Irreducible Representation Space Research Articles

Fractal Weyl law for skew extensions of expanding maps

We consider compact Lie group extensions of expanding maps of the circle, essentially restricting to SU(2) extensions. The main objective of the paper is the associated Ruelle transfer (or pull-back) operator . Harmonic analysis yields a natural decomposition , where j indexes irreducible representation spaces. Using semi-classical techniques we extend a previous result by Faure proving an asymptotic spectral gap for the family when restricted to adapted spaces of distributions. Our main result is a fractal Weyl upper bound for the number of eigenvalues (the Ruelle resonances) of these operators out of some fixed disc centred on 0 in the complex plane.

CLIFFORD MODULES AND SYMMETRIES OF TOPOLOGICAL INSULATORS

We complete the classification of symmetry constraints on gapped quadratic fermion hamiltonians proposed by Kitaev. The symmetry group is supposed compact and can include arbitrary unitary or antiunitary operators in the Fock space that conserve the algebra of quadratic observables. We analyze the multiplicity spaces of real irreducible representations of unitary symmetries in the Nambu space. The joint action of intertwining operators and antiunitary symmetries provides these spaces with the structure of Clifford module: we prove a one-to-one correspondence between the ten Altland–Zirnbauer symmetry classes of fermion systems and the ten Morita equivalence classes of real and complex Clifford algebras. The antiunitary operators, which occur in seven classes, are projectively represented in the Nambu space by unitary "chiral symmetries". The space of gapped symmetric hamiltonians is homotopically equivalent to the product of classifying spaces indexed by the dual object of the group of unitary symmetries.

Discrete series of unitary irreducible representations of the U q (u(3, 1)) and U q (u(n, 1)) noncompact quantum algebras

The structure of all discrete series of unitary irreducible representations of the U q (u(3, 1)) and U q (u(n, 1)) noncompact quantum algebras are investigated with the aid of extremal projection operators and the q-analog of the Mickelsson-Zhelobenko algebra Z(g, g′) q . The orthonormal basis constructed in the infinite-dimensional space of irreducible representations of the U q (u(n, 1)) ⊇ U q (u(n)) algebra is the q-analog of the Gelfand-Graev basis in the space of the corresponding irreducible representations of the u(n, 1) ⊇ u(n) classical algebra.

Mean field theory for U(n) dynamical groups

Algebraic mean field theory (AMFT) is a many-body physics modeling tool which firstly, is a generalization of Hartree–Fock mean field theory, and secondly, an application of the orbit method from Lie representation theory. The AMFT ansatz is that the physical system enjoys a dynamical group, which may be either a strong or a weak dynamical Lie group G. When G is a strong dynamical group, the quantum states are, by definition, vectors in one irreducible unitary representation (irrep) space, and AMFT is equivalent to the Kirillov orbit method for deducing properties of a representation from a direct geometrical analysis of the associated integral co-adjoint orbit. AMFT can be the only tractable method for analyzing some complex many-body systems when the dimension of the irrep space of the strong dynamical group is very large or infinite. When G is a weak dynamical group, the quantum states are not vectors in one irrep space, but AMFT applies if the densities of the states lie on one non-integral co-adjoint orbit. The computational simplicity of AMFT is the same for both strong and weak dynamical groups. This paper formulates AMFT explicitly for unitary Lie algebras, and applies the general method to the Lipkin–Meshkov–Glick (2) model and the Elliott (3) model. When the energy in the (3) theory is a rotational scalar function, Marsden–Weinstein reduction simplifies AMFT dynamics to a two-dimensional phase space.

The Polynomial Null Solutions of a Higher Spin Dirac Operator in Two Vector Variables

The polynomial null solutions are studied of the higher spin Dirac operator Q k,l acting on functions taking values in an irreducible representation space for Spin(m) with highest weight $${(k + \frac{1}{2},l+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})}$$ , with $${k,l \in \mathbb {N}}$$ and k = l.

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

The higher spin Dirac operator \(\mathcal{Q}_{k,l}\) acting on functions taking values in an irreducible representation space for \(\mathfrak{so}(m)\) with highest weight \((k+\frac{1}{2},l+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})\), with k, l ∈ \(\mathbb{N}\) and \(k\geqslant l\), is constructed. The structure of the kernel space containing homogeneous polynomial solutions is then also studied.

A toy model for higher spin Dirac operators

This paper deals with the higher spin Dirac operator Q2,1 acting on functions taking values in an irreducible representation space for so(m) with highest weight \( (\tfrac{5} {2},\tfrac{3} {2},\tfrac{1} {2},...,\tfrac{1} {2}) \). This operator acts as a toy model for generalizations of the classical Rarita—Schwinger equations in Clifford analysis. Polynomial null solutions for this operator are studied in particular.

