Quantized affine Lie algebras and diagonalization of braid generators

Let U _q be a quantized affine Lie algebra. It is proven that the universal R-matrix R of U _q satisfies the celebrated conjugation relationR ⁺ =TR withT the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight U _q -module and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin and Gould forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found.     

Diagonalization of the braid generator on unitary irreps of quantum supergroups

It is shown that the braid generator associated with the universalR-matrix is diagonalizable on all unitary representations of quantum supergroups. An example is considered using U _q (gl(2|1)) and a family of eight-dimensional typical representations.     

Quantum affine algebras and universalR-matrix with spectral parameter

Using the previously obtained universalR-matrix for the quantized nontwisted affine Lie algebras U _q (A ₁ ⁽¹⁾ ) and U _q (A ₂ ⁽¹⁾ ), we determine the explicitly spectral dependent universalR-matrix for the corresponding quantum Lie algebras U _q (A ₁) and U _q (A ₂). As applications, we reproduce the well known results in the fundamental representations and we also derive an extremely explicit formula of the spectral-dependentR-matrix for the adjoint representation of U _q (A ₂), the simplest nontrivial case when the tensor product decomposition of the representation with itself has nontrivial multiplicity.     

Quantum algebras for maximal motion groups ofN-dimensional flat spaces

An embedding method to getq-deformations for the nonsemisimple algebras generating the motion groups ofN-dimensional flat spaces is presented. This method gives a global and simultaneous scheme ofq-deformation for all iso(p, q) algebras and for those obtained from them by some Inönü-Wigner contractions, such as theN-dimensional Euclidean, Poincaré, and Galilei algebras.     

Quantum Heisenberg groups and sklyanin algebras

New bialgebra structures on the Heisenberg-Lie algebra and their quantizations are introduced. Some of these quantizations give rise to new multiplications in homogeneous coordinate rings of Abelian varieties, via the well-known identification of theta functions with suitable matrix coefficients of the Stone-von Neumann representations.N.A.: Forschungsstipendiat der Alexander von Humboldt-Stiftung. J. D. and A. T.: Postdoctoral fellowship, ICTP. N. A. and A. T.: This work was also partially supported by CONICET, CONICOR and FAMAF, Argentina.     

A natural and rigid model of quantum groups

We introduce a natural (Fréchet-Hopf) algebra A containing all generic Jimbo algebras U _t (sl(2)) (as dense subalgebras). The Hopf structures on A extend (in a continuous way) the Hopf structures of generic U _t (sl(2)). The Universal R-matrices converge in A A. Using the (topological) dual of A, we recover the formalism of functions of noncommutative arguments. In addition, we show that all these Hopf structures on A are isomorphic (as bialgebras), and rigid in the category of bialgebras.     

Construction of a new quantum double and new solutions to the Yang-Baxter equation

A new quantum double is established from a new Hopf algebra and a new kind of quantum R-matrix is obtained.     

Eigenvalues of Casimir invariants for type I quantum superalgebras

We present the eigenvalues of the Casimir invariants for the type I quantum superalgebras on any irreducible highest weight module.     

On universalR-matrix for quantized nontwisted rank 3 affine KM algebras

Explicit formulas of the universalR-matrix are given for all quantized nontwisted rank 3 affine KM algebras U _q (A ₂ ⁽¹⁾ ), U _q (C ₂ ⁽¹⁾ ) and U _q (G ₂ ⁽¹⁾ ).     

Baxterization of ther-matrix for the adjoint representation of U q [D(2, 1;α)]

The Baxterization procedure is applied to the Braid group representation arising from the adjoint representation of the quantum supergroup U _q [D(2, 1;)]. This yields a two-parameter family of solutions to the (parameter dependent) Yang-Baxter equation of interest in supersymmetric lattice models.     

Casimir invariants for quantized affine Lie algebras

Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest-weight representation.     

Two variable link polynomials from quantum supergroups

New two variable link polynomials are constructed corresponding to a one-parameter family of representations of the quantum supergroup U _q [gl(2 | 1)]. Their connection with the Kauffman polynomials is also investigated.     

Left regular representation and contraction of sl q (2) to e q (2)

The left regular representation of the quantum algebras sl _q (2) and e _q (2) are discussed and shown to be related by contraction. The reducibility is studied andq-difference intertwining operators are constructed.     

Fusion structures from quantum groups: II. Why truncation is necessary

It is shown that a finite, reflection positive, and nontruncated fusion structure on an arbitrary Hopf algebra is trivial in the sense thatq-traces coincide with ordinary traces andq-dimensions coincide with ordinary dimensions. Thus, nontruncated fusion structures are ruled out to describe the fusion rules of quantum field theories with noninteger statistical dimensions and a finite number of superselection sectors.Work supported in part by DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

A *-product on SL(2) and the corresponding nonstandard (\mathfrak{s}\mathfrak{l}({\text{2}}))

We obtain Zakrzewski's deformation of Fun SL(2) through the construction of a *-product on SL(2). We then give the deformation of dual to this, as well as a Poincaré basis for both algebras.Aspirant au Fonds National belge de la Recherche Scientifique. Partially supported by EEC contract SC1-0105-C.     

Quantum double and differential calculi

We show that bicovariant bimodules as defined by Woronowicz are in one-to-one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is connected to the existence of a particular (n+1)-dimensional representation of the double. An example of bicovariant differential calculus on the nonquasitriangular quantum group E _q (2) is developed. The construction is studied in terms of Hochschild cohomology and a correspondence between differential calculi and 1-cocycles is proved. Some differences of calculi on quantum and finite groups with respect to Lie groups are stressed.     

On a possible algebra morphism of U q [osp(1/2n)] onto the deformed oscillator algebra W q (n)

We formulate a conjecture stating that the algebra ofn pairs of deformed Bose creation and annihilation operators is a factor algebra of U _q [osp(1/2n)], considered as a Hopf algebra, and prove it for then = 2 case. To this end, we show that for any value ofq, U _q [osp(1/4)] can be viewed as a superalgebra freely generated by two pairsB ₁ ^± ,B ₂ ^± of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the commutation relations between the generators and a basis in U _q [osp(1/2n)] entirely in terms ofB ₁ ^± ,B ₂ ^± .     

A superalgebra morphism of U q [osp(1/2N)] onto the deformed oscillator superalgebra W q (N)

We prove that the deformed oscillator superalgebra W _q (n) (which in the Fock representation is generated essentially byn pairs ofq-bosons) is a factor algebra of the quantized universal enveloping algebra U _q [osp(1/2n)]. We write down aq-analog of the Cartan-Weyl basis for the deformed osp(1/2n) and also give an oscillator realization of all Cartan-Weyl generators.