Infinite-dimensional Representations Research Articles

Products and symmetrized powers of irreducible representations of

The calculation of branching rules, tensor products and plethysms of the infinite-dimensional harmonic series unitary irreducible representations of the non-compact group is considered and the duality between and Sp(2k) exploited. The branching rule for the restriction of an arbitrary harmonic series irreducible representation of to U(n) is derived, and the decomposition is given explicitly for each of the infinite number of fundamental harmonic series irreducible representations, , of whose direct sum constitutes the metaplectic representation, H, of . A concise expression for the decomposition of tensor products is derived and a complete analysis of the terms in both and is given. A general formula for plethysms of arbitrary irreducible representations of is derived and its implementation illustrated both by means of a detailed generic example and by a complete determination of the symmetric and antisymmetric terms of . Finally, relationships that arise from the embedding of the product groups and in the metaplectic group Mp(4nk) are discussed.

Geometry of growth: approximation theorems for $L^2$ invariants

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Luck, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invariants, corresponding to infinite dimensional representations and sequences of finite dimensional representations, assuming that their normalized characters converge. Also, we find an approximation theorem for residually finite $p$-groups ($p$ is a prime), where we use the homology with coefficients in a finite field $\fp$. We view sequences of finite dimensional flat bundles of growing dimension as examples of growth processes. We study a von Neumann category with a Dixmier type trace, which allows to describe the asymptotic invariants of growth processes. We introduce a new invariant of torsion objects, the torsion dimension. We show that the torsion dimension appears in general as an additional correcting term in the approximation theorems; it vanishes under some arithmeticity assumptions. We also show that the torsion dimension allows to establish non-triviality of the Grothendieck group of torsion objects.

Universal R-matrix and graded Hopf algebra structure of

An explicit formula for the universal R-matrix of is given using Drinfeld's basis. Its generators are chosen according to a non-standard set of positive simple roots. The expression implies an extension of the standard graded Hopf algebra defined in terms of its Chevalley generators to . In addition to a four-dimensional vector representation, an infinite-dimensional representation of suitable for the description of a model related to the integer quantum Hall transition is considered.

Exact solutions in the Yang-Mills-Wong theory

This paper discusses in a systematical way exact retarded solutions to the classical SU(N) Yang-Mills equations with the source composed of several colored point particles. A new method of finding such solutions is reviewed. Relying on features of the solutions, a toy model of quark binding is suggested. According to this model, quarks forming a hadron are influenced by no confining force in spite of the presence of a linearly rising term of the potential. The large-N dynamics of quarks conforms well with Witten's phenomenology. On the semiclassical level, hadrons are color neutral in the Gauss law sense. Nevertheless, a specific multiplet structure is observable in the form of the Regge sequences related to infinite-dimensional unitary representations of SL(4,R) which is shown to be the color gauge group of the background field generated by any hadron. The simultaneous consideration of SU(N), SO(N), and Sp(N) as gauge groups offers a plausible explanation of the fact that clusters containing two or three quarks are more stable than multiquark clusters.

Ordered products, W∞-algebra, and two-variable, definite-parity, orthogonal polynomials

It has been shown that the Cartan subalgebra of W∞-algebra is the space of the two-variable, definite-parity polynomials. Explicit expressions of these polynomials, and their basic properties are presented. It also has been shown that they carry the infinite dimensional irreducible representation of the su(1,1) algebra having the spectrum bounded from below. A realization of this algebra in terms of difference operators is also obtained. For particular values of the ordering parameter s they are identified with the classical orthogonal polynomials of a discrete variable, such as the Meixner, Meixner–Pollaczek, and Askey–Wilson polynomials. With respect to variable s they satisfy a second order eigenvalue equation of hypergeometric type. Exact scattering states with zero energy for a family of potentials are expressed in terms of these polynomials. It has been put forward that it is the İnönü–Wigner contraction and its inverse that form a bridge between the difference and differential calculus.

State space modeling of long-memory processes

This paper develops a state space modeling for long-range dependent data. Although a long-range dependent process has an infinite-dimensional state space representation, it is shown that by using the Kalman filter, the exact likelihood function can be computed recursively in a finite number of steps. Furthermore, an approximation to the likelihood function based on the truncated state space equation is considered. Asymptotic properties of these approximate maximum likelihood estimates are established for a class of long-range dependent models, namely, the fractional autoregressive moving average models. Simulation studies show rapid converging properties of the approximate maximum likelihood approach.

Algebraic approach to scattering theory of the Fokker–Planck equation

The one-variable Fokker–Planck equation is studied in a scattering formalism. Scattering processes are represented by paths in one-dimensional space, and the paths are treated as algebraic objects that constitute infinite-dimensional representations of SL(2, C). Various expressions for the scattering coefficients are derived in a systematic way by means of algebraic methods with considerations on symmetries.

Lorentz-invariant Hamiltonian and Riemann hypothesis

We have given some arguments that a two-dimensional Lorentz-invariant Hamiltonian may be relevant to the Riemann hypothesis concerning zero points of the Riemann zeta function. Some eigenfunction of the Hamiltonian corresponding to infinite-dimensional representation of the Lorentz group have many interesting properties. Especially, a relationship exists between the zero zeta function condition and the absence of trivial representations in the wave function.

Nonstandard q-deformation of the Euclidean Lie algebra and its representations

A nonstandard q-deformed Euclidean algebra Uq(ison), based on the definition of the twisted q-deformed algebra Uqson) (different from the Drinfeld–Jimbo algebra Uq(son)), is defined. Infinite dimensional representations R of Uq(ison) are described. Explicit formulas for operators of these representations in the orthonormal basis are given. The spectra of the operators R(Tn) corresponding to a q-analogue of the infinitesimal operator of shifts along the n-th axis are described. Contrary to the case of the classical Euclidean Lie algebra ison, these spectra are discrete and spectral points have one point of accumulation.

