Showing papers on "Lie group published in 1975"

Prolongation structures of nonlinear evolution equations

Abstract: The prolongation structure of a closed ideal of exterior differential forms is further discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubically nonlinear Schrodinger equation. The prolongation structure in this case is explicitly given, and recurrence relations derived which support the conjecture that the structure is open—i.e., does not terminate as a set of structure relations of a finite‐dimensional Lie group. We introduce the use of multiple pseudopotentials to generate multiple Backlund transformation, and derive the double Backlund transformation. This symmetric transformation concisely expresses the (usually conjectured) theorem of permutability, which must consequently apply to all solutions irrespective of asymptotic constraints.

Classification of quaternionic spaces with a transitive solvable group of motions

Abstract: A complete classification of quaternionic Riemannian spaces (that is, spaces with the holonomy group , ) which admit a transitive solvable group of motions is given. It turns out that the rank of these spaces does not exceed four and that all spaces whose rank is less than four are symmetric. The spaces of rank four are in natural one-to-one correspondence with the Clifford modules of Atiyah, Bott and Shapiro. In this correspondence, the simplest Clifford modules, which are connected with division algebras, are mapped to symmetric spaces of exceptional Lie groups. Other Clifford modules, which are obtained from the simplest with help of tensor products, direct sums and restrictions, correspond to nonsymmetric spaces.Bibliography: 17 items.

Branching rules for classical Lie groups using tensor and spinor methods

Abstract: Tensor and spinor methods are used to derive branching rule formulae for the embedding of one classical Lie group in another. These formulae involve operations on S-functions. By the judicious use of identities satisfied by certain infinite series of S-functions, they are reduced to forms which may be used very efficiently. Eleven sets of the branching rule formulae derived are as simple as possible, in that they involve only a sum of positive terms, whilst four other sets involve some negative terms which ultimately cancel. The advantage of using a composite notation, both for mixed tensor and for spinor representations, is made apparent. A comparison is made with methods used to derive branching rules based on mapping from one weight space to another.

A note on coherent state representations of Lie groups

Abstract: The analyticity properties of coherent states for a semisimple Lie group are discussed. It is shown that they lead naturally to a classical ’’phase space realization’’ of the group.

Commutative Formal Groups

Abstract: Formal varieties.- Formal groups and buds.- The general equivalence of categories.- The special equivalences of categories.- The structure theorem and its consequences.- On formal groups in characteristic p.- Extending and lifting some formal groups.

Some properties of square-integrable representations of semisimple Lie groups

Abstract: In the theory of irreducible representations of a compact Lie group, the formula for the multiplicity of a weight and the so-called theorem of the highest weight are among the most important results. At least conjecturally, both of these statements have analogues for the discrete series of representations of a semisimple Lie group. Let G be a connected, semisimple Lie group, K c G a maximal compact subgroup, and suppose that rk K = rk G. Exactly in this situation, G has a non-empty discrete series [8]. Blattner's conjecture predicts how a given discrete series representation should break up under the action of K; precise statements can be found in [10], [15], [16]. Formally, the conjectured multiplicity formula looks just like the formula for the multiplicity of a weight. Partial results toward the conjecture have been proved in [10], [15]. More recently, the full conjecture was established for those linear groups G, whose quotient G/K admits a Hermitian symmetric structure [16]. As this paper was being completed, H. Hecht and I succeeded in proving Blattner's conjecture for all linear groups, by extending the arguments of [16]. According to Blattner's conjecture, any particular discrete series representation 7t contains a distinguished irreducible K-module V, with multiplicity one; moreover, w contains no irreducible K-module with a highest weight which is lower, in the appropriate sense, than that of V,:. For "most" discrete series representations, it was known that these two properties characterize at, up to infinitesimal equivalence, among all irreducible representations of G [10], [15]. In this paper, I shall give an infinitesimal characterization, by lowest K-type, for all discrete series representations. The result, which is stated as Theorem (1.3) below, closely resembles the theorem of the highest weight. I shall also draw a number of conclusions from it. The methods of this paper have some further, less immediate consequences, which will be taken up elsewhere. For the remainder of the introduction, I assume that G is a linear group.

An alternative class of supersymmetries

Abstract: The supersymmetry of Wess and Zumino is generalized to square roots of a class of ordinary Lie groups. The case of O2,3 is studied in detail. The authors obtain supersymmetric equations of motion for fields on a variant of de-Sitter space. The only dimensional parameter is the radius of space-time.

The characters of the holomorphic discrete series.

Abstract: This paper summarizes the derivation of an explicit and global formula for the character of any holomorphic discrete series representation of a reductive Lie group G which satisfies certain conditions. The only very restrictive condition is that G/K be a Hermitian symmetric space. (Here K is the maximal compact subgroup of G.)

