Narrow Results

In this paper, we give a pedagogical presentation of the irreducible unitary representations of ${ℂ}^4⋊Spin(4,ℂ)$, that is, of the universal cover of the complexified Poincaré group ${ℂ}^4⋊SO(4,ℂ)$. These representations were first investigated by Roffman in 1967. We provide a modern formulation of his results together with some facts from the general Wigner–Mackey theory which are relevant in this context. Moreover, we discuss different ways to realize these representations and, in the case of a non-zero “complex mass”, we give a detailed construction of a more explicit realization. This explicit realization parallels and extends the one used in the classical Wigner case of ${ℝ}^4⋊{Spin}^0(1,3)$. Our analysis is motivated by the interest in the Euclidean formulation of Fermionic theories.

Let G = K ⋉ ℝⁿ, where K is a compact connected subgroup of O(n) acting on ℝⁿ by rotations. Let 𝔤 ⊃ 𝔨 be the respective Lie algebras of G and K, and pr : 𝔤* → 𝔨* the natural projection. For admissible coadjoint orbits and , we denote by the number of K-orbits in , which is called the Corwin–Greenleaf multiplicity function. Let π ∈ Ĝ and be the unitary representations corresponding, respectively, to and by the orbit method. In this paper, we investigate the relationship between and the multiplicity m(π, τ) of τ in the restriction of π to K. If π is infinite-dimensional and the associated little group is connected, we show that if and only if m(π, τ) ≠ 0. Furthermore, for K = SO(n), n ≥ 3, we give a sufficient condition on the representations π and τ in order that .

A smoothing operator for a unitary representation $\pi :G\to \mathrm{\text{U}}({\mathscr{H}})$ of a (possibly infinite dimensional) Lie group $G$ is a bounded operator $A:{\mathscr{H}}\to {\mathscr{H}}$ whose range is contained in the space ${{\mathscr{H}}}^{\infty }$ of smooth vectors of $(\pi ,{\mathscr{H}})$. Our first main result characterizes smoothing operators for Fréchet–Lie groups as those for which the orbit map ${\pi }^A:G\to B({\mathscr{H}}),g\mapsto \pi (g)A$ is smooth. For unitary representations $(\pi ,{\mathscr{H}})$ which are semibounded, i.e. there exists an element $x_0\in 𝔤$ such that all operators $id\pi (x)$ from the derived representation, for $x$ in a neighborhood of $x_0$, are uniformly bounded from above, we show that ${{\mathscr{H}}}^{\infty }$ coincides with the space of smooth vectors for the one-parameter group ${\pi }_{x_0}(t)=\pi (expt{x}_0)$. As the main application of our results on smoothing operators, we present a new approach to host $C^{\ast }$-algebras for infinite dimensional Lie groups, i.e. $C^{\ast }$-algebras whose representations are in one-to-one correspondence with certain continuous unitary representations of $G$. We show that smoothing operators can be used to obtain host algebras and that the class of semibounded representations can be covered completely by host algebras. In particular, the latter class permits direct integral decompositions.

Let G be a unipotent algebraic group defined over a $p$-adic field of characteristic zero. We denote by $G$ the set of rational points of G. It is a $p$-adic Lie group with Lie algebra denoted by $𝔤$. Let $\pi$ be an irreducible unitary representation of $G$ in a Hilbert space ${{\mathscr{H}}}_{\pi }$, $f$ be a linear form on $𝔤$ and $𝔥$ be a polarization at $f$. We denote by ${\chi }_f$ a character of $H=exp(𝔥)$ related to $f$. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors $({{\mathscr{H}}}_{\pi }^{-\infty }{)}^{H,{\chi }_f}$ of $\pi$.

In this paper, it is shown that the covariant representation (CR) transforming the Dirac field under de Sitter isometries is equivalent to a direct sum of two unitary irreducible representations (UIRs) of the Sp(2, 2) group transforming alike the particle and antiparticle field operators in momentum representation. Their basis generators and Casimir operators are written down for the first time finding that these representations are equivalent to an UIR from the principal series whose canonical labels are determined by the fermion mass and spin. The properties of the conserved observables (i.e. one-particle operators) associated to the de Sitter isometries via Noether theorem and of the corresponding Pauli–Lubanski type operator are also pointed out.

We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of p-groups of lower p-series of class 2. Furthermore, we calculate Kazhdan constants of the tame automorphism groups of the free nilpotent groups.

A non-singular sesquilinear form is constructed that is preserved by the Lawrence–Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence–Krammer representation are chosen to be appropriate algebraically independent unit complex numbers, then the form is negative-definite Hermitian. Using the fact that non-invertible knots exist this result implies that there are matrices in the image of the Lawrence–Krammer representation that are conjugate in the unitary group, yet the braids that they correspond to are not conjugate as braids. The two primary tools involved in constructing the sesquilinear form are Bigelow's interpretation of the Lawrence–Krammer representation, together with the Morse theory of functions on manifolds with corners.

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

In this paper, utilizing the theory of Watson transform and Watson convolution, we explore the Watson wavelet convolution product and its related properties. The relation between the Watson Wavelet convolution product and Watson convolution is also computed. Watson wavelet transform and its inversion formula are analyzed heuristically. Watson two-wavelet multipliers and its trace class are derived from Watson wavelet convolution product

By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.

This paper proposes a method through which a family of wavelets can be obtained. This is done by choosing each member based on a random variable. The method is preferred in situations where a single mother wavelet proves inadequate and an evolving sequence of mother wavelets is needed but a priori the next member in the sequence is uncertain. The adopted approach is distinct from the way spatiotemporal wavelets are used or even the way stochastic processes have been studied using spatiotemporal wavelets.

In this paper, the boundedness and compactness of the Hankel wavelet multiplier associated with the unitary representation on $L^p$ space for $1\le p\le \infty$, are obtained by using the Hankel transform technique. The application of Hankel wavelet multipliers in form of the Landau–Pollak–Slepian operator is given. With the help of the Hankel wavelet multiplier, the Sobolev space associated with the Hankel transform is constructed and its various properties are studied.

A unitary representation of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators idπ(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra 𝔤. In the first part of the present paper we explain how this concept leads to a fruitful interaction between the areas of infinite dimensional convexity, Lie theory, symplectic geometry (momentum maps) and complex analysis. Here open invariant cones in Lie algebras play a central role and semibounded representations have interesting connections to C*-algebras which are quite different from the classical use of the group C*-algebra of a finite dimensional Lie group. The second half is devoted to a detailed discussion of semibounded representations of the diffeomorphism group of the circle, the Virasoro group, the metaplectic representation on the bosonic Fock space and the spin representation on fermionic Fock space.

In this note we introduce the concept of a semi-bounded unitary representations of an infinite-dimensional Lie group G. Semi-boundedness is defined in terms of the corresponding momentum set in the dual 𝔤' of the Lie algebra 𝔤 of G. After dealing with some functional analytic issues concerning certain weak-*-locally compact subsets of dual spaces, called semi-equicontinuous, we characterize unitary representations which are bounded in the sense that their momentum set is equicontinuous, we characterize semi-bounded representations of locally convex spaces in terms of spectral measures, and we also describe a method to compute momentum sets of unitary representations of reproducing kernel Hilbert spaces of holomorphic functions.

We investigate faithful unitary representations for Polishable ideals, or more generally abelian Polish groups. We give various characterizations for the existence of such representations. The main technical result is to show that the density ideal does not admit any faithful unitary representations.