Dimensional Unitary Representations Research Articles

Quantum Rajeev–Ranken model as an anharmonic oscillator

The Rajeev–Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber k in a scalar field theory pseudodual to the 1 + 1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotational energy. The quantum theory has two dimensionless parameters. Upon separating variables in the Schrödinger equation, we find that the radial equation has a four-term recurrence relation. It is of type [0, 1, 16] and lies beyond the ellipsoidal Lamé and Heun equations in Ince’s classification. At strong coupling λ, the energies of highly excited states are shown to depend on the scaling variable λk. The energy spectrum at weak coupling and its dependence on k in a double-scaling strong coupling limit are obtained. The semi-classical Wentzel-Kramers-Brillouin (WKB) quantization condition is expressed in terms of elliptic integrals. Numerical inversion enables us to establish a (λk)2/3 dispersion relation for highly energetic quantized “screwons” at moderate and strong coupling. We also suggest a mapping between our radial equation and the one of Zinn-Justin and Jentschura that could facilitate a resurgent WKB expansion for energy levels. In another direction, we show that the equations of motion of the RR model can also be viewed as Euler equations for a step-3 nilpotent Lie algebra. We use our canonical quantization to uncover an infinite dimensional reducible unitary representation of this nilpotent algebra, which is then decomposed using its Casimir operators.

Obstructions to matricial stability of discrete groups and almost flat K-theory

A discrete countable group G is matricially stable if the finite dimensional approximate unitary representations of G are perturbable to genuine representations in the point-norm topology. For large classes of groups G, we show that matricial stability implies the vanishing of the rational cohomology of G in all nonzero even dimensions. We revisit a method of constructing almost flat K-theory classes of BG which involves the dual assembly map and quasidiagonality properties of G. The existence of almost flat K-theory classes of BG which are not flat represents an obstruction to matricial stability of G due to continuity properties of the approximate monodromy correspondence.

Super-zeta functions and regularized determinants associated with cofinite Fuchsian groups with finite-dimensional unitary representations

Let M be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $$\chi $$ denote a finite dimensional unitary representation of the fundamental group of M. Let $$\Delta $$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over M associated with $$\chi $$ . From the spectral theory of $$\Delta $$ , there are three distinct sequences of numbers: the first coming from the eigenvalues of $$L^{2}$$ eigenfunctions, the second coming from resonances associated with the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, $$\mathcal {Z}_-(s,z)$$ and $$\mathcal {Z}_+(s,z)$$ that encode the spectrum of $$\Delta $$ in such a way that they can be used to define the regularized determinant of $$\Delta -z(1-z)I$$ . The resulting formula for the regularized determinant of $$\Delta -z(1-z)I$$ in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $$z\leftrightarrow 1-z$$ .

On the Unitary Representations of the Braid Group B6

We consider a non-abelian leakage-free qudit system that consists of two qubits each composed of three anyons. For this system, we need to have a non-abelian four dimensional unitary representation of the braid group B 6 to obtain a totally leakage-free braiding. The obtained representation is denoted by ρ . We first prove that ρ is irreducible. Next, we find the points y ∈ C * at which the representation ρ is equivalent to the tensor product of a one dimensional representation χ ( y ) and μ ^ 6 ( ± i ) , an irreducible four dimensional representation of the braid group B 6 . The representation μ ^ 6 ( ± i ) was constructed by E. Formanek to classify the irreducible representations of the braid group B n of low degree. Finally, we prove that the representation χ ( y ) ⊗ μ ^ 6 ( ± i ) is a unitary relative to a hermitian positive definite matrix.

Infinite characters on GLn(Q), on SLn(Z), and on groups acting on trees

Answering a question of J. Rosenberg from [21], we construct the first examples of infinite characters on GLn(K) for a global field K and n≥2. The case n=2 is deduced from the following more general result. Let G a non amenable countable group acting on a locally finite tree X. Assume either that the stabilizer in G of every vertex of X is finite or that the closure of the image of G in Aut(X) is not amenable. We show that G has uncountably many infinite dimensional irreducible unitary representations (π,H) of G which are traceable, that is, such that the C⁎-subalgebra of B(H) generated by π(G) contains the algebra of the compact operators on H. In the case n≥3, we prove the existence of infinitely many characters for G=GLn(R), when R is an integral domain such that G is not amenable. In particular, the group SLn(Z) has infinitely many such characters for n≥2.

On the characteristic polynomials of multiparameter pencils

In this paper we study the characteristic polynomial of multiparameter pencil z1A1+z2A2+⋯+zsAs. The main theorem states that a unitary representation of a finitely generated group contains a one-dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor. It follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible. The result is of kin to the Dedekind and Frobenius theorem on finite group determinant.

