 6. Trace Formulas.- Remarks.- VIII. Lie Algebras, Manifolds and Lie Groups.- 1. Lie Algebras.- 2. Finite Dimensional Representations of Lie Algebras. Cartan's Criteria and the Theorems of Engel and Lie.- 3. Presheaves and Sheaves.- 4. Differentiable Manifolds.- 5. Lie Groups and their Lie Algebras.- 6. The Exponential Map and Canonical Coordinates.- 7. Lie Subgroups and Subalgebras.- 8. Invariant Lie Subgroups and Quotients of Lie Groups. The Projective Groups and the Lorentz Group.- Remarks.- Index of Notations and Special Symbols.
179 citations
TL;DR: In this article, the authors studied square integrable irreducible unitary representations of simply connected nilpotent Lie groups and determined which such groups have such representations, and gave a simple direct formula for the formal degrees of such representations and also an explicit simple version of the Plancherel formula.
Abstract: We study square integrable irreducible unitary representations (i.e. matrix coefficients are to be square integrable mod the center) of simply connected nilpotent Lie groups N, and determine which such groups have such representations. We show that if N has one such square integrable representation, then almost all (with respect to Plancherel measure) irreducible representations are square integrable. We present a simple direct formula for the formal degrees of such representations, and give also an explicit simple version of the Plancherel formula. Finally if r is a discrete uniform subgroup of N we determine explicitly which square integrable representations of N occur in L2(Nr/J), and we calculate the multiplicities which turn out to be formal degrees, suitably normalized.
170 citations
TL;DR: The structure of solvable Lie groups and the splitting theorems of compact solvmanifolds are discussed in detail in this paper, where a global theorem concerning the uniqueness of compact Lie groups is given.
Abstract: CHAPTER III SOLVABLE LIE GROUPS AND SOLVMANIFOLDS 1 Examples 2 The structure of solvable Lie groups—the splitting theorems 3 Splitting theorems and a global theorem concerning solvmanifolds 4 Abstract splitting theorems and uniqueness of compact solvmanifolds 5 Existence of compact solvmanifolds 6 Presentations of compact solvmanifolds 7 Compact solvmanifolds with discrete isotropy groups
146 citations
TL;DR: In this article, the Fourier transformation is shown to be an intertwining operator between ρ(ƒ0,b−) and ρ (ƒ 0,) for a suitable real polarization h.
TL;DR: In this article, the asymptotic behavior of the Haar measure of Un = U · U ··· U, where U is a compact neighborhood of the identity in a separable, connected locally compact group, is considered.
TL;DR: Gel'fand as mentioned in this paper introduced a new language to describe many problems of differential geometry: for example, problems connected with the theory of pseudogroups, Lie equations, foliations, characteristic classes.
Abstract: In this article we introduce a new language to describe many problems of differential geometry: for example, problems connected with the theory of pseudogroups, Lie equations, foliations, characteristic classes, etc. This is the language of infinite-dimensional Lie algebras and their homogeneous spaces. It is closely connected with the general idea of formal differential geometry set forth by I. M. Gel'fand in his lecture at the International Congress of Mathematicians in Nice. In addition to a detailed account of the theory of homogeneous spaces of infinite-dimensional Lie algebras, this article contains applications of this theory to the characteristic classes of foliations. It also includes results on these questions from earlier papers by I. M. Gel'fand, D. A. Kazhdan, D. B. Fuks, and the authors.
TL;DR: In this paper, the p-part of the multiplier of the Chevalley groups G A4, G 2(3) and F.(2) was determined, along with the Steinberg variations and the Tits simple group F (2).
Abstract: This paper presents results on Schur multipliers of finite groups of Lie type. Specifically, let p denote the characteristic of the finite field over which such a group is defined. We determine the p-part of the multiplier of the Chevalley groups G A4), G2(3) and F .(2); the Steinberg variations; the Ree groups of type F , and the Tits simple group F (2) . Introduction. For a general account of work on the Schur multipliers of the known finite simple groups, the reader is referred to [8]. The gaps in the multiplier situation as discussed in [8] have since been filled (specifically, rr2(.l) = 2, 772(.2) = 1, 772(M(24) ) = 1) and these results will appear in a paper dealing with multipliers of sporadic simple groups [9]. In this paper we prove the following theorems about multipliers of groups of Lie type. Theorem 1 will follow from the theorems of Chapter I, the statements of which contain detailed information about generators and relations for the (unique) covering groups of these groups. The results of Chapter II establish Theorems 2 and 3, and the results of Chapter III establish Theorems 4 and 5. Theorem 1. 7222(G2(4)) = 2, ttz3(G2(3)) = 3, zt22(F4(2)) = 2. Theorem 2. M(2A2(q)) SZ(SU(3, q)>, i.e., m(SU(3, »)) = 1. Theorem 3. Let G be a Steinberg variation defined over a finite field of characteristic p, i.e., G = 2a„(i)> n>2, 2Dnil), rz > 4, 3D^(q) or 2E&(q), where q is a power of p. Then M (G) = 1 except for M2(2A3(2))=* Z2, M3(2A3(3)) ^ Z3 xZ3, M2(2A5(2)) Si Z2 x Z2, M2(2E6(2)) Si Z2x Zy Theorem 4. // G is a Ree group of type F , then m(G) = 1. Theorem 5. The Tits simple group 2F.(2)' has trivial multiplier. Received by the editors December 30, 1971. AMS (MOS) subject classifications (1970). Primary 20C25, 20D05, 20G40; Secondary 20G05, 2CF25.
