Infinite-dimensional Representations Research Articles

Identification of a 2D turbulent wind spectrum

Turbulent wind spectrum models usually involve nonrational terms. For such models the development of a Markovian time simulator relies on a rational approximation obtained by the way of an identification stage. This paper presents a general method to manage this identification stage, that provides us with a family of stable rational approximations which is proved to converge towards the true model as the dimension increases. We first give an exact but infinite dimensional state space representation of the spectrum. It is based on the use of a diffusive equation. For bi-dimensional (2D) signals it also uses decoupling by spatial Fourier transformation. Then the discretization of this exact model leads easily to stable finite dimensional approximations over a prescribed frequency range. This approach is applied to the identification of a theoretical 2D turbulent wind spectrum, and of a 1D turbulent wind spectrum estimated from in flight recorded data.

Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants

Let $k$ be a local field of characteristic not $2$, and let $G$ be the group of $k$-rational points of a connected reductive linear algebraic group defined over $k$ with a simple derived group of $k$-rank at least $2$. We construct new uniform pointwise bounds for the matrix coefficients of all infinite-dimensional irreducible unitary representations of $G$. These bounds turn out to be optimal for ${\rm SL}\sb n(k), n\geq 3$, and ${\rm Sp}\sb {2n}(k),n\geq 2$. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of semisimple $G$.

A New Class of Two-Dimensional Noncommutative Spaces

We find an infinite number of noncommutative geometries which posses a differential structure. They generalize the two dimensional noncommutative plane, and have infinite dimensional representations. Upon applying generalized coherent states we are able to take the continuum limit, where we recover the punctured plane with non constant Poisson structures.

THE RIESZ REPRESENTATION THEOREM ON INFINITE DIMENSIONAL SPACES AND ITS APPLICATIONS

Let H be a real separable Hilbert space and let E⊂H be a nuclear space with the chain {Em: m=1,2,…} of Hilbert spaces such that E = ∩m∈ℕEm. Let E* and E-m denote the dual spaces of E and Em, respectively. For γ > 0, let [Formula: see text] be the collection of complex-valued continuous functions f defined on E* such that [Formula: see text] is finite for every m. Then [Formula: see text] is a complete countably normed space equipping with the family {‖·‖m,γ : m = 1,2,…} of norms. Using a probabilistic approach, it is shown that every continuous linear functional T on [Formula: see text] can be represented uniquely by a complex Borel measure νT satisfying certain exponential integrability condition. The results generalize an infinite dimensional Riesz representation theorem given previously by the first author for the case γ = 2. As applications, we establish a Weierstrass approximation theorem on E* for γ≥1 and show that the space [Formula: see text] spanned by the class { exp [i(x,ξ)] : ξ ∈ E} is dense in [Formula: see text] for γ>0. In the course of the proof we give sufficient conditions for a function space on which every positive functional can be represented by a Borel measure on E*.

C*-Envelopes and Interpolation Theory

The C*-envelope of an operator algebra A is the minimal C*-algebra that can be generated by a completely isometric representation of A. In this paper we compute the C*-envelopes of two operator algebras that arise in interpolation theory. The first example is the quotient of H∞(D 2 ) by the ideal of functions vanishing at 3 particular points. We prove, among other results, that the C*-envelope of this 3 dimensional operator algebra is infinite dimensional and, moreover, possesses infinite dimensional irreducible representations. The second example is the quotients of H∞(A), where A denotes an annulus, by the ideal of functions vanishing at n ≥ 3 points. We prove that the C*-envelope of this n-dimensional operator algebra is *-isomorphic to M n ⊗ C(T), where T denotes the unit circle.

Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and separation of variables

We analyze a completely integrable two-dimensional quantum-mechanical model that emerged in the recent studies of the compound gluonic states in multi-color QCD at high energy. The model represents a generalization of the well-known homogeneous Heisenberg spin magnet to infinite-dimensional representations of the SL(2, C) group and can be reformulated within the quantum inverse scattering method. Solving the Yang–Baxter equation, we obtain the R-matrix for the SL(2, C) representations of the principal series and discuss its properties. We explicitly construct the Baxter Q-operator for this model and show how it can be used to determine the energy spectrum. We apply Sklyanin's method of the separated variables to obtain an integral representation for the eigenfunctions of the Hamiltonian. We demonstrate that the language of Feynman diagrams supplemented with the method of uniqueness provide a powerful technique for analyzing the properties of the model.

The tensor product of exceptional representations on the general linear group

The exceptional representations are certain infinite-dimensional projective representations of the general linear group over a local field, somewhat analogous to the Weil representations of the symplectic group. We examine the decomposition of the tensor product of two exceptional representations. Our main results concern the multiplicity with which a given representation may occur in this product and the restrictions imposed upon a representation of the principal series by the assumption that it does occur.