The heat equation on compact Lie group

Abstract: McKean and Singer [9] posed the problem of the existence of an analogue of the Poisson's summation formula for manifolds other than flat tori. Y. Colin de Verdiere [3] gave an answer to it in the case of a 2-dimensional compact Riemannian manifold with negative sectional curvature. The purpose of this paper is to determine the Minakshisundaram's expansion (Theorem 3) related to the heat equation and to give an analogue of the Poisson's summation formula of the fundamental solution of this equation on a simply-connected compact semi-simple Lie group (Theorem 2). The author expresses his gratitude to Prof. K. Okamoto who suggested him to study the Poisson's formula when he obtained Theorem 3.

Estimation for rotational processes with one degree of freedom--Part I: Introduction and continuous-time processes

Abstract: A class of bilinear estimation problems involving single-degree-of-freedom rotation is formulated and resolved. Continuous-time problems are considered here, and discrete-time analogs will be studied in a second paper. Error criteria, probability densities, and optimal estimates on the circle are studied. An effective synthesis procedure for continuous-time estimation is provided, and a generalization to estimation on arbitrary Abelian Lie groups is included. Applications of these results to a number of practical problems including frequency demodulation will be considered in a third paper.

The Bondi-Metzner-Sachs Group in the Nuclear Topology

Abstract: The Bondi-Metzner-Sachs group is topologized as a nuclear Lie group, and it is shown th at irreducible representations arise from either (i) transitive SL{2,C) actions on supermomentum space, or (ii) cylinder measures in supermomentum space with respect to which the SL(2,C) action is strictly ergodic. The irreducibles arising from transitive actions are shown to be induced, and most of the theorems from a previous analysis (in which the group was given a Hilbert topology) are generalized so as to apply here. All non-discrete closed subgroups of 2, C) are found, and this analysis is used to construct all induced representations whose little groups are not both discrete and infinite. In the previous analysis, there were exactly two connected little groups, SU{2) and r (T double covers $0(2)). In the present analysis, exactly one additional connected little group A (which double covers E{2)) arises for faithful representations (that is, those for which the mass squared is defined); the associated mass squared value is zero. Exactly one further connected little group arises;

Extension theorems for reductive group actions on compact Kaehler manifolds

Abstract: It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

what is the Hamiltonian Formalism

Abstract: In this paper the basic concepts of the classical Hamiltonian formalism are translated into algebraic language. We treat the Hamiltonian formalism as a constituent part of the general theory of linear differential operators on commutative rings with identity. We take particular care in motivating the concepts we introduce. As an illustration of the theory presented here, we examine the Hamiltonian formalism in Lie algebras. We conclude by presenting a version of the orbit method in the theory of representations of Lie groups, which is a natural corollary of our view of the Hamiltonian formalism.

Recent advances in the Einstein--Cartan theory of gravity

Abstract: It is pointed out that the commutation relations among the fundamental vector fields defined on a Lorentz bundle with a connection generalize the corresponding relations for the Lie algebra of the Poincare group. It is apparent from these commutators that there is an analogy between torsion and curvature with regard to their relation to translations and rotations in the tangent spaces of the manifold. A detailed analysis of the variational principle that underlies the Einstein--Cartan theory leads to a theorem on the compatibility of the metric and affine structures in space-time. 29 references. (auth)

Unified Geometry of Internal Space with Space-Time

Abstract: The unified theory of gravitation with gauge fields proposed earlier is shown to be a (4 + n) -dimensional Einstein theory with n linearly independent complete spacelike Killing vector fields which span an n-dimensional metric submanifold and form a closed Lie algebra. In this picture the internal space is combined with space-time and the internal symmetry becomes a symmetry of the metric of the (4 + n) -dimensional combined space. The 4-dimensional space-time appears as the quotient space of the (4 + n) -dimensional enlarged space by the equivalence relation of the group of motions of the n Killing vector fields. (AIP)

Identical Relations in Finite Lie Rings

Abstract: The article is devoted to proving that the identical relations of a finite Lie ring are consequences of a finite number of such relations. Some of the results on algebraic Lie algebras may be of independent interest.Bibliography: 13 titles.

Coherent states and partition function

Abstract: The concept of coherent states for arbitrary Lie group is suggested as a tool for explicitly obtaining an integral representation of the partition function, whenever the Hamiltonian has a dynamical group. Two examples are thoroughly discussed: the case of the nilpotent group of Weyl related to a generic many-body problem with two-body interactions, and the case of\(\mathop \Pi \limits_{k^ \otimes }\)SU(1, 1)(κ) relevant for a superfluid system.