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At least 793 records · Page 44Linked to original sources
arXiv1996

Two-parameter deformation of the Poincaré algebra

We examine a two-parameter ($\hbar ,$ $λ$) deformation of the Poincarè algebra which is covariant under the action of $SL_q(2,C).$ When $λ\rightarrow 0$ it yields the Poincarè algebra, while in the $\hbar\rightarrow 0$ limit we recover the classical quadratic algebra discussed previously in \cite{ssy95}, \cite{sy95}. The analogues of the Pauli-Lubanski vector $w$ and Casimirs $p^2$ and $w^2$ are found and a set of mutually commuting operators is constructed.

q-alg↗
arXiv1996

Riemannian Geometry on Quantum Spaces

An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended to complex manifolds. Examples include the quantum sphere, the complex quantum projective spaces and the two-sheeted space.

q-alg↗
arXiv1996

On ${\cal U}_h(sl(2))$, ${\cal U}_h(e(3))$ and their Representations

By solving a set of recursion relations for the matrix elements of the ${\cal U}_h(sl(2))$ generators, the finite dimensional highest weight representations of the algebra were obtained as factor representations. Taking a nonlinear combination of the generators of the two copies of the ${\cal U}_h(sl(2))$ algebra, we obtained ${\cal U}_h(so(4))$ algebra. The latter, on contraction, yields ${\cal U}_h(e(3))$ algebra. A nonlinear map of ${\cal U}_h(e(3))$ algebra on its classical analogue $e(3)$ was obtained. The inverse mapping was found to be singular. It signifies a physically interesting situation, where in the momentum basis, a restricted domain of the eigenvalues of the classical operators is mapped on the whole real domain of the eigenvalues of the deformed operators.

q-alg↗
arXiv1996

Geometrical foundations of fractional supersymmetry

A deformed $q$-calculus is developed on the basis of an algebraic structure involving graded brackets. A number operator and left and right shift operators are constructed for this algebra, and the whole structure is related to the algebra of a $q$-deformed boson. The limit of this algebra when $q$ is a $n$-th root of unity is also studied in detail. By means of a chain rule expansion, the left and right derivatives are identified with the charge $Q$ and covariant derivative $D$ encountered in ordinary/fractional supersymmetry and this leads to new results for these operators. A generalized Berezin integral and fractional superspace measure arise as a natural part of our formalism. When $q$ is a root of unity the algebra is found to have a non-trivial Hopf structure, extending that associated with the anyonic line. One-dimensional ordinary/fractional superspace is identified with the braided line when $q$ is a root of unity, so that one-dimensional ordinary/fractional supersymmetry can be viewed as invariance under translation along this line. In our construction of fractional supersymmetry the $q$-deformed bosons play a role exactly analogous to that of the fermions in the familiar supersymmetric case.

hep-th↗
arXiv1996

Fusion of twisted representations

The comultiplication formula for fusion products of untwisted representations of the chiral algebra is generalised to include arbitrary twisted representations. We show that the formulae define a tensor product with suitable properties, and determine the analogue of Zhu's algebra for arbitrary twisted representations. As an example we study the fusion of representations of the Ramond sector of the N=1 and N=2 superconformal algebra. In the latter case, certain subtleties arise which we describe in detail.

hep-th↗
arXiv1997

Poisson Algebra of Differential Forms

We give a natural definition of a Poisson Differential Algebra. Consistence conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on differential calculus in a simple canonical form by a coordinate transformation. This is in analogy with the standard Darboux's theorem for symplectic geometry. For certain cases there exists a realization of the exterior derivative through a certain canonical one-form. All the above are carried out similarly for the case of a complex Poisson Differential Algebra. The case of one complex dimension is treated in details and interesting features are noted. A conclusion is made in the last section.

q-alg↗
arXiv1997

q-Uncertainty Relations

As one would anticipate from the realization of the q-commutators by difference operators, the states of maximum localization are smeared delta functions depending on q.

q-alg↗
arXiv1997

Differential Calculus on Fuzzy Sphere and Scalar Field

We find that there is an alternative possibility to define the chirality operator on the fuzzy sphere, due to the ambiguity of the operator ordering. Adopting this new chirality operator and the corresponding Dirac operator, we define Connes' spectral triple on the fuzzy sphere and the differential calculus. The differential calculus based on this new spectral triple is simplified considerably. Using this formulation the action of the scalar field is derived.

q-alg↗
arXiv1997

2+1 Einstein Gravity as a Deformed Chern-Simons Theory

The usual description of 2+1 dimensional Einstein gravity as a Chern-Simons (CS) theory is extended to a one parameter family of descriptions of 2+1 Einstein gravity. This is done by replacing the Poincare' gauge group symmetry by a q-deformed Poincare' gauge group symmetry, with the former recovered when q-> 1. As a result, we obtain a one parameter family of Hamiltonian formulations for 2+1 gravity. Although formulated in terms of noncommuting dreibeins and spin-connection fields, our expression for the action and our field equations, appropriately ordered, are identical in form to the ordinary ones. Moreover, starting with a properly defined metric tensor, the usual metric theory can be built; the Christoffel symbols and space-time curvature having the usual expressions in terms of the metric tensor, and being represented by c-numbers. In this article, we also couple the theory to particle sources, and find that these sources carry exotic angular momentum. Finally, problems related to the introduction of a cosmological constant are discussed.