Fractional helicity, Lorentz symmetry breaking, compactification and anyons

We construct the covariant, spinor sets of relativistic wave equations for a massless field on the basis of the two copies of the R-deformed Heisenberg algebra. For the finite-dimensional representations of the algebra they give a universal description of the states with integer and half-integer helicity. The infinite-dimensional representations correspond formally to the massless states with fractional (real) helicity. The solutions of the latter type, however, break down the (3+1)D Poincaré invariance to the (2+1)D Poincaré invariance, and via a compactification on a circle a consistent theory for massive anyons in d=2+1 is produced. A general analysis of the “helicity equation” shows that the (3+1)D Poincaré group has no massless irreducible representations with the trivial non-compact part of the little group constructed on the basis of the infinite-dimensional representations of sl(2, C) . This result is in contrast with the massive case where integer and half-integer spin states can be described on the basis of such representations, and means, in particular, that the (3+1)D Dirac positive energy covariant equations have no massless limit.

κ-Deformed Wigner construction of relativistic wave functions and free fields on κ-Minkowski space

We describe the extension of the Wigner's infinite-dimensional unitary representations of Poincaré group to the case of κ-deformed Poincaré group. We show that the corresponding coordinate wave functions on noncommutative space-time are described by free field equations on κ-deformed Minkowski space. The cases of Klein—Gordon, Dirac, Proca and Maxwell fields are considered. Finally some aspects of second quantization are also discussed.

Induced Representations of the Two-Parameter Jordanian Quantum Algebra

We construct induced infinite-dimensional representations of the two-parameter quantum algebraUg,h(gl(2)) which is in duality with the deformationGLg,h(2). The representations depend on two representation parameters, but split into one-parameter representations of a one-generator central subalgebra and the three-generator Jordanian quantum subalgebraU\(_{\tilde g} \) (sl(2)),\(\tilde g\) =g + h. The representations of the latter can be mapped to representations in one complex variable, which give anew deformation of the standard one-parameter vector-field realization ofsl(2). These infinite-dimensional representations are reducible for some values of the representation parameters, and then we obtain canonically the finite-dimensional representations ofU\({\tilde g}\) (sl(2)).

Classical Integrable Systems and Automorphisms of Representations of gl(2)

Outer automorphisms of infinite-dimensional representations of sl(2)algebra are applied to produce some classical integrable systems with continuous and discrete time. The associated Lax pairs and r-matrix algebras are constructed. Bibliography: 14 titles.

Algebraic Perturbation Theory for Hydrogen Atom in Weak Electric Fields

An algorithm for symbolic calculation of eigenvalues and eigenfunctions of a hydrogen atom in weak electric fields is suggested. A perturbation theory scheme is constructed that is based on an irreducible infinite-dimensional representation of algebra so(4, 2) of the group of dynamical symmetry for the hydrogen atom l1r. The scheme implementation does not rely on the assumption that the independent variables of the perturbation operator can be separated, and fractional powers of parabolic quantum numbers are not used in the recurrent relations determining the operation of algebra generators on the corresponding basis of the irreducible representation l2r. A seventh-order correction to the energy spectrum of the hydrogen atom in a uniform electric field is given. The algorithm suggested is implemented in REDUCE 3.6 l4r.

Indecomposable representations of the nonlinear Lie algebras

On the two-state Heisenberg–Weyl basis, the indecomposable representations of the nonlinear Lie algebras of vector type, Rνλ (ν,λ=0,±1), one of which is generated by two angular momentum operators J0, J2 and any component of an irreducible tensor operator of rank 1, T(1νλ) (ν,λ=0,±1), of an SO(3) group, are studied in detail. We give the explicit expressions for the infinite-dimensional indecomposable representations of Rνλ, which further subduce the infinite-dimensional indecomposable representations on the invariant subspaces and induce the finite-dimensional or infinite-dimensional (indecomposable, irreducible) representations on the quotient spaces. As special cases, the standard angular momentum representations of Rνλ are given also.

Analogies between finite-dimensional irreducible representations of SO(2n) and infinite-dimensional irreducible representations of Sp(2n,R). II. Plethysms

The basic spin difference character Δ″ of SO(2n) is a useful device in dealing with characters of irreducible spinor representations of SO(2n). It is shown here that its kth-fold symmetrized powers, or plethysms, associated with partitions κ of k factorize in such a way that Δ″⊗{κ}=(Δ″)r(κ)Πκ, where r(κ) is the Frobenius rank of κ. The analogy between SO(2n) and Sp(2n,R) is shown to be such that the plethysms of the basic harmonic or metaplectic character Δ̃ of Sp(2n,R) factorize in the same way to give Δ̃⊗{κ}=(Δ̃)r(κ)Π̃κ. Moreover, the analogy is shown to extend to the explicit decompositions into characters of irreducible representations of SO(2n) and Sp(2n,R) not only for the plethysms themselves, but also for their factors Πκ and Π̃κ. Explicit formulas are derived for each of these decompositions, expressed in terms of various group–subgroup branching rule multiplicities, particularly those defined by the restriction from O(k) to the symmetric group Sk. Illustrative examples are included, as well as an extension to the symmetrized powers of certain basic tensor difference characters of both SO(2n) and Sp(2n,R).