Quantum expanders and growth of group representations

Let π be a finite dimensional unitary representation of a group G with a generating symmetric n-element set S⊂G. Fix ε>0. Assume that the spectrum of |S| -1 ∑ s∈S π(s)⊗π(s) ¯ is included in [-1,1-ε] (so there is a spectral gap ≥ε). Let r N ' (π) be the number of distinct irreducible representations of dimension ≤N that appear in π. Then let R n,ε ' (N)=supr N ' (π) where the supremum runs over all π with n,ε fixed. We prove that there are positive constants δ ε and c ε such that, for all sufficiently large integer n (i.e. n≥n 0 with n 0 depending on ε) and for all N≥1, we have expδ ε nN 2 ≤R n,ε ' (N)≤expc ε nN 2 . The same bounds hold if, in r N ' (π), we count only the number of distinct irreducible representations of dimension exactly =N.

Quadratic algebra for superintegrable monopole system in a Taub-NUT space

We introduce a Hartmann system in the generalized Taub-NUT space with Abelian monopole interaction. This quantum system includes well known Kaluza-Klein monopole and MIC-Zwanziger monopole as special cases. It is shown that the corresponding Schrödinger equation of the Hamiltonian is separable in both spherical and parabolic coordinates. We obtain the integrals of motion of this superintegrable model and construct the quadratic algebra and Casimir operator. This algebra can be realized in terms of a deformed oscillator algebra and has finite dimensional unitary representations (unirreps) which provide energy spectra of the system. This result coincides with the physical spectra obtained from the separation of variables.

Ultraproducts of quasirandom groups with small cosocles

Abstract A D-quasirandom group is a group without any non-trivial unitary representation of dimension less than D. Given a sequence of groups with increasing quasirandomness, it is natural to ask if the ultraproduct will end up with no finite dimensional unitary representation at all. This is not true in general, but we answer this question in the affirmative when the groups in question have uniform small cosocles, i.e., their quotients by small kernels are direct products of finite simple groups. Two applications of our results are given, one in triangle patterns inside quasirandom groups and one in self-bohrifying groups. Our main tools are some variations of the covering number for groups, different kinds of length functions on groups, and the classification of finite simple groups.

Recurrence approach and higher rank cubic algebras for the N-dimensional superintegrable systems

By applying the recurrence approach and coupling constant metamorphosis, we construct higher order integrals of motion for the Stackel equivalents of the N-dimensional superintegrable Kepler–Coulomb model with non-central terms and the double singular oscillators of type (). We show how the integrals of motion generate higher rank cubic algebra with structure constants involving Casimir operators of the Lie algebras L1 and L2. The realizations of the cubic algebras in terms of deformed oscillators enable us to construct finite dimensional unitary representations and derive the degenerate energy spectra of the corresponding superintegrable systems.

Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum

We generalize the construction of integrals of motion for quantum superintegrable models and the deformed oscillator algebra approach. This is presented in the context of 1D systems admitting ladder operators satisfying a parabosonic algebra involving reflection operators and more generally extended oscillator algebras with grading. We apply the construction on two-dimensional oscillators. We also introduce two new superintegrable Hamiltonians that are the anisotropic Dunkl and the singular Dunkl oscillators. Integrals are constructed by extending the approach of Daskaloyannis to include grading. An algebraic derivation of the energy spectra of the two models is presented, making use of finite dimensional unitary representations. We show how the spectra divide into sectors, and make comparisons with the physical case.

Family of N-dimensional superintegrable systems and quadratic algebra structures

Classical and quantum superintegrable systems have a long history and they possess more integrals of motion than degrees of freedom. They have many attractive properties, wide applications in modern physics and connection to many domains in pure and applied mathematics. We overview two new families of superintegrable Kepler-Coulomb systems with non-central terms and superintegrable Hamiltonians with double singular oscillators of type (n, N — n) in N-dimensional Euclidean space. We present their quadratic and polynomial algebras involving Casimir operators of so(N + 1) Lie algebras that exhibit very interesting decompositions Q(3) ⊕ so(N — 1), Q(3) ⊕ so(n) ⊕ so(N — n) and the cubic Casimir operators. The realization of these algebras in terms of deformed oscillator enables the determination of a finite dimensional unitary representation. We present algebraic derivations of the degenerate energy spectra of these systems and relate them with the physical spectra obtained from the separation of variables.

Realizing the analytic surgery group of Higson and Roe geometrically, part I: the geometric model

© 2015, Tbilisi Centre for Mathematical Sciences. We construct a geometric analog of the analytic surgery group of Higson and Roe for the assembly mapping for free actions of a group with values in a Banach algebra completion of the group algebra. We prove that the geometrically defined group, in analogy with the analytic surgery group, fits into a six term exact sequence with the assembly mapping and also discuss mappings with domain the geometric group. In particular, given two finite dimensional unitary representations of the same rank, we define a map in the spirit of η-type invariants from the geometric group (with respect to assembly for the full group C ∗ -algebra) to the real numbers.

