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At least 1,279 records · Page 71Linked to original sources
arXiv1996

Non Abelian Sugawara Construction and the q-deformed N=2 Superconformal Algebra

The construction of a q-deformed N=2 superconformal algebra is proposed in terms of level 1 currents of ${\cal{U}}_{q} ({\widehat{su}}(2))$ quantum affine Lie algebra and a single real Fermi field. In particular, it suggests the expression for the q-deformed Energy-Momentum tensor in the Sugawara form. Its constituents generate two isomorphic quadratic algebraic structures. The generalization to ${\cal{U}}_{q} ({\widehat{su}}(N+1))$ is also proposed.

q-alg↗
arXiv1996

Deformed and extended Galilei group Hopf algebras

The $\hatκ$-deformed extended Galilei Hopf group algebra, ${\rm Fun}_{\hatκ}(\tilde G_{(m)})$, is introduced. It provides an example of a cocycle bicrossproduct structure, and is shown to be the contraction limit of a pseudoextension of the $κ$-Poincaré group algebra. The possibility of obtaining another deformed extended Galilei groups is discussed, including one obtained from a non-standard Poincaré deformation.

q-alg↗
arXiv1996

On a Deformation of $sl(2)$ with Paragrassmannian Variables

We propose a new structure ${\cal U}^{r}_{\displaystyle{q}}(sl(2)) $. This is realized by multiplying $δ$ ($q=e^δ$, $δ\in \CC$) by $θ$, where $θ$ is a real nilpotent -paragrassmannian- variable of order $r$ ($θ^{r+1}=0$) that we call the order of deformation, the limit $r\rightarrow \infty$ giving back the standard ${\cal U}_{\displaystyle {q}}(sl(2))$. In particular we show that, for $r=1$, there exists a new ${\cal R}$-matrix associated with $sl(2)$. We also proof that the restriction of the values of the parameters of deformation give nonlinear algebras as particular cases.

q-alg↗
arXiv1996

Holomorphic Quantization on the Torus and Finite Quantum Mechanics

We construct explicitly the quantization of classical linear maps of $SL(2, R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of $SL(2, Z)$ to the subgroup $SL(2, Z)/Γ_l$, $Γ_l$ being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, $O_{l}(2)\subset SL(2,l)$, for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.

hep-th↗
arXiv1995

Twisted Yang-Baxter equations for linear quantum (super)groups

We consider the modified (or twisted) Yang-Baxter equations for the $SL_{q}(N)$ groups and $SL_{q}(N|M)$ supergroups. The general solutions for these equations are presented in the case of the linear quantum (super)groups. The introduction of spectral parameters in the twisted Yang-Baxter equation and its solutions are also discussed.

q-alg↗
arXiv1996

Bethe Ansatz for Higher Spin XYZ Models --- Low-lying Excitations ---

A higher spin generalization of the XYZ spin chain defined by the fusion procedure is considered. Energy and momentum of the ground state and low-lying excited states of the model are computed by means of an isomorphism between a space of theta functions on which the Sklyanin algebra acts and a space obtained by the fusion procedure.

q-alg↗
arXiv1996

Common Algebraic Structure for the Calogero-Sutherland Models

We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland models by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis for the rational case.

solv-int↗
arXiv1996

The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures

Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we determine the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.

hep-th↗
arXiv1996

On the Two q-Analogue Logarithmic Functions

There is a simple, multi-sheet Riemann surface associated with e_q(z)'s inverse function ln_q(w) for 0< q < 1. A principal sheet for ln_q(w) can be defined. However, the topology of the Riemann surface for ln_q(w) changes each time "q" increases above the collision point of a pair of the turning points of e_q(x). There is also a power series representation for ln_q(1+w). An infinite-product representation for e_q(z) is used to obtain the ordinary natural logarithm ln{e_q(z)} and the values of sum rules for the zeros "z_i" of e_q(z). For |z|<|z_1|, e_q(z)=exp{b(z)} where b(z) is a simple, explicit power series in terms of values of these sum rules. The values of the sum rules for the q-trigonometric functions, sin_q(z) and cos_q(z), are q-deformations of the usual Bernoulli numbers.

q-alg↗
arXiv1995

New Solutions of the Yang-Baxter Equation Based on Root of 1 Representations of the Para-Bose Superalgebra U$_q$[osp(1/2)]

New solutions of the quantum Yang-Baxter equation, depending in general on three arbitrary parameters, are written down. They are based on the root of unity representations of the quantum orthosymplectic superalgebra \\U, which were found recently. Representations of the braid group $B_N$ are defined within any $N^{th}$ tensorial power of root of 1 \\U modules.

q-alg↗
arXiv1995

On Quantum Groups in the Hubbard Model with Phonons

The correct Hamiltonian for an extended Hubbard model with quantum group symmetry as introduced by A. Montorsi and M. Rasetti is derived for a D-dimensional lattice. It is shown that the superconducting SUq(2) holds as a true quantum symmetry only for D = 1 and that terms of higher order in the fermionic operators in addition to phonons are required for a quantum symmetric hamiltonian. The condition for quantum symmetry is "half filling" and there is no local electron-phonon coupling. A discussion of Quantum symmetries in general is given in a formalism that should be readily accessible to non Hopf-algebraists.

cond-mat↗
arXiv1996

Quantum Group Symmetry in Multiple Chern-Simons Theory on a Torus

We discuss $w_\infty$ and $sl_q(2)$ symmetries in multiple Chern-Simons theory on a torus. It is shown that these algebraic structures arise from the dynamics of the non-integrable phases of the Chern-Simons fields. The generators of these algebras are constructed from the Wilson line operators corresponding to these phases. The vacuum states form the basis of cyclic representation of $sl_q (2)$.

hep-th↗
arXiv1995

Recursion relations and branching rules for simple Lie algebras

The branching rules between simple Lie algebras and its regular (maximal) simple subalgebras are studied. Two types of recursion relations for anomalous relative multiplicities are obtained. One of them is proved to be the factorized version of the other. The factorization property is based on the existence of the set of weights $Γ$ specific for each injection. The structure of $Γ$ is easily deduced from the correspondence between the root systems of algebra and subalgebra. The recursion relations thus obtained give rise to simple and effective algorithm for branching rules. The details are exposed by performing the explicit decomposition procedure for $A_{3} \oplus u(1) \to B_{4}$ injection.

q-alg↗
arXiv1995

Atypical Representations of $U_{q}(sl(N))$ at Roots of Unity

We show how to adapt the Gelfand-Zetlin basis for describing the atypical representation of ${\cal U}_{\displaystyle{q}}(sl(N))$ when $q$ is root of unity. The explicit construction of atypical representation is presented in details for $N=3$.

q-alg↗
arXiv1995

Deformed Minkowski spaces: clasification and properties

Using general but simple covariance arguments, we classify the `quantum' Minkowski spaces for dimensionless deformation parameters. This requires a previous analysis of the associated Lorentz groups, which reproduces a previous classification by Woronowicz and Zakrzewski. As a consequence of the unified analysis presented, we give the commutation properties, the deformed (and central) length element and the metric tensor for the different spacetime algebras.

q-alg↗
arXiv1996

New Generalized Poisson Structures

New generalized Poisson structures are introduced by using suitable skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are provided by conditions on these tensors, which may be understood as cocycle conditions. As an example, we provide the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.

q-alg↗