A New q-Deformed Boson Realization of the Quantum Algebra Cq(l) and Its Representations

A new q-deformed boson realization of the quantum algebra C_q(l) is given and some of its finite dimensional representations are explicitly presented when q is a root of unity.     

Commutator Representations of Covariant Differential Calculi on Quantum Groups

Let (, d) be a first-order differential *-calculus on a *-algebra . We say that a pair (, F) of a *-representation of on a dense domain of a Hilbert space and a symmetric operator F on gives a commutator representation of if there exists a linear mapping : L( ) such that (adb) = (a)i[F, (b) ], a, b . Among others, it is shown that each left-covariant *-calculus of a compact quantum group Hopf *-algebra has a faithful commutator representation. For a class of bicovariant *-calculi on , there is a commutator representation such that F is the image of a central element of the quantum tangent space. If is the Hopf *-algebra of the compact form of one of the quantum groups SL _q (n+1), O _q (n), Sp _q (2n) with real trancendental q, then this commutator representation is faithful.     

Spinor Representations of and Quantum Boson-Fermion Correspondence

This is an extension of quantum spinor construction in [DF2]. We define quantum affine Clifford algebras based on the tensor product of a finite dimensional representation and an infinite highest weight representation of and the solutions of q-KZ equations, construct quantum spinor representations of and explain classical and quantum boson-fermion correspondence. Received: 22 November 1995 / Accepted: 21 July 1998     

Representations of Multiparameter Quantum Groups

We construct representations of the quantum algebras U_q,q(gl(n)) and U_q,q(sl(n)) which are in duality with the multiparameter quantum groups GL_qq(n), SL_qq(n), respectively. These objects depend on n(n − 1)/2+ 1 deformation parameters q, q_ij (1 ≤ i< j ≤ n) which is the maximal possible number in the case of GL(n). The representations are labelled by n − 1 complex numbers r_i and are acting in the space of formal power series of n(n − 1)/2 non-commuting variables. These variables generate quantum flag manifolds of GL_qq(n), SL_qq(n). The case n = 3 is treated in more detail.

     

Reducibility of Quantum Representations of Mapping Class Groups

In this paper we provide a general condition for the reducibility of the Reshetikhin–Turaev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular tensor category are reducible. In particular, for SU(N) we get reducibility for certain levels and ranks. For the quantum SU(2) Reshetikhin–Turaev theory we construct a decomposition for all even levels. We conjecture this decomposition is a complete decomposition into irreducible representations for high enough levels.     

On Minimal Affinizations of Representations of Quantum Groups

In this paper we study minimal affinizations of representations of quantum groups (generalizations of Kirillov-Reshetikhin modules of quantum affine algebras introduced in [Cha1]). We prove that all minimal affinizations in types A, B, G are special in the sense of monomials. Although this property is not satisfied in general, we also prove an analog property for a large class of minimal affinizations in types C, D, F. As an application, the Frenkel-Mukhin algorithm [FM1] works for these modules. For minimal affinizations of type A, B we prove the thin property (the l-weight spaces are of dimension 1) and a conjecture of [NN1] (already known for type A). The proof of the special property is extended uniformly for more general quantum affinizations of quantum Kac-Moody algebras.     

Langlands Duality for Representations and Quantum Groups at a Root of Unity

We give a representation-theoretic interpretation of the Langlands character duality of [FH], and show that the “Langlands branching multiplicities” for symmetrizable Kac-Moody Lie algebras are equal to certain tensor product multiplicities. For finite type quantum groups, the connection with tensor products can be explained in terms of tilting modules.     

Inhomogeneous q-Differential Realization,Inhomogeneous q-Boson Realization of Quantum Group SU(n)q and Its Indecomposable Representations

An inhomogeneous q-differential realization (q-IHDR) and its corresponding inhomogeneous q-boson realization (q-IHBR) of quantum group SU(n)_q are studied. By making use of the q-IHBR a kind of infinite dimensional indecomposable and irreducible representations of SU(n)_q is studied on the q-Fcck space. The finite dimensional irreducible representations of SU(n), are obtained on the finite dimensional subspaces of q-Fock space. As an example, these representations of SU(2)_q are studied in detail and Jimbo standard irreducible representations are obtained.     

Realization of Quantum Circuits in Fock Space

In this letter, by using the method we offered in our paper [L. Ma and Y.D. Zhang, Commun. Theor. Phys. (Beijing, China) 36 (2001) 119], some extended quantum logic gates, such as quantum counter, quantum adder, are studied and their expressions are given. It may be useful for us to study the more complicated quantum logic circuits deeply.     

Canonical Differential Calculi for Finitely Generated Abelian Groups and Their Fermionic Representations

Canonical differential calculi are defined for finitely generated Abelian groups with involutions existing consistemtly.Two such the canonical calculi are presented.Fermionic representations for canonical calculi are defined based on quantized calculi.Fermionic representations for aforementioned two canonical calculi are searched out.     

