Theoretical Physics

Employing the method of generating functions and making use of some infinite product identities like Euler, Jacobi's triple product and pentagon identities we derive recursion relations for Kostant's partition functions for the twisted Kac-Moody algebras.

By solving a set of recursion relations for the matrix elements of the 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 U h (sl(2)) algebra, we obtained U h (so(4)) algebra. The latter, on contraction, yields U h (e(3)) algebra. A nonlinear map of 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.

Using the corepresentation of the quantum supergroup OSp q (1/2) a general method for constructing noncommutative spaces covariant under its coaction is developed. In particular, a one-parameter family of covariant algebras, which may be interpreted as noncommutative superspheres, is constructed. It is observed that embedding of the supersphere in the OSp q (1/2) algebra is possible. This realization admits infinitesimal characterizationà la Koornwinder. A covariant oscillator realization of the supersphere is also presented.

We obtain a closed form expression of the universal T -matrix encapsulating the duality between the quantum superalgebra U q [osp(1/2)] and the corresponding supergroup OSp q (1/2). The classical q → 1 limit of this universal T matrix yields the group element of the undeformed OSp(1/2) supergroup. The finite dimensional representations of the quantum supergroup OSp q (1/2) are readily constructed employing the said universal T -matrix and the known finite dimensional representations of the dually related deformed U q [osp(1/2)] superalgebra. Proceeding further, we derive the product law, the recurrence relations and the orthogonality of the representations of the quantum supergroup OSp q (1/2). It is shown that the entries of these representation matrices are expressed in terms of the little Q-Jacobi polynomials with Q = −q. Two mutually complementary singular maps of the universal T -matrix on the universal R-matrix are also presented.

Modified braid equations satisfied by generalizedR matrices ( for a given set of group relations obeyed by the elements of T matrices ) are constructed for q-deformed quantum groups GL q (N ), SO q (N ) and Sp q (N ) with arbitrary values of N . The Baxterization ofR matrices, treated as an aspect complementary to the modification of the braid equation, is obtained for all these cases in particularly elegant forms. A new class of braid matrices is discovered for the quantum groups SO q (N ) and Sp q (N ). TheR matrices of this class, while being distinct from restrictions of the universalR matrix to the corresponding vector representations, satisfy the standard braid equation. The modified braid equation and the Baxterization are obtained for this new class ofR matrices. Diagonalization of the generalizedR matrices is studied. The diagonalizers are obtained explicitly for some lower dimensional cases in a convenient way, giving directly the eigenvalues of the correspondingR matrices. Applications of such diagonalization are then studied in the context of associated covariantly quantized noncommutative spaces. *

A nonlinear realization of the nonstandard (super-Jordanian) version of U(sl(N |1)) is given, for all N . * The recent work shows that there exist three distinct bialgebra structure on osp(1|2) and all of them are coboundary. We therefore have three distinct quantization of osp(1|2).

Representations of the quantum superalgebra U q [osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch-Gordan decomposition for the superalgebra U q [osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch-Gordan coefficients are derived, and it is observed that they may be expressed in terms of the Q-Hahn polynomials. We next investigate representations of the quantum supergroup OSp q (1/2) which are not well-defined in the classical limit. Employing the universal T -matrix, the representation matrices are obtained explicitly, and found to be related to the little Q-Jacobi polynomials. Characteristically, the relation Q = −q is satisfied in all cases. Using the Clebsch-Gordan coefficients derived here, we construct new noncommutative spaces that are covariant under the coaction of the even dimensional representations of the quantum supergroup OSp q (1/2).

We note that the recently introduced fuzzy torus can be regarded as a q-deformed parafermion. Based on this picture, classification of the Hermitian representations of the fuzzy torus is carried out. The result involves Fock-type representations and new finite dimensional representations for q being a root of unity as well as already known finite dimensional ones.

We introduce and investigate a one parameter family of quantum spaces invariant under the left (right) coactions of the group-like element T (j=1) h of the Jordanian function algebra F un h (SL(2)). These spaces may be regarded as Jordanian quantization of the two-dimensional spheres.

Maps and twists relating U (sl(2)) and the nonstandard U h (sl(2)): unified construction Abstract A general construction is given for a class of invertible maps between the classical U (sl(2)) and the Jordanian U h (sl(2)) algebras. Different maps are directly useful in different contexts. Similarity trasformations connecting them, in so far as they can be explicitly constructed, enable us to translate results obtained in terms of one to the other cases. Here the role of the maps is studied in the context of construction of twist operators between the cocommutative and noncocommutative coproducts of the U (sl(2)) and U h (sl(2)) algebras respectively. It is shown that a particular map called the 'minimal twist map' implements the simplest twist given directly by the factorized form of the R h -matrix of Ballesteros-Herranz. For other maps the twist has an additional factor obtainable in terms of the similarity transformation relating the map in question to the minimal one. The series in powers of h for the operator performing this transformation may be obtained up to some desired order, relatively easily. An explicit example is given for one particularly interesting case. Similarly the classical and the Jordanian antipode maps may be interrelated by a similarity transformation. For the 'minimal twist map' the transforming operator is determined in a closed form. * invite@elbahia.cerist.dz † chakra@orphee.polytechnique.fr ‡ Laboratoire Propre du CNRS UPR A.0014 R h = exp(−hX ⊗ e hX H) exp(e hX ⊗ hX).

A class of transformations of R q -matrices is introduced such that the q → 1 limit gives explicit nonstandard R h -matrices. The transformation matrix is singular itself at q → 1 limit. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. Our method can be implemented systematically for R-matrices of all dimensions and not only for sl(2) but also for algebras of higher dimensions. Explicit constructions are presented starting with U q (sl(2)) and U q (sl(3)), while choosing R q for (fund. rep.)⊗(arbitrary irrep.). The treatment for the general case and various perspectives are indicated. Our method yields nonstandard deformations along with a nonlinear map of the h-Borel subalgebra on the corresponding classical Borel subalgebra. For U h (sl(2)) this map is extended to the whole algebra and compared with another one proposed by us previously.

The generators of the Jordanian quantum algebra U h (sl(2)) are expressed as nonlinear invertible functions of the classical sl(2) generators. This permits immediate explicit construction of the finite dimensional irreducible representations of the algebra U h (sl(2)). Using this construction, new finite dimensional solutions of the Yang-Baxter equation may be obtained.