A multiplicity theorem for representations of inhomogeneous compact groups

The problem of finiteness of multiplicities of irreducible unitary representations of a compact subgroup is considered for decompositions of irreducible unitary representations of locally compact groups. A simple solution is found for inhomogeneous compact groups and for a physically interesting class of groups with a non-abelian radical.On leave of absence from Faculty of Nuclear Science, Czech Technical University, Prague, Czechoslovakia.     

Non-linear representations of extended supersymmetry with central charges

A new class of non-linear representations ofN extended supersymmetry is found. These non-linear realizations extend the Volkov-Akulov non-linear realizations to include central charges which can also be non-linearly realized. The general construction of the corresponding non-linear Lagrangians is presented. The coupling toN extended supergravity provides the simultaneous occurrence of the Higgs and super-Higgs effects. The corresponding formulae for spin-3/2 gravitinos and spin-1 vector boson masses are derived.     

Contribution of the inhomogeneous waves in angular-spectrum representations

Angular-spectrum representations of wave fields separate naturally into two parts, V_h(R,t) containing only homogeneous plane waves and V_i(R,t) containing only inhomogeneous plane waves. Some properties of V_i(R,t) are presented here. We conclude that, for some problems, V_i(R,t) has several unphysical properties, and that under certain specified conditions, V_i(R,t) can not be neglected compared to V_h(R,t), even far from the sources of the field.     

Unusual representations of local groups

By means of duality transformations we obtain representations of Yang-Mills groups which are neither tensors nor gauge fields, and fields which under general coordinate transformations are neither tensors nor tensor densities.     

Physical applications of inhomogeneous quantum groups

It is shown that the real formE _q (1, 1) is the kinematical symmetry of phonons. It is also indicated how a quantum symmetry works in the non-relativistic 1D lattice systemsPresented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.     

The Casimir operators of inhomogeneous groups

We have found the number of invariant operators for the inhomogeneous groups IGL(n, R), ISL(n, R), ISO(p, q), IU(p, q), ISU(p, q), ISp(2n), i.e. the inhomogeneous groups with the classical homogeneous subgroups, and also for the Weyl group W(p, q). For some special cases explicit forms of the invariant operators are obtained. We also discuss the methods applied, together with problems in some cases, possible further developments and relevance for the supersymmetric theories.     

Irreducible representations of subperiodic rod groups

A procedure for taking the irreducible representations of subperiodic rod groups from tables of irreducible representations of three-periodical space groups is derived. Examples demonstrating the use of this procedure and derivation of selection rules for direct and phonon assisted electrical dipole transitions are presented. The text was submitted by the authors in English.     

Unitary and non-unitary representations of the complex inhomogeneous Lorentz group

We construct unitary and non-unitary representations of the complex inhomogeneous Lorentz group, including all its unitary, irreducible representations. We discuss the decomposition of these representations when they are restricted to the real inhomogeneous Lorentz group. We also discuss the representations of the Poincaré group for which the translation subgroup transforms under a not necessarily unitary representation. We summarize briefly the physical motivation for this study.Supported by the National Science Foundation (NSF-GP-58) while at Brandeis University and by the National Science Foundation, Grant NSF-GP-3465, while at the Courant Institute of Mathematical Sciences.     

Contractions of representations of de Sitter groups

In order to construct the quantum field theory in a curved space with no old infinities as the curvature tends to zero, the problem of contraction of representations of the corresponding group of motions is studied. The definitions of contraction of a local group and of its representations are given in a coordinate-free manner. The contraction of the principal continuous series of the de Sitter groupsSO ₀(n, 1) to positive mass representations of both the Euclidean and Poincaré groups is carried out in detail. It is shown that all positive mass continuous unitary irreducible representations of the resulting groups can be obtained by this method. For the Poincaré groups the contraction procedure yields reducible representations which decompose into two non-equivalent irreducible representations.On leave of absence from the Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.     