APPROACH OF THE AZIMUTHAL AND MAGNETIC QUANTUM NUMBERS l AND m TO REPRESENTATION OF A CUBIC DEFORMATION OF su(2) ALGEBRA

It is shown that the space of spherical harmonics [Formula: see text] whose 2l - m = p - 1 is given, represent irreducibly a cubic deformation of su(2) algebra, the so-called su Φp(2), with deformation function as [Formula: see text]. The irreducible representation spaces are classified in three different bunches, depending on one of values 3k - 2, 3k - 1 and 3k, with k as a positive integer, to be chosen for p. So, three different methods for generating the spectrum of spherical harmonics are presented by using the cubic deformation of su(2). Moreover, it is shown that p plays the role of deformation parameter.

A generalized Wigner function on the space of irreducible representations of the Weyl–Heisenberg group and its transformation properties

A natural extension of the Wigner function to the space of irreducible unitary representations of the Weyl–Heisenberg group is discussed. The action of the automorphisms group of the Weyl–Heisenberg group onto Wigner functions and their generalizations and onto symplectic tomograms is elucidated. Some examples of physical systems are considered to illustrate some aspects of the characterization of the Wigner functions as solutions of differential equations.

New ladder operators for the monopole harmonics

Using the ladder operators shifting the index m of the associated Jacobi functions, for a given n, the monopole harmonics and their corresponding angular momentum operators are, respectively, extracted as the irreducible representation space and generators of su(2) Lie algebra. The indices n and m play the role of principal and azimuthal quantum numbers. By introducing the ladder operators shifting the index n of the same associated Jacobi functions, we also get a new type of the raising and lowering relations which are realized by the operators shifting only the index n of the monopole harmonics. Moreover, other symmetries, including the transformation of the irreducible representation spaces into each other, are derived based on the operators that shift the indices n and m of the monopole harmonics simultaneously and agreeably as well as simultaneously and inversely. Our results are reduced to spherical harmonics by eliminating magnetic charge of the monopole.

Algorithms for Computing U(N) Clebsch Gordan Coefficients

If V (m) is an irreducible representation space for the unitary group U(N), then the r-fold tensor product space, $V^{(m)_1}\otimes...\otimes V^{(m)_r}$ is in general reducible. Such a reducible representation can be reduced to a direct sum of irreducible representation spaces, albeit with multiplicity. Clebsch–Gordan coefficients are the overlap coefficients between an orthonormal basis in the tensor product space and an orthonormal basis in the direct sum space. Since such coefficients are basis dependent, bases in the U(N) irrep spaces must be chosen. In this paper we use the Gelfand–Zetlin basis, built out of the chain of subgroups U(N)⊃...⊃U(1). Our first result is to derive algorithms for generating Gelfand–Zetlin bases from Gelfand–Zetlin tableaux. Given these concrete basis realizations we develop algorithms for computing the Clebsch–Gordan coefficients themselves.

THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS

In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings. In this paper, a concise and elementary description of this refined Zariski topology is presented, under certain hypotheses, for the space of simple left modules over a ring R. Namely, if R is left noetherian (or satisfies the ascending chain condition for semiprimitive ideals), and if R is either a countable dimensional algebra (over a field) or a ring whose (Gabriel-Rentschler) Krull dimension is a countable ordinal, then each closed set of the refined Zariski topology is the union of a finite set with a Zariski closed set. The approach requires certain auxiliary results guaranteeing embeddings of factor rings into direct products of simple modules. Analysis of these embeddings mimics earlier work of the first author and Zimmermann-Huisgen on products of torsion modules.

Rarita–Schwinger fields in the half space†

This article deals with the Hardy space related to solutions of generalized Rarita–Schwinger operators in the ball and the half space. These operators generalize the Rarita–Schwinger operator for spin-fields to the case of functions taking values in irreducible representation spaces with weight. †Dedicated to Richard Delanghe on the occasion of his 65th birthday.

Mean field theory forsu(4) simeso(6)

Algebraic mean field theory constructs a group theoretical model of quantum systems that have a weak dynamical symmetry but may break dynamical symmetry. The strong defining condition for dynamical symmetry is that states belong to one irreducible representation space. Weak dynamical symmetry demands that the densities corresponding to the states have a constant value for each Casimir. Quantum phase transitions and other complex systems exhibit weak dynamical symmetry. Furthermore mean field theory often yields analytic formulae for expectations and energy spectra that are not feasible in representation theory. This paper develops mean field theory on any coadjoint orbit of su(4) densities. The simple Lie algebra su(4) so(6) is a 15-dimensional algebra that contains the subalgebra usp(4) so(5) and the angular momentum algebra su(2). The su(4) dual space consists of density matrices which are defined by the expectations of the su(4) generators. A coadjoint orbit is a common level surface in the dual space of the three su(4) Casimirs. A Lax pair determines the dynamics of these densities on each coadjoint orbit. Analytic solutions are reported for rotating su(4) densities in equilibrium for a particular energy function.