hep-th↗
arXiv1997

A Finite Quantum Symmetry of M(3,C)

The 27-dimensional Hopf algebra A(F), defined by the exact sequence of quantum groups A(SL(2,C))->A(SL_q(2))->A(F), q^3=1, is studied as a finite quantum group symmetry of the matrix algebra M(3,C), describing the color sector of Alain Connes' formulation of the Standard Model. The duality with the Hopf algebra H,investigated in a recent work by Robert Coquereaux, is established and used to define a representation of H on M(3,C) and two commuting representations of H on A(F).

hep-th↗
arXiv1997

Dynamics in a noncommutative phase space

Dynamics has been generalized to a noncommutative phase space. The noncommuting phase space is taken to be invariant under the quantum group $GL_{q,p}(2)$. The $q$-deformed differential calculus on the phase space is formulated and using this, both the Hamiltonian and Lagrangian forms of dynamics have been constructed. In contrast to earlier forms of $q$-dynamics, our formalism has the advantage of preserving the conventional symmetries such as rotational or Lorentz invariance.

hep-th↗
arXiv1997

A Burge tree of Virasoro-type polynomial identities

Using a summation formula due to Burge, and a combinatorial identity between partition pairs, we obtain an infinite tree of q-polynomial identities for the Virasoro characters χ^{p, p'}_{r, s}, dependent on two finite size parameters M and N, in the cases where: (i) p and p' are coprime integers that satisfy 0 < p < p'. (ii) If the pair (p', p) has a continued fraction (c_1, c_2, ... , c_{t-1}, c_t+2), where t >= 1, then the pair (s, r) has a continued fraction (c_1, c_2, ... , c_{u-1}, d), where 1 =< u =< t, and 1 =< d =< c_{u}. The limit M -> infinity, for fixed N, and the limit N -> infinity, for fixed M, lead to two independent boson-fermion-type q-polynomial identities: in one case, the bosonic side has a conventional dependence on the parameters that characterise the corresponding character. In the other, that dependence is not conventional. In each case, the fermionic side can also be cast in either of two different forms. Taking the remaining finite size parameter to infinity in either of the above identities, so that M -> infinity and N -> infinity, leads to the same q-series identity for the corresponding character.

q-alg↗
arXiv1996

Bethe Ansatz and Classical Hirota Equation

We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. Namely, the eigenvalues of the quantum transfer matrix and the scattering $S$-matrix itself are identified with a certain $τ$-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems, hep-th/9604080.

cond-mat.stat-mech↗
arXiv1996

A Higher-level Bailey Lemma

We propose a generalization of Bailey's lemma, useful for proving $q$-series identities. As an application, generalizations of Euler's identity, the Rogers-Ramanujan identities, and the Andrews-Gordon identities are derived. This generalized Bailey lemma also allows one to derive identities for the branching functions of higher-level $A^{(1)}_1$ cosets.

q-alg↗
arXiv1995

An $U_{qp}(u_2)$ Rotor Model for Rotational Bands of Superdeformed Nuclei

A nonrigid rotor model is developed from the two-parameter quantum algebra $U_{qp}({\rm u}_2)$. [This model presents the $U_{qp}({\rm u}_2)$ symmetry and shall be referred to as the qp-rotor model.] A rotational energy formula as well as a qp-deformation of E2 reduced transition probabilities are derived. The qp-rotor model is applied (through fitting procedures) to twenty rotational bands of superdeformed nuclei in the $A \sim 130$, 150 and 190 mass regions. Systematic comparisons between the qp-rotor model and the q-rotor model of Raychev, Roussev and Smirnov, on one hand, and a basic three-parameter model, on the other hand, are performed on energy spectra, on dynamical moments of inertia and on B(E2) values. The physical signification of the deformation parameters q and p is discussed.

nucl-th↗
arXiv1997

Elementary Excitations and Dynamical Correlation Functions of the Calogero-Sutherland Model with Internal Symmetry

We consider the physical properties of elementary excitations of the Calogero-Sutherland (CS) model with SU(K) internal symmetry. From the results on the thermodynamics of this model, we obtain the charge, spin, and statistics of elementary excitations. Combining this knowledge and the known results on the dynamics in the spinless CS model, we propose the expression for the dynamical correlation functions of the SU(K) CS model. In the asymptotic region, we confirm the consistency of our results with predictions from conformal field theory.

cond-mat.str-el↗
arXiv1997

Orthogonal Symmetric Polynomials Associated with the Calogero Model

The Calogero model is a one-dimensional quantum integrable system with inverse-square long-range interactions confined in an external harmonic well. It shares the same algebraic structure with the Sutherland model, which is also a one-dimensional quantum integrable system with inverse-sine-square interactions. Inspired by the Rodrigues formula for the Jack polynomials, which form the orthogonal basis of the Sutherland model, recently found by Lapointe and Vinet, we construct the Rodrigues formula for the Hi-Jack (hidden-Jack) polynomials that form the orthogonal basis of the Calogero model.

cond-mat.stat-mech↗