TL;DR: In this article, the authors focus from the beginning on the symmetry Lie group of canonical transformations in the classical picture and derive its representation in quantum mechanics, though they limit their discussion to the anisotropic oscillator in two dimensions, and indicate possible extensions of the reasoning to other problems in which we have accidental degeneracy.
Abstract: The problem of accidental degeneracy in quantum mechanical systems has fascinated physicists for many decades. The usual approach to it is through the determination of the generators of the Lie algebra responsible for the degeneracy. In these papers we want to focus from the beginning on the symmetry Lie group of canonical transformations in the classical picture. We shall then derive its representation in quantum mechanics. In the present paper we limit our discussion to the anisotropic oscillator in two dimensions, though we indicate possible extensions of the reasoning to other problems in which we have accidental degeneracy.
TL;DR: In this paper, the eigenstates of the annihilation type operator U = C + iS, where C and S are the cosine and sine operators for harmonic oscillator phase, are shown to be closely related to thermal equilibrium states of the oscillator and to provide a new interpretation of the thermal equilibrium density operator.
Abstract: Eigenstates of the annihilation type operator U = C + iS, where C and S are the ``cosine'' and ``sine'' operators for harmonic oscillator phase, are shown to be closely related to thermal equilibrium states of the oscillator and to provide a new interpretation of the thermal equilibrium density operator. The problem of creating such states is considered and a general theorem is established leading to the construction of interaction Hamiltonians which transform the eigenstates of U among themselves and, in particular, create them from the oscillator ground state. These Hamiltonians lead to representations of the Lie algebras of O(2,1) and O(3). It is suggested that the mathematical technique used, in which generalized U‐type operators provide the link between a group and its representations, has its own intrinsic interest for the study of Lie groups.
TL;DR: In this article, Hoel et al. showed that a differential operator D on a manifold M is globally hypoelliptic (GH) if when D/= g (with f E '(M), g E C"c(M)) then / E coc (M).
Abstract: We study differential operators D which commute with a fixed normal elliptic operator E on a compact manifold M. We use eigenfunction expansions relative to E to obtain simple conditions giving global hypoellipticity. These conditions are equivalent to D having parametrices in certain spaces of functions or distributions. An example is given by M = compact Lie group and and E = Casimir operator, with D any invariant differential operator. The connections with global subelliptic estimates are investigated. 0. Introduction. We say a differential operator D on a manifold M is globally hypoelliptic (GH) if when D/= g (with f E '(M), g E C"c(M)) then / E coc (M). We begin this paper by recalling some Fourier analysis relative to an elliptic operator E on a compact manifold M, and apply this to obtain simple conditions on the rate of growth of the Fourier transform (relative to E) of D which are equivalent to (GH). The growth conditions are interpreted as global solvability conditions. We apply these theorems to the case: M = compact homogeneous space, E = invariant Laplace-Beltrami operator, and D = any invariant differential operator. Some new examples are discussed. We note that global hypoellipticity seems to be quite directly connected with questions of number theory-unlike the (analytically) more delicate questions of local hypoellipticity. We connect the eigenfunction estimates of E-Fourier analysis with global subelliptic estimates. Almost every result in this paper can be extended to differential operators on vector bundles (see Wallach [11 I for the basic ideas). We thank Carl Hoel and William Sweeney for patiently teaching us about differential equations. We also thank Richard Bumby for suggesting the use of Pell's equation in ?3. 1. Fourier analysis relative to an elliptic operator. Let M be a compact manifold without boundary of dimension n with a fixed volume element dv. Let E be an elliptic, normal (EE =E E) differential operator of order e on M. We Received by the editors July 10, 1972. AMS (MOS) subject classifications (1970). Primary 35H05; Secondary 43A80. (1) Partially supported by NSF GP-20647. (2) Partially supported by Alfred P. Sloan Fellowship. Copyright
TL;DR: In this article, a two-level model is introduced and developed in order to describe the nuclear rotation at high angular momenta, and the implications of this symmetry including various subalgebras of R(8) corresponding to different particular cases are studied.
01 Nov 1973
TL;DR: In this paper, the following theorem was proved: if G is a Lie group which acts transitively on a compact pseudo-riemannian manifold, then G is geodesically complete.
Abstract: The following theorem answers a question raised by J. A. Wolf.
Theorem. Let M be a compact pseudo-riemannian manifold. Let G be a Lie group which acts transitively on M by isometries. Then M is geodesically complete
TL;DR: In this paper, it was shown that the kernel of the representation associated with an element of a unitary equivalence class of factor representations of the group C* algebra C*(G) can be used to obtain a bijection between M and the primitive ideal space of C* (G).