An algebraic approach to linear-optical schemes for deterministic quantum computing

Linear-Optical Passive (LOP) devices and photon counters are sufficient to implement universal quantum computation with single photons, and particular schemes have already been proposed. In this paper we discuss the link between the algebraic structure of LOP transformations and quantum computing. We first show how to decompose the Fock space of N optical modes in finite-dimensional subspaces that are suitable for encoding strings of qubits and invariant under LOP transformations (these subspaces are related to the spaces of irreducible unitary representations of U(N)). Next we show how to design in algorithmic fashion LOP circuits which implement any quantum circuit deterministically. We also present some simple examples, such as the circuits implementing a CNOT gate and a Bell-State Generator/Analyzer.

Landau levels on the hyperbolic plane

The quantum states of a spinless charged particle on a hyperbolic plane in the presence of a uniform magnetic field with a generalized quantization condition are proved to be the bases of the irreducible Hilbert representation spaces of the Lie algebra u(1, 1). The dynamical symmetry group U(1, 1) with the explicit form of the Lie algebra generators is extracted. It is also shown that the energy has an infinite-fold degeneracy in each of the representation spaces which are allocated to the different values of the magnetic field strength. Based on the simultaneous shift of two parameters, it is also noted that the quantum states realize the representations of Lie algebra u(2) by shifting the magnetic field strength.

Tensor operators and Wigner–Eckart theorem for the quantum superalgebraUq[osp(1|2)

Tensor operators in graded representations of Z_{2}-graded Hopf algebras are defined and their elementary properties are derived. Wigner-Eckart theorem for irreducible tensor operators for U_{q}[osp(1\mid 2)] is proven. Examples of tensor operators in the irreducible representation space of Hopf algebra U_{q}[osp(1\mid 2)] are considered. The reduced matrix elements for the irreducible tensor operators are calculated. A construction of some elements of the center of U_{q}[osp(1\mid 2)] is given.

Collective States in Highly Symmetric Atomic Configurations and Single-Photon Traps

We study correlated states in circular and linear-chain configurations of identical two-level atoms containing the energy of a single quasi-resonant photon in the form of a collective excitation, where the collective behavior is mediated by exchange of transverse photons between the atoms. For a circular atomic configuration containing N atoms, the collective energy eigenstates can be determined by group-theoretical means making use of the fact that the configuration possesses a cyclic symmetry group Z N . For these circular configurations, the carrier spaces of the various irreducible representations of the symmetry group are at most two-dimensional, so that the effective Hamiltonian on the radiationless subspace of the system can be diagonalized analytically. As a consequence, the radiationless energy eigenstates carry a Z N quantum number p = 0, 1, …, N, which is analogous to the angular momentum quantum number l = 0, 1, … carried by particles propagating in a central potential, such as a hydrogen-like system. Just as the hydrogen s states are the only electronic wave functions that can occupy the central region of the Coulomb potential, the quasi-particle corresponding to a collective excitation of the circular atomic sample can occupy the central atom only for vanishing Z N quantum number p. When a central atom is present, the p = 0 state splits into two, showing level crossing at certain radii; in the regions between these radii, damped oscillations between two “ extreme” p = 0 states occur, where the excitation occupies either the outer atoms or the central atom only. For large numbers of atoms in a maximally subradiant state, a critical interatomic distance of λ/2 emerges both in the linear-chain and in the circular configuration of atoms. The spontaneous decay rate of the linear configuration exhibits a jumplike “critical” behavior for next-neighbor distances close to a half-wavelength. Furthermore, both the linear-chain and the circular configurations exhibit exponential photon trapping once the next-neighbor distance becomes less than a half-wavelength, with the suppression of spontaneous decay being particularly pronounced in the circular system. In this way, circular configurations containing sufficiently many atoms may be natural candidates for single-photon traps.

Integral varieties of the canonical cone structure on G/P

The canonical cone structure on a compact Hermitian symmetric space G/P is the fiber bundle Open image in new window where Open image in new window is the cone of the highest weight vectors under the action of the reductive part of P. It is known that the cone Open image in new window coincides with the cone Open image in new window of the vectors tangent to the lines in G/P passing through x, when we consider G/P as a projective variety under its homogeneous embedding into the projective space Open image in new window of the irreducible representation space Vλ of G with highest weight λ associated to P. A subvariety X of G/P is said to be an integral variety of Open image in new window at all smooth points x∈G/P. Equivalently, an integral variety of Open image in new window is a subvariety of G/P whose embedded projective tangent space at each smooth point is a linear space Open image in new window We prove a kind of rigidity of the integral varieties under some dimension condition. After making a uniform setting to study the problem, we apply the theory of Lie algebra cohomology as a main tool. Finally we show that the dimension condition is necessary by constructing counterexamples.

Noetherianity of the space of irreducible representations

LetR be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple leftR-modules (or, more generally, simple objects in a complete abelian category). Under this topology the points are closed, and whenR is left noetherian the corresponding topological space is noetherian. IfR is commutative (or PI, or FBN) the corresponding topological space is naturally homeomorphic to the maximal spectrum, equipped with the Zariski topology. WhenR is the first Weyl algebra (in characteristic zero) we obtain a one-dimensional irreducible noetherian topological space. Comparisons with topologies induced from those on A. L. Rosenberg’s spectra are briefly noted.