Cλ-extended oscillator algebra and parasupersymmetric quantum mechanics

The Cλ-extended oscillator algebra is generated by {1, a, a†, N, T}, where T is the generator of the cyclic group Cλ of order λ. It can be realized as a generalized deformed oscillator algebra (GDOA). Its unirreps can thus be easily exhibited using the representation theory of GDOAs and their carrier spaces show a Zλ-grading structure. Within its infinite-dimensional Fock space representation, this algebra provides a bosonization of parasupersymmetric quantum mechanics of order p = λ − 1.

A link between two elliptic quantum groups

We construct a fully faithful functor from the category C_F of finite-dimensional representations of Felder's (dynamical) elliptic quantum group E_{tau,gamma}(gl(n)) to a cretain category D_B of (infinite-dimensional) representations of Belavin's quantum elliptic algebra B by difference operators, and a fully faithful functor from the category C_B of finite-dimensional representations of B to D_B. As a corollary, we show that the abelian subcategories of C_B and C_F generated by tensor products of vector representations are equivalent.

A Homological Bridge Between Finite and Infinite-Dimensional Representations of Algebras ?

Given a finite-dimensional algebra Λ, we show that a frequently satisfied finiteness condition for the category $$P^\infty$$ Λ-mod) of all finitely generated (left) Λ-modules of finite projective dimension,namely contravariant finiteness of $$P^\infty$$ (Λ-mod) in Λ-mod, forces arbitrary modules of finite projective dimension to be direct limits of objects in $$P^\infty$$ (Λ-mod). Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimensionconjecture, that is, for the little finitistic dimension of Λ to coincide with the big (this is well known to fail overfinite-dimensional algebras in general).

Automorphisms of sl(2) and dynamical r-matrices

Two outer automorphisms of the space of infinite-dimensional representations of sl(2) algebra are considered. The similar constructions for the loop algebras and Yangians are presented. The corresponding linear and quadratic R-brackets include the dynamical r-matrices.

Nonlinear Schrödinger Model with Boundary, Integrability and Scattering Matrix Based on the Degenerate Affine Hecke Algebra

The δ-function interacting many-body systems (nonlinear Schrödinger models) on an infinite interval and with boundary are studied by use of the integrable differential-difference operators, so-called Dunkl operators. These models are related with the classical root systems of type A and BC, and we give a systematic method to construct these integrable operators. This method is based on the infinite-dimensional representation for solutions of the classical Yang–Baxter equation and the classical reflection equation. In addition the scattering matrices of the boundary nonlinear Schrödinger model are investigated.

On Bessel Distributions forGL2over ap-Adic Field

We show that the Bessel distribution attached by Gelfand and Kazhdan to an infinite-dimensional irreducible admissible representation ofG=GL2(F), whereFis a non-Archimedean local field, is given by a locally integrable function which is locally constant on an open and dense set ofG.

Boson representations, non-standard quantum algebras and contractions

A Gelfan'd--Dyson mapping is used to generate a one-boson realization for the non-standard quantum deformation of $sl(2,\R)$ which directly provides its infinite and finite dimensional irreducible representations. Tensor product decompositions are worked out for some examples. Relations between contraction methods and boson realizations are also explored in several contexts. So, a class of two-boson representations for the non-standard deformation of $sl(2,\R)$ is introduced and contracted to the non-standard quantum (1+1) Poincar\'e representations. Likewise, a quantum extended Hopf $sl(2,\R)$ algebra is constructed and the Jordanian $q$-oscillator algebra representations are obtained from it by means of another contraction procedure.

On representations of Uqosp(1|2) when q is a root of unity

The infinite dimensional highest weight representations of Uqosp(1|2) for the deformation parameter q being a root of unity are investigated. As in the cases of q-deformed nongraded Lie algebras, we find that every irreducible representation is isomorphic to the tensor product of a highest weight representation of sl2(R) and a finite dimensional one of Uqosp(1|2). The structure is investigated in detail.

Boundary K-operator for the elliptic R-operator acting on functional space

An infinite-dimensional representation of the K-operator satisfying the reflection equation is given. Using the vertex-IRF correspondence, this K-operator corresponds to the diagonal solution of the face reflection equation.

Deformed Heisenberg algebra with reflection

A universality of the deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional representations of parafermionic nature. We demonstrate that finite-dimensional representations are representations of the deformed parafermionic algebra with internal Z 2-grading structure. On the other hand, any finite- or infinite-dimensional representation of the algebra supplies us with irreducible representations of the osp(1|2) superalgebra. We show that the normalized form of the deformed Heisenberg algebra with reflection has the structure of a guon algebra related to the generalized statistics.

A note on quantum Liouville theory via the quantum group An approach to strong coupling Liouville theory

Quantum Liouville theory is analysed in terms of the infinite dimensional representations of U q sl(2, C) with q a root of unity. Making full use of the characteristic features of the representations, we show that the vertex operators in this Liouville theory are factorized into classical vertex operators and those which are constructed from the finite dimensional representations of U q sl(2, C) We further show explicitly that the fusion rules in this model also enjoy such a factorization. Upon the conjecture that the Liouville action effectively decouples into the classical Liouville action and that of a quantum theory, correlation functions and transition amplitudes are discussed, especially an intimate relation between our model and geometric quantization of the moduli space of Riemann surfaces is suggested. The most important result is that our Liouville theory is in the strong coupling region, i.e. the central charge CL satisfies 1 < C L < 25. An interpretation of the quantum space-time is also given within this formulation.