Realization of the N(odd)-Dimensional Quantum Euclidean Space by Differential Operators

The quantum Euclidean space R_q^N is a kind of noncommutative space that is obtained from ordinary Euclidean space R^N by deformation with parameter q. When N is odd, the structure of this space is similar to R_q³. Motivated by realization of R_q³ by differential operators in R³, we give such realization for R_q⁵ and R_q⁷ cases and generalize our results to R_q^N (N odd) in this paper, that is, we show that the algebra of R_q^N can be realized by differential operators acting on C^∝ functions on undeformed space R^N.     

One-Parameter Homogeneous Differential Realization and Boson–Fermion Realization of the SPL(2,1) Superalgebra

One-parameter homogeneous differential realization of the SPL(2,1) superalgebra on the space of homogeneous polynomials and the corresponding boson–fermion realization are studied. The parameter α may be related to the interaction parameter U in one exactly solvable model for correlated electrons.The author was supported financially by Shenzhen Institute of Information Technology grant No. LG-060109. PACS: 12.60.Jv, 03.65.FD     

Boson Representations of Subalgebra Chain for Quantum Algebra (Cl)q with q a Root of Unity

The method of q-deformed boson realization is extended to the caee that q is a root of unity. By using thb method, many irreducible and indecomposable representations of each su balgebra in a su balgebra chain of the quantum universal enveloping algebra (quan tum algebra) (C_l)_q are obtained on the q-deformed Fock space and its quotient space. AB an example, the representations of (C₃)_q are constructed in explicit forms. Besides, by embedding (A₁)_q = (C_l)_q, into the subalgebra chain, a new realisation of (A₁)_q and its representations are obtained.     

Phase Space Optimization of Quantum Representations: Non-Cartesian Coordinate Spaces

In an earlier article [Found. Phys. 30, 1191 (2000)], a quasiclassical phase space approximation for quantum projection operators was presented, whose accuracy increases in the limit of large basis size (projection subspace dimensionality). In a second paper [J. Chem. Phys. 111, 4869 (1999)], this approximation was used to generate a nearly optimal direct-product basis for representing an arbitrary (Cartesian) quantum Hamiltonian, within a given energy range of interest. From a few reduced-dimensional integrals, the method determines the optimal 1D marginal Hamiltonians, whose eigenstates comprise the direct-product basis. In the present paper, this phase space optimized direct-product basis method is generalized to incorporate non-Cartesian coordinate spaces, composed of radii and angles, that arise in molecular applications. Analytical results are presented for certain standard systems, including rigid rotors, and three-body vibrators.     

Quantization of Boson Fields in Quantum Geometry

It is shown that the introduction of an upper limit on the acceleration of particles provides a natural cutoff on momenta, avoiding the problem of ultraviolet divergencies in local quantum field theory. Such a cutoff turns out to be related to Planck energy.     

A Hilbert Space Realization of Nonlinear Quantum Mechanics as Classical Extension of Its Linear Counterpart

This paper studies the state-effect-probability structure associated with thequantum mechanics of nonlinear (homogeneous, in general nonadditive) operatorson a Hilbert space. Its aim is twofold: to provide a concrete representation ofthe features of nonlinear quantum mechanics on a Hilbert space, and to showthat the properties of the nonlinear version of quantum mechanics here describedhave the structure of a classical logic.     

Correspondence Rules for SU(1,1) Quasidistribution Functions and Quantum Dynamics in the Hyperbolic Phase Space

We derive the explicit differential form for the action of the generators of the SU(1,1) group on the corresponding s-parametrized symbols. This allows us to obtain evolution equations for the phase-space functions on the upper sheet of the two-sheet hyperboloid and analyze their semiclassical limits. Dynamics of quantum systems with SU(1,1) symmetry governed by compact and non-compact Hamiltonians are discussed in both quantum and semiclassical regimes.     

Finite Dimensional Unitary Representations of Quantum Anti–de Sitter Groups at Roots of Unity

We study irreducible unitary representations of U _q (SO(2,1)) and U _q (SO(2,?3)) for q a root of unity, which are finite dimensional. Among others, unitary representations corresponding to all classical one-particle representations with integral weights are found for , with M being large enough. In the “massless” case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only after factoring out a subspace of “pure gauges” as classically. A truncated associative tensor product describing unitary many-particle representations is defined for . Received: 27 November 1996 / Accepted: 28 July 1997     

New Operator Realization of Superoperators and Its Application in Quantum Information Theory

By exposing deficiency of the usual superoperators that have no explicit operator-expression in quantum information theory we introduce thermo entangled state representation to endow each of these superoperators a definite operator-expression in an enlarged space in which one mode is a fictitious. This helps us to directly derive the role of exponential of superoperators and the solutions of some master equations.     

New Operator Realization of Superoperators and Its Application in Quantum Information Theory

By exposing deficiency of the usual superoperators that have no explicit operator-expression in quantum information theory we introduce thermo entangled state representation to endow each of these superoperators a definite operator-expression in an enlarged space in which one mode is a fictitious. This helps us to directly derive the role of exponential of superoperators and the solutions of some master equations.