On the Analytic Structure of Commutative Nilmanifolds

On a class of unitary representations of the braid groups B3 and B4

We revisit the classifications of classical and quantum galilean particles: that is, we fully classify homogeneous symplectic manifolds and unitary irreducible projective representations of the Galilei group. Equivalently, these are coadjoint orbits and unitary irreducible representations of the Bargmann group, the universal central extension of the Galilei group. We provide an action principle in each case, discuss the nonrelativistic limit, as well as exhibit, whenever possible, the unitary irreducible representations in terms of fields on Galilei spacetime. Motivated by a forthcoming study of planons we pay close attention to the mobility of the less familiar massless Galilei particles.

A complete description of all topological groups admitting a separating system of continuous irreducible finite-dimensional unitary representations of uniformly bounded dimensions is obtained. A similar description is obtained for the formally more general class of all unitarily representable topological groups whose von Neumann algebra is a sum of homogeneous summands of finite and uniformly bounded degrees. Related results are obtained; in particular, a description is given of all locally bounded topological groups all of whose irreducible unitary representations are finite-dimensional.

Introduction. A great deal of work has been done in the last fifteen years on the theory and classification of irreducible unitary representations of locally compact groups. It is also generally recognized (see for example [12, ? 8]) that, at least for some classes of groups, it is artificial to confine oneself to unitary representations; one ought to study the irreducible Banach space representations also. One cogent reason for this is that continuation of representations inevitably leads us outside the unitary domain. Thus one ought to try to extend to the case as much as possible of the theory developed for the unitary situation. The topology of the dual of C*-algebras and locally compact groups (that is, the of their irreducible unitary representations) has been studied in several papers (see [2], [3], [4], [6]). In the present paper we begin the study of the topology of the nonunitary dual spaces of algebras and groups. Two closely related motives suggest this: First, one would like to generalize the now classical theory of commutative Banach algebras to the noncommutative case even in the absence of an involution in particular to obtain a noncommutative version of the Banach-algebraic approach to the theory of analytic functions (see [ 14]). Secondly, one would like to use this to develop an abstract theory of continuation of group representations. Unfortunately, the space of all irreducible Banach space representations of an arbitrary Banach algebra or locally compact group seems too vast and unstructured to submit to a thorough analysis at present. We do not even know what we ought to mean by the equivalence of two such representations. Some happy of our subject-matter is called for. The work of Harish-Chandra and Godement [8] suggests what this restriction ought to be. Among Banach algebras A, it suggests that we should look first at those whose irreducible representations are all of bounded finite dimension, that is, whose dual are of bounded degree. Among locally compact groups, it suggests that we first examine groups G having a compact subgroup. The group algebra of such a group will have a large stock of subalgebras whose duals are of bounded degree; and the study of the irreducible Banach space representations of G reduces to that of the (finite-dimensional) irreducible representations of these subalgebras. The class of groups having compact subgroups includes all Euclidean

The Bondi-Metzner-Sachs groupBis the common asymptotic group of all asymptotically flat (lorentzian) space-times, and is the best candidate for the universal symmetry group of general relativity. However, in quantum gravity, complexified or euclidean versions of general relativity are frequently considered, and the question arises: Are there similar symmetry groups for these versions of the theory? In this paper it is shown that there are such analogues ofBand a variety of further ones, either real in any signature, or complex. The relationships between these various groups are described. Irreducible unitary representations (IRS) of the complexificationCBofBitself are analysed. It is proved that all induced IRS ofCBarise from IRS ofcompact'little groups’. It follows that some IRS ofCBare controlled by the IRS of the ‘A,D,E' series of finite symmetry groups of regular polygons and polyhedra in ordinary euclidean 3-space. Possible applications to quantum gravity are indicated.

Weyl–Pedersen calculus for some semidirect products of nilpotent Lie groups

Moment sets and unitary dual for the diamond group

ABSTRACTThe unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.

ABSTRACTThe inverse matrix of a Vandermonde matrix with arbitrary field entries of dimension with is well-studied in the literature. Especially, an explicit inversion formula does exist. However in theory as e.g. for determining (analytically) the eigenvalues of (arbitrary) matrix polynomials, Bernstein-Vandermonde matrices must be inverted. This note derives an explicit low-cost inversion formula for this matrix class with entries over a field whose characteristic equals zero. Its derivation requires at most additions and multiplications.

This is a survey of several models (including new models) of irreducible complementary series representations and their limits, special representations, for the groups and . These groups, whose geometrical meaning is well known, exhaust the list of simple Lie groups for which the identity representation is not isolated in the space of irreducible unitary representations (that is, which do not have the Kazhdan property) and hence there exist irreducible unitary representations of these groups, so-called `special representations', for which the first cohomology of the group with coefficients in these representations is non-trivial. For technical reasons it is more convenient to consider the groups and , and most of this paper is devoted to the group .The main emphasis is on the so-called commutative models of special and complementary series representations: in these models, the maximal unipotent subgroup is represented by multipliers in the case of , and by the canonical model of the Heisenberg representations in the case of . Earlier, these models were studied only for the group . They are especially important for the realization of non-local representations of current groups, which will be considered elsewhere.Substantial use is made of the `denseness' of the irreducible representations under study for the group : their restrictions to the maximal parabolic subgroup are equivalent irreducible representations. Conversely, in order to extend an irreducible representation of to a representation of , it is necessary to determine only one involution. For the group , the situation is similar but slightly more complicated.

On the irreducible representations of the lorentz group

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.

We classify the unitary representations of the extended Poincaré supergroups in three dimensions. Irreducible unitary representations of any spin can appear, which correspond to supersymmetric anyons. Our results also show that all irreducible unitary representations necessarily have physical momenta. This is in sharp contrast to the ordinary Poincaré group that admits in addition irreducible unitary representations with nonphysical momenta, which are discarded on physical grounds.

In this chapter we introduce semi-direct products such as the Poincare group, the Galilei group and the Bargmann group. We describe their irreducible unitary representations, which are induced from representations of their translation subgroup combined with a so-called little group. We interpret these representations as particles propagating in space-time and having definite transformation properties under the corresponding symmetry group. This picture will be instrumental in our study of the BMS\(_3\) group.

Two main results are obtained. First, for any unimodular type I almost connected group, it is proven that almost all of its irreducible unitary representations have global distribution characters. Second, for a certain class of semidirect products, these characters are computed and shown to be given by a function on an open dense subset, the function however not being locally integrable on the whole group.