Projective unitary antiunitary representations of locally compact groups

We give here a systematic presentation of the theory of projective representations when antiunitary operators are present. In particular the imprimitivity theorem of Mackey is proved in this situation and all the unitary antiunitary representations of the extended Poincaré group are derived.     

Fourier analysis on multiplier representations of locally compact Abelian groups

Generalising known results [2] for vector groups, it is shown that, for an arbitrary multiplier ω for an arbitrary locally compact Abelian group G, there is a faithful normal semifinite trace on the von Neuman algebra generated by the regular ω-representation of G which is translation invariant in a certain sense. Analogues of the Fourier transformation, the Plancherel identity and the Fourier inversion formula are obtained in which this trace replaces the Haar measure on the dual group.     

Continuity conditions for finite-dimensional representations of connected locally compact groups

We obtain simple necessary and sufficient conditions for the continuity of a locally bounded finite-dimensional representation of a connected locally compact group. We also prove that the discontinuity group of a locally bounded finite-dimensional representation of a connected locally compact group is a central subgroup in the closure of the image.     

Fourier-Stieltjes localization in neighborhoods of finite-dimensional irreducible representations of locally compact groups

An analog of the localization principle for the almost convergence for any element of the Fourier-Stieltjes algebra (Fourier-Stieltjes transforms of bounded measures) is established for any unimodular amenable locally compact groups in neighborhoods of finite-dimensional representations of these groups. To the blessed memory of Yuri Solov’ev Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by the INTAS grant.     

Connected Lie groups having faithful locally bounded (not necessarily continuous) finite-dimensional representations

In the paper, it is proved that a connected Lie group admits a (possibly discontinuous) faithful finite-dimensional representation if and only if it admits a continuous faithful finite-dimensional representation, i.e., if and only if it is a linear Lie group.     

Properties of pseudocharacters on connected locally compact groups

Several specific properties of pseudocharacters on connected locally compact groups are indicated.     

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.     

Almost periodic functions on connected locally compact groups

Some automatic continuity conditions for almost periodic functions on connected Lie groups and connected locally compact groups are discussed.     

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.     

Invariant star products and representations of compact semisimple Lie groups

Starting from the work by F. A. Berezin, and earlier paper by the author defined an invariant star product on every nonexceptional Kähler symmetric space. In this Letter a recursion formula is obtained to calculate the corresponding invariant Hochschild 2-cochains for spaces of types II and III. An invariant star product is defined on every integral symplectic (Kähler) homogeneous space of simply-connected compact Lie groups (on every integral orbit of the coadjoint representation). The invariant 2-cochains are obtained from the Bochner-Calabi function of the space. The leading term of the lth-2-cochain is determined by the l-power of the Laplace operator.     

Continuity conditions for finite-dimensional representations of some locally bounded groups

Necessary and sufficient conditions are obtained for the continuity of finite-dimensional representations of “almost divisible” locally pseudocompact groups in terms of the oscillations of these representations at a point. Applications to the continuity problem for finite-dimensional representations of almost connected locally compact groups and to the theory of quasirepresentations are given. To the blessed memory of Eckehart Hotzel Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by an INTAS grant.     

Faithful actions of locally compact quantum groups on classical spaces

We construct examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on connected classical (noncompact) smooth manifolds. However, none of these actions can be isometric in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009), leading to the conjecture that the result obtained by Goswami and Joardar (Rigidity of action of compact quantum groups on compact, connected manifolds, 2013. arXiv:1309.1294) about nonexistence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the noncompact case as well.     

Algebras of observables with continuous representations of symmetry groups

Concrete C*-algebras, interpreted physically as algebras of observables, are defined for quantum mechanics and local quantum field theory.Aquantum mechanical system is characterized formally by a continuous unitary representation up to a factorU _g of a symmetry group in Hilbert space and a von Neumann algebra on invariant with respect toU _g . The set of all operatorsX such thatU _g X U _g ^–1 , as a function ofg , is continuous with respect to the uniform operator topology, is aC*-algebra called thealgebra of observables. The algebra is shown to be the weak (or strong) closure of .Infield theory, a unitary representation up to a factorU(a, ) of the proper inhomogeneous Lorentz group and local von Neumann algebras _C for finite open space-time regionsC are assumed, with the usual transformation properties of underU(a, ). The collection of allX_C giving uniformly continuous functionsU (a, )X U ^–1 (a, ) on is then a localC*-algebra , called thealgebra of local observables. The algebra is again weakly (or strongly) dense in _c . The norm-closed union of the for allC is calledalgebra of quasilocal observables (or quasilocal algebra).In either case, the group is represented by automorphisms V _g resp. V(a, ) — with V _g X=U _g X U _g ^–1 — of theC*-algebra , and this is astrongly continuous representation of on the Banach space . Conditions for V (a, ) can then be formulated which correspond to the usualspectrum condition forU (a, ) in field theory.Work supported in part by the Deutsche Forschungsgemeinschaft.     

Matrix elements of representations of non-compact groups in a continuous basis

Explicit formulas are obtained by a simple algebraic method for the representations of the finite group transformations ofO(2,1) in a continuous basis when a non-compact generator is diagonalized. Compact and non-compact cases are treated in a unified form and the nature of analytic continuation is determined. The transformation function between the discrete and the continuous bases is also given. These explicit formulas have not been obtained in the literature before.Supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under Grant No. AF-AFOSR- 30–67.     

Some tips on the decomposition of tensor product representations of compact connected lie groups

Let ?₁, ?₂, ?₃, ?₄ be irreducible representations of a compact connected semisimple Lie group G with highest weights Λ₁, Λ₂, Λ₃, Λ₄, respectively. Let ?₁?₂ (resp. ?₃?₄) be irreducible representations of G with highest weights Λ₁·Λ₂ (resp. Λ₃·Λ₄). It is assumed that one knows the Clebsch-Gordan series of ?₁??₃ and ?₂??₄ (resp. of S²?₁, S²?₂, A²?₁ and A²?₂). Then we formulate a result (Theorem 2) which gives information on the decomposition of ?₁?₂??₃?₄ (resp. of S²(?₁?₂) and A²(?₁?₂)). Though this result is not complete, it is useful because it delivers the information very quickly and in a very simple manner.     

Extension of K.M.S. states with respect to some locally compact groups

Given a state ω on the C^*-system { , R×G, α} which is K.M.S. with respect to time on the sub-algebra of of fixed points under the action of some locally compact group G, we study its K.M.S.-extension to the whole algebra. In this work, we study more precisely the cases where G is a class of locally compact abelian groups (such that { , G, α|G} is almost periodic) or a product of these groups with a compact group.     

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.