On the type and the standardform of induced UA representations

We derive to which of the three types an irreducible UA representation which is obtained with the procedure of generalised induction belongs. We analyse the question whether or not the irreducible induced UA representations are on standardform. The results hold as well for induced PUA representations.     

On a method of constructing irreducible multiplier representations

A systematic method is presented for constructing the irreducible multiplier representations (ray representations) of a class of finite groups corresponding to a given factor system. The method consists in first identifying a normal subgroup of prime index, classifying its irreducible multiplier representations into orbits and then inducing the required representations from these orbits. The proposed technique finds useful application in solid state physics where irreducible multiplier representations of the point group underlying the group of the wave vector are frequently required.     

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.     

Coherent states and induced representations

A generalization of the notion of coherent states is given. The following one-to-one correspondences are pointed out: (1) between covariant overcomplete systems of coherent states and a class of covariant semi-spectral measures; (2) between covariant semispectral measures and unitary irreducible subrepresentations of induced representations of Lie groups; (3) between unitary irreducible representations of Lie groups with covariant overcomplete systems of coherent states and unitary irreducible subrepresentations of induced representations, whose representation spaces are reproducing kernel Hilbert spaces.     

Projective Systems of Imprimitivity and Applications for the Galilei Group in 1+2 and 1+3 Dimensions

The study of the projective unitary irreducible representations of the Galilei group (in 1+3 and 1+2 dimensions) is usually done using firstly some group extensions techniques (in this way one is reduced to the study of true unitary representations) and then Mackey induction procedure. In this paper we reobtain these results using a different approach based on the notion of projective systems of imprimitivity due also to Mackey. This extension of the usual Mackey procedure is presented rather extensively and illustrated by detailed computations concerning the classification of the projective unitary irreducible representations.     

The representation theory of the symmetry group of lattice fermions as a basis for kinematics in lattice QCD

The symmetry group of staggered lattice fermions is discussed as a discrete subgroup of the symmetry group of the Dirac-Kähler equation. For the representation theory of this group, G. Mackey's generalization of E.P. Wigner's procedure for the construction of unitary representations of groups with normal subgroups is used. A complete classification of these irreducible representations by “momentum stars”, “flavour orbits” and “reduced spins” is given.     

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.     

Unitary Representations of Nilpotent Super Lie Groups

We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups.     

Algebraic investigations in landau model of structural phase transitions

First we introduce the basic notions of the theory of permutation representations: stabilizers, orbits, stable subsets and strata. Then we consider the relation between permutation and linear representations which lead to some formulae connecting subduction coefficients, Clebsch-Gordan coefficients, and dimensions of stability spaces. This relation also leads to the concept of suborbits. Epikernels, the subgroups which are stabilizers of vectors of irreducible subspaces (either on the complex or on the real field) — are, studied and several theorems about them are proved. Further we consider the relation between epikernels, stability spaces and strata for subspaces irreducible on the real field as compared with subspaces irreducible on the complex field. Finally, the exomorphism is defined with use of permutation representations. The vectors of irreducible subspaces and corresponding epikernels (their stabilizers) for real ireps (representations irreducible on the real) of the classical crystal point groups are given in the appendix.     

空间群的选择定则

本文在J.L.Birman方法的基础上,对布里渊区内跃迁和散射的选择定则,作了进一步的理论分析,在满足准动量守恒条件的情况下,提出了空间羣简约系数的比较简单的计算方法。并用这个方法计算了六角密积结构及纤维锌矿结构的空间羣简约系数,从而决定了这两个结构的布里渊区内跃迁和散射的选择定则。     

Positive operators and automorphism groups of bounded symmetric domains

The spectrum of self-adjoint operators arising from unitary representations of semi-simple Lie groups is investigated. A series of irreducible unitary representations in which certain generators of non-compact one-parameter subgroups are realized by positive operators is described. These representations occur only for groups of automorphisms of bounded symmetric domains.     