Basis of irreducible representations of orthogonal groups

A method for constructing a basis of irreducible representations of unitary groupsU _n is generalized to orthogonal groupsO _n. The corresponding algorithm is given for the chain for groupsU _n O _n and also for the chain which is of great relevance in physics.The constructed basis is used for the calculation of the fractional six-particle parentage coefficients for separation of one-, two- and three-particle wave functions. All possible states with excitation quantaN2 are taken into account. The present results, together with previous work, are used for the investigation of the six-quark system, in particular for calculating theD-state probability in the ground state of this system.     

Discrete degenerate representations of non-compact unitary groups

Two degenerate principal series of irreducible unitary representations of an arbitrary non-compact unitary groupU(p,1) are derived. These series are determined by the eigenvalues of the first and second-order invariant operators, which are shown to possess a discrete spectrum. The explicit form of the corresponding harmonic functions is derived and the properties of the discrete representations are discussed in detail. Moreover, in the Appendix, we derive the properties of the corresponding degenerate representations of an arbitrary compactU(p) group.On leave of absence from Institute of Nuclear Research, Warsaw, Poland.On leave of absence from Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.     

Coherent state representations of nilpotent Lie groups

The coherent state representations of a connected and simply connected nilpotent Lie group are characterized in terms of the Kirillov correspondence, as being those irreducible unitary representations whose associated orbits under the coadjoint representation are linear varieties.     

Some monodromy representations of generalized braid groups

A flat connection on the trivial bundle over the complement inC ⁿ of the complexification of the system of the reflecting hyperplanes of theB _n,D _n Coxeter groups is built from a simple Lie algebra and its representation. The corresponding monodromy representations of the generalized braid groupsXB _n,XD _n are computed in the simplest case.     

Some double-valued representations of the linear groups

We review the mathematical theory ofSL(n, R) and its double-covering group , especially forn = 2, 3, 4. After discussing a variety of physical applications, we show that provides holonomic curved space (world) spinors with an infinite number of components. We construct the relevant holonomic manifield and discuss the gravitational interaction of a proton as an example.Invited paper, dedicated to Eugene Paul Wigner on the occasion of his eightieth birthday.Supported in part by a U.S. Department of Energy Research Grant.     

Norm continuous representations of locally compact groups

The structure of norm continuous representations of locally compact groups in Banach spaces is specified. Partially supported by the RFBR grant no. 08-01-00034 and by the Russian Government Program of Supporting Leading Scientific Schools under grant no. NSh-1562.2008.1. To the Memory of S. A.     

Lie algebras and representations of continuous groups

It is shown that every finitely generated continuous group has a subgroup generated by its infinitesimal transformations. This subgroup has a group algebra which is the Lie algebra of the group. By obtaining complete systems in the Lie algebra and complete rectangular arrays, it is shown that these can yield matrix representations of the continuous group. Illustrative examples are given for the rotation groups and for the full linear groups. It would seem that all the finite motion representations can be obtained by these methods, including spin representations of rotation groups. But the completeness of the method is not here demonstrated.     

Unitary representations of some infinite dimensional groups

We construct projective unitary representations of (a) Map(S ¹;G), the group of smooth maps from the circle into a compact Lie groupG, and (b) the group of diffeomorphisms of the circle. We show that a class of representations of Map(S ¹;T), whereT is a maximal torus ofG, can be extended to representations of Map(S ¹;G),     

Some double-valued representations of the linear groups

We review the mathematical theory of SL(n, R) and its double-covering group \(\overline {SL}\) (n, R), especially for n=2, 3, 4. After discussing a variety of physical applications, we show that \(\overline {SL}\) (3, R) provides holonomic curved space (“world”) spinors with an infinite number of components. We construct the relevant holonomic “manifield” and discuss the gravitational interaction of a proton as an example.     

On braided poisson and quantum inhomogeneous groups

The well known incompatibility between inhomogeneous quantum groups and the standardq-deformation is shown to disappear (at least in certain cases) when admitting the quantum group to be braided. Braided quantumISO(p, N - p) containingSO _q (p, N - p) with |q|=1 are constructed forN=2p, 2p + 1, 2p + 2. Their Poisson analogues (obtained first) are presented as an introduction to the quantum case. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.