Abstract: LetG be a connected and simply connected solvable Lie group. In a previous paper (cf.[22]) we associated withG a familyM of geometrical objects (“generalized orbits”), and with each elementO ofM a unitary equivalence classF(O) of factor representations. IfG is nilpotent,M coincides with the orbit space of the coadjoint representation, and the mapO→F(O) reproduces essentially the Kirillov isomorphism betweenM and the dual ofG. As a fargoing extension of this, the principal result of the present paper asserts, that upon assigning to 0∈M the kernel of the representation, associated with some element ofF(O), of the groupC* algebraC*(G), we obtain a bijection betweenM and the primitive ideal space ofC*(G).
TL;DR: In this article, the authors discuss the accidental degeneracy in the problem of a particle in two-dimensional oscillator potential constrained to move in a sector of angle π/q,q integer.
Abstract: In this paper we discuss the accidental degeneracy in the problem of a particle in two dimensional oscillator potential constrained to move in a sector of angle π/q,q integer. The degeneracy is due to both the Hamiltonian and the boundary conditions. The symmetry Lie group of canonical transformations is suggested by the explicit form of a complete nonorthonormal set of states expressed in terms of the creation operators. This group is complex and the corresponding representation in quantum mechanics is nonunitary. We discuss briefly the appearance of complex canonical transformations in physical problems.
TL;DR: In this article, Coulomb integrals for electrons with hydrogenic wavefunctions are expressed in terms of operations of the Lie algebra of the group O(4,2) × O( 4,2).
TL;DR: In this paper, the problem of computing L((p) is studied in the case of holomorphic diffeomorphisms, and the problem is solved in the degenerate case by Atiyah and Singer.
Abstract: We are concerned with the problem of computing L((p). REMARK. Let G be a compact Lie group acting on X as a group of holomorphic diffeomorphisms and cp e G. The problem in this case has been solved by Atiyah and Singer, see [2]. Also in the case cp has isolated fixed points, the problem was solved in the nondegenerate case (see §2 for definition) by Atiyah and Bott in [1] and by Toledo and Tong ii>[6] and [7] in the degenerate case.
01 Jan 1973
TL;DR: In this paper, it was shown that the major limitation of Willsky and Lo's work is that it deals only with abelian Lie groups and this limitation prevents their techniques from being directly applicable to a number of problems.
Abstract: Recently a number of results [1]–[8] related to stochastic bilinear systems have been obtained. In particular, the work of Willsky and Lo [l]-[6] on optimal estimation on certain Lie groups leads to concrete and simple solutions to several bilinear estimation and stochastic control problems. The major limitation of Willsky and Lo’s work is that it deals only with abelian Lie groups. Although there are a number of interesting problems in that setting [4], this limitation prevents their techniques from being directly applicable to a number of problems.
01 May 1973
TL;DR: The monograph introduces the concepts and methods of the Lie theory used in current applications in a form accessible to the non-specialist by keeping mathematical pre-requisites to a minimum.
Abstract: : The monograph introduces the concepts and methods of the Lie theory used in current applications in a form accessible to the non-specialist by keeping mathematical pre-requisites to a minimum. Chapter 1 deals with the basic properties of Lie groups and Lie algebras and Chapter 2 covers representation theory. Chapter 3, on constructive methods, presents the computational aspects of the subject and includes material developed by the authors for using various algorithms in the Lie theory in programs for electronic digital computers. (Author)
TL;DR: In this paper, the authors studied the geodesic flow on homogeneous spaces with a Riemannian metric invariant under the group action, as one more application of Smale's theory.
Abstract: The geodesic flow on a homogeneous space with an invariant metric can be naturally considered within the framework of Smale's mechanical systems with symmetry. In this way we have at our disposal the whole method of Smale for studying such systems. We prove that certain sets E', E, Im, a and Re which play an important role in the global behavior of those systems, have a particularly simple structure in our case, and we also find some geometrical implications about the geodesics. The results obtained are especially powerful for the case of Lie groups, as in the rigid body problem. I. Introduction. Since Smale developed his theory of mechanical systems with symmetry to study the plane n-body problem from a global viewpoint (121, (131, a few applications have been considered to other mechanical systems admitting some symmetry. We study here from the global analysis viewpoint geodesics on homogeneous spaces with a Riemannian metric invariant under the group action, as one more application of Smale's theory. Roughly speaking, the paper is carried out as follows. The sets 2', 2, Im, a and Re associated with any mechanical system with symmetry furnish an important part of its global structure (?2). We show that those sets have particularly simple structure for our problem under consideration, and then we find some geometri- cal implications about the corresponding geodesics. The rigid body problem is the most classical example where our general results apply, and lacob (61 has already studied some aspects of it by using Smale's theory (see end of ?3). ?2 is devoted to notation and a quick review of Smale's theory. In ?3 we study properties of structure and invariance under the group action for the five above- mentioned sets. By exploiting the transitivity we find that in some precise sense they can be generated from much simpler sets. In particular, that simpler set Re'