Duals of crystallographic groups. Band and quasi-band representations

Band representations are analyzed from a pure group theoretical point of view, with the aid of the dual of the crystallographic group (the set of equivalence classes of unitary irreducible representations). It is shown on the examples of the onedimensional crystallographic groups that we have to introduce a distinction between band and quasi-band representations, the wordband being reserved for induced representations.The dual of the groupF222 is explicitly constructed. It permits to show that two elementary band representations which have the same decompositions into unitary irreducible representations are not equivalent.     

The cyclic representations of the quantum superalgebra U q osp(2, 1) withq a root of unity

By generalizing De Concini and Kac's cyclic representation theory of quantum groups at roots of unity, the cyclic representations of the quantum superalgebra U _q osp(2, 1) are constructed in three classes: irreducible representations with single multiplicities, irreducible representations with the multiplicities larger than one, and indecomposable representations.This work is supported in part by the National Sciene Foundation in China.     

On the irreducible representations of the lorentz group

All continuous irreducible representations of the SL(2, C) group (as given by Naimark) are obtained by means of methods developed by Harish-Chandra and Kihlberg. The analysis is done in the SU(2) basis and a single closed expression for the matrix elements of the noncompact generators for an arbitrary irreducible representation of SL(2, C) is given. For the unitary irreducible representations the scalar product for each irreducible Hilbert space is found explicitly. The connection between the unitary irreducible representations of SL(2, C) and those of

is discussed by means of Inönü and Wigner contraction procedure and the Gell-Mann formula. Finally, due to physical interest, the addition of a four-vector operator to SL(2, C) unitary irreducible representations in a minimal way is considered; and all group extensions of the parity and time reversal operators by SL(2, C) are explicitly obtained and some aspects of their representations are treated.     

空间羣不可约表示对称冪及反对称冪的简约

本文将作者们对空间羣不可约表示直接乘积的简约所发展的方法推广到空间羣不可约表示对称幂及反对称幂的简约,列出了对六角密积与纤维锌矿结构对称化平方的计算结果;利用这些结果,对组态不隐定性作了初步讨论。     

Multiplet classification of the reducible elementary representations of real semisimple Lie groups: the SO e (p,q) example

A multiplet classification of the reducible elementary representations (=generalized principal series representations) is proposed as a useful intermediate step in the important problem of determining the irreducible composition factors of the elementary representations and their multiplicities and, thus, of the distribution characters of all irreducible representations of real semisimple Lie groups. As an example, the results for the elementary representations of SO _e (p, q) induced from its minimal parabolic subgroup are given. Further research is also indicated. Presented at the 1st National Congress of Bulgarian Physicists, Sofia, September 1983.     

Super Multiplicative Integrals

The globally supported order 0 irreducible unitary representations of gauge groups are known as multiplicative integrals, and arise as cocycle representations. We generalize them to the setting of gauge supergroups.     

Branching rules and replicating representations

The problem of finding irreducible representations of a simple Lie group that decompose into just pure replications of a representation of a subgroup is considered. Replications involving tensor representations of the orthogonal groups are studied in detail using counting groups whose representation dimensions correspond to branching multiplicities. Replications involving spinor and mixed tensor representations are briefly considered.     

Discrete Hamiltonian symmetries and semiclassical quantization

Most quantum Hamiltonian systems exhibit discrete symmetries. Allowing for these is crucial when properly calculating the fluctuation properties of the quantal spectrum. These properties are then employed to distinguish between classically chaotic or non-chaotic quantum systems. In general, semiclassical quantization procedures do not take into account irreducible representations of the Hamiltonian. A procedure is presented to take these into account in semiclassical quantization schemes and calculate some of the energy eigenvalues belonging to a specific irreducible representation.     

Representations of canonical anticommutation relations and implementability of canonical transformations

It is proved that irreducible representations of CAR are determined by the groups of implementable automorphisms of the correspondingC*-algebra. This is done by a study of implementable canonical transformations. Some results in the same directions for factor representations are given.