The representations of the oscillator group

Using the Mackey theory of induced representations all the unitary continuous irreducible representations of the 4-dimensional Lie groupG generated by the canonical variables and a positive definite quadratic hamiltonian are found. These are shown to be in a one to one correspondence with the orbits underG in the dual spaceG to the Lie algebraG ofG, and the representations are obtained from the orbits by inducing from one-dimensional representations provided complex subalgebras are admitted. Thus a construction analogous to that ofKirillov andBernat gives all the representations of this group.The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research OAR through the European Office Aerospace Research, United States Air Force.     

Nonlinear group representations and evolution equations

The theory of nonlinear evolution equations developed by M. Flato, J. Simon and a few others is reviewed. The method of construction of global solutions is described and the cohomological and analytic properties of linearizability of these equations are described.Invited talk at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Yang-Baxterization of braid group representations

For a given braid group representation (BGR), a process of the Yang-Baxterization is formulated to generate solutions of the Yang-Baxter equation (YBE). When a BGR admits the Birman-Wenzl (BW) algebraic structure, this process can be explicitly passed through and two types of trigonometric solutions of YBE are generated from such a BGR. These two solutions have, the essential difference to each other and both of them, preserve the crossing symmetry property if the given BGR has. By taking certain, reduction on the BW algebra, the rational solution is also generated. A practical condition to judge whether a BGR satisfies the BW algebra is given, from which one finds that not only the familiar BGRs of [5,7,9], but also some new, ones obtained recently in [12] have the BW structure. Thus they can be explicitly Yang-Baxterized to solutions of the YBE.     

Notes on quaternionic group representations

We study quaternionic group representations of finite groups systematically and obtain some basic tools of the theory, such as orthogonality relations and the Clebsch-Gordan series for reducible representations. We also derive all irreducible inequivalentQ-representations of a groupG, classifying them according to a suitable generalization of the Wigner-Frobenius-Schur classification.     

Analytic continuation of group representations

Some general methods for analytically continuing group representations are presented. In favorable cases, this enables one to recognize the generalization of the principal, discrete and supplementary series.Work performed under the auspices of the United States Atomic Energy Commission.     

Superconformal current algebras and their unitary representations

A natural supersymmetric extension is defined of the current (= affine Kac-Moody Lie) algebra ; it corresponds to a superconformal and chiral invariant 2-dimensional quantum field theory (QFT), and hence appears as an ingredient in superstring models. All unitary irreducible positive energy representations of are constructed. They extend to unitary representations of the semidirect sumS (G) of with the superconformal algebra of Neveu-Schwarz, for , or of Ramond, for =0.On leave of absence from the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences, BG-1184 Sofia, Bulgaria     

Possible representations of local current algebra

Mass operators which lead to possible representations of local (vector) current algebra are discussed.     

Mechanical models for Lorentz group representations

Simple classical mechanical models are constructed to help understand the natures of certain unitary representations of the Lorentz groupSO(3, 1) associated with its action on spacetime. In particular, different kinds of Principal Series unitary irreducible representations ofSO(3, 1) with positive or negative quadratic Casimir invariant are seen to correspond to bounded and unbounded motions, respectively, in the mechanical models.     

Infinite-dimensional representations of the Einstein group

The representations of the general linear group GL(4.R) are described. This group corresponds to the space-time transformations discussed in the general theory of relativity. Besides the well-known tensor representations, the group is also characterized by infinite-dimensional representations with integral and half-integral spins. This fact opens up a natural possibility, in principle, of constructing a covariant theory of particle fields. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 5, 309–312 (10 September 1996)     

Analytic continuation of group representations II

The paper deals with the general background connecting the ideas of analytic continuation and contraction of Lie algebras and their representations. A connection is also established between the Kodaira-Spencer deformation theory and the theory of cohomology of Lie algebras.Supported by the Office of Naval Research, NONR 3656(09).     

Infinite-dimensional representations of the Lsorentz group: The complete series

The method used by Carmeli to obtain a new form for the principal series of representations of the groupSL(2, C) is further generalized to all completely irreducible (finite and infinite-dimensional) representations of that group. This is done, following Naimark, by extending the meaning of one of the parameters appearing in the formula for the operators of the principal series of representations. As a result a new form for the complete series of representations of the groupSL(2, C) is obtained which describes the transformation law of an infinite set of quantities under the group translation in a way which is very similar, but as a generalization, to the way spinors appear in the finite-dimensional case. The finite-dimensional representation is then discussed in details and the relation between the new set of quantities (which becomes finite in this case) and 2-component spinors is found explicitly.     

The six-point families of exceptional representations of the conformal group

Six-point families of topological reducible (indecomposable) representations of the conformal group of space-time M ₄ are studies. The structure of invariant subspaces is analysed and the full set of equivalence relations among the suprepresentations is derived.     

Deformation and contraction of Poincaré group representations

The irreducible representations of the Poincaré group with p ≠ 0 and discrete spin are analytically deformed to representations of the Galilei group. Via decomposition, reducible representations are also deformed. Consequences for physical interpretation of taking the limit c → ∞ are discussed.     

Highest weight representations of quantum current algebras

We study the highest weight and continuous tensor product representations ofq-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of theq-deformed algebra sl_q(2,) is calculated in detail.Alexander von Humboldt-Stiftung fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing, P.R. China.     

Analytic continuation of group representations. VI

The Gell-Mann formula for analytically continuing group representations is worked out explicitly for more cases than in previous work, and extended to certain pseudo-Riemannian symmetric spaces. The method of finding the asymptotic behavior of matrix elements of group representations introduced in Part V is developed in more detail and it is shown how it leads to new mathematical problems in the theory of dynamical systems and Hilbert space theory.Work supported by the U.S. Atomic Energy Commission.     

Irreducible unitary representations of quantum Lorentz group

A complete classification of irreducible unitary representations of a one parameter deformationS _q L(2,C) (0<q<1) ofSL(2,C) is given. It shows that in spite of a popular belief the representation theory forS _q L(2,C) is not a smooth deformation of the one forSL(2,C).     

Analytic continuation of group representations. III

The connection between the ideas of contraction and analytic continuation of Lie algebras and their representations is discussed, with particular emphasis on the contraction of the Poincaré to the Galilean group.This research was supported in part by the Office of Air Force Scientific Research AF 49 (638)-1440.     

A new method for constructing factorisable representations for current groups and current algebras

LetC _e (R ⁿ ,G) denote the group of infinitely differentiable maps fromn-dimensional Euclidean space into a simply connected and connected Lie group, which have compact support. This paper introduces a class of factorisable unitary representations ofC _e (R ⁿ ,G) with the property that the unitary operatorU _f corresponding to a functionf inC _e (R ⁿ ,G) depends not only onf, but also on the derivatives off up to a certain order. In particular these representations can not be extended to the group of all continuous functions fromR ⁿ toG with compact support.