Analogies between finite-dimensional irreducible representations of SO(2n) and infinite-dimensional irreducible representations of Sp(2n,R). I. Characters and products

The analogy between the finite-dimensional spin representation Δ of SO(2n) and the infinite-dimensional representation Δ̃ of Sp(2n,R) is made precise. It is then shown that this analogy can be extended so as to provide a precise link between each finite dimensional unitary irreducible representation of SO(2n) and a corresponding infinite-dimensional unitary irreducible representation of Sp(2n,R). The analogy shows itself at the level of the corresponding characters and difference characters, and involves the use of Schur function methods to express both characters and difference characters of SO(2n) and Sp(2n,R) in terms of characters of irreducible representations of their common subgroup U(n). The analogy is extended still further to cover the explicit decomposition of not only tensor products of Δ and Δ̃ with other unitary irreducible representations of SO(2n) and Sp(2n,R), respectively, but also arbitrary tensor powers of Δ and Δ̃.

The concept of a noncommutative Riemann surface

We consider the compactification M(atrix) theory on a Riemann surface Σ of genus g>1. A natural generalization of the case of the torus leads to construct a projective unitary representation of π 1( Σ), realized on the Hilbert space of square integrable functions on the upper half-plane. A uniquely determined gauge connection, which in turn defines a gauged sl 2( R) algebra, provides the central extension. This has a geometric interpretation as the gauge length of a geodesic triangle, and corresponds to a 2-cocycle of the 2nd Hochschild cohomology group of the Fuchsian group uniformizing Σ. Our construction can be seen as a suitable double-scaling limit N→∞, k→−∞ of a U( N) representation of π 1( Σ), where k is the degree of the associated holomorphic vector bundle, which can be seen as the higher-genus analog of 't Hooft's clock and shift matrices of QCD. We compare the above mentioned uniqueness of the connection with the one considered in the differential-geometric approach to the Narasimhan–Seshadri theorem provided by Donaldson. We then use our infinite dimensional representation to construct a C ★-algebra which can be interpreted as a noncommutative Riemann surface Σ θ . Finally, we comment on the extension to higher genus of the concept of Morita equivalence.

Thermodynamic properties of a solid exhibiting the energy spectrum given by the logistic map

We show that the infinite-dimensional representation of the recently introduced logistic algebra can be interpreted as a nontrivial generalization of the Heisenberg or oscillator algebra. This allows us to construct a quantum Hamiltonian having the energy spectrum given by the logistic map. We analyze the Hamiltonian of a solid whose collective modes of vibration are described by this generalized oscillator and compute the thermodynamic properties of the model in the two-cycle and r=3.6785 chaotic region of the logistic map.

Commuting difference operators with elliptic coefficients from Baxter's vacuum vectors

For quantum integrable models with elliptic R-matrix, we construct the Baxter Q-operator in infinite-dimensional representations of the algebra of observables.

Highest weights, projective geometry, and the classical limit: II. Coherent states and integrable systems

Coherent states of the commutation relations, like highest weight vectors for compact semi-simple Lie groups, satisfy quadratic equations. This paper explores the situation for quadratic varieties of vectors in some other infinite-dimensional representations, the tau functions for loop groups providing one example. Other generalisations are discussed.

Exact results for a fully asymmetric exclusion process with sequential dynamics and open boundaries

An exact and rigorous calculation of the current and density profile in the steady state of the one-dimensional fully asymmetric simple-exclusion process with open boundaries and forward-ordered sequential dynamics is presented. The method is based on a matrix product representation of the steady-state probability distribution. The main idea is to choose a suitable representation in which the scalar products describing the current and local density profile for a chain of arbitrary finite size depend only on the elements in a finite number of rows and columns. This makes possible the use of a truncated finite-dimensional representation of the matrices and vectors involved. After performing the calculations, we lift the truncation by letting its dimensionality go to infinity. In this limit the results become exact for any size of the chain. By rescaling one of the infinite-dimensional matrix representations found in the work of Derrida et al. [J. Phys. A 26, 1493 (1993)] for their algebra, we obtain a symmetric ``propagator'' matrix. Its truncated version is diagonalized by orthogonal transformation for easy calculation of the relevant scalar products. An interpretation of the phase transitions between the different phases is given in terms of eigenvalue splitting from a bounded quasicontinuous spectrum. A precise description of the local density profiles is given for all values of the parameters. It is shown that the leading-order asymptotic form of the position-dependent terms in the local density changes within the low- and high-density phases, signaling the presence of a higher-order transition.