A new family of N dimensional superintegrable double singular oscillators and quadratic algebra Q(3) ⨁ so(n) ⨁ so(N-n)

We introduce a new family of N dimensional quantum superintegrable models consisting of double singular oscillators of type (n, N-n). The special cases (2,2) and (4,4) have previously been identified as the duals of 3- and 5-dimensional deformed Kepler–Coulomb systems with u(1) and su(2) monopoles, respectively. The models are multiseparable and their wave functions are obtained in (n, N-n) double-hyperspherical coordinates. We obtain the integrals of motion and construct the finitely generated polynomial algebra that is the direct sum of a quadratic algebra Q(3) involving three generators, so(n), so(N-n) (i.e. Q(3) ⨁ so(n) ⨁ so(N-n)). The structure constants of the quadratic algebra itself involve the Casimir operators of the two Lie algebras so(n) and so(N-n). Moreover, we obtain the finite dimensional unitary representations (unirreps) of the quadratic algebra and present an algebraic derivation of the degenerate energy spectrum of the superintegrable model.

Finite dimensional compact and unitary Lie superalgebras

Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is achieved by describing the classification of real finite dimensional compact simple Lie superalgebras, and analyzing, in a rather elementary and direct way, the decomposition of reductive Lie superalgebras (g is a semisimple g0¯-module) over fields of characteristic zero into ideals.

Dihedral group frames which are maximally robust to erasures

Let be a natural number larger than two. Let be the Dihedral group, and an -dimensional unitary representation of acting in as follows. and for . For any representation which is unitarily equivalent to we prove that when is prime, there exists a Zariski open subset of such that for any vector any subset of cardinality of the orbit of under the action of this representation is a basis for . However, when is even, there is no vector in which satisfies this property. As a result, we derive that if is prime, for almost every (with respect to Lebesgue measure) vector in , the -orbit of is a frame which is maximally robust to erasures. We also consider the case where is equivalent to an irreducible unitary representation of the Dihedral group acting in a vector space and we provide conditions under which it is possible to find a vector such that has the Haar property.

New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies

Four new families of two-dimensional quantum superintegrable systems are constructed from k-step extension of the harmonic oscillator and the radial oscillator. Their wavefunctions are related with Hermite and Laguerre exceptional orthogonal polynomials (EOP) of type III. We show that ladder operators obtained from alternative construction based on combinations of supercharges in the Krein-Adler and Darboux Crum (or state deleting and creating) approaches can be used to generate a set of integrals of motion and a corresponding polynomial algebra that provides an algebraic derivation of the full spectrum and total number of degeneracies. Such derivation is based on finite dimensional unitary representations (unirreps) and doesn't work for integrals build from standard ladder operators in supersymmetric quantum mechanics (SUSYQM) as they contain singlets isolated from excited states. In this paper, we also rely on a novel approach to obtain the finite dimensional unirreps based on the action of the integrals of motion on the wavefunctions given in terms of these EOP. We compare the results with those obtained from the Daskaloyannis approach and the realizations in terms of deformed oscillator algebras for one of the new families in the case of 1-step extension. This communication is a review of recent works.

On the remainder term in the Weyl law for cofinite Kleinian groups with finite dimensional unitary representation

We obtain a new bound for the remainder term in the Weyl law for general cofinite Kleinian groups with finite dimensional unitary representation by applying a Tauberian theorem for the Laplace transform and a suitable truncation argument.

Periodicity in the stable representation theory of crystallographic groups

Deformation K -theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G . In all known examples, this spectrum is 2-periodic above the rational cohomological dimension of G (minus 2), in the sense that T. Lawson's Bott map is an isomorphism on homotopy in these dimensions. We establish a periodicity theorem for crystallographic subgroups of the isometries of k -dimensional Euclidean space. For a certain subclass of torsion-free crystallographic groups, we prove a vanishing result for the homotopy groups of the stable moduli space of representations, and we provide examples relating these homotopy groups to the cohomology of G . These results are established as corollaries of the fact that for each , the one-point compactification of the moduli space of irreducible n -dimensional representations of G is a CW complex of dimension at most k . This is proven using real algebraic geometry and projective representation theory.

Su(2) -expansion of the Lorentz algebra so(3,1 )

We exploit the Iwasawa decomposition to construct coherent state representations of [Formula: see text], the Lorentz algebra in 3 + 1 dimensions, expanded on representations of the maximal compact subalgebra [Formula: see text]. Examples of matrix elements computation for finite dimensional and infinite-dimensional unitary representations are given. We also discuss different base vectors and the equivalence between these different choices. The use of the [Formula: see text]-matrix formalism to truncate the representation or to enforce unitarity is discussed.