Quantum Algebra And Topology

We consider Hopf bimodules and crossed modules over a Hopf algebra H in a braided category. They are the key-stones for braided bicovariant differential calculi and their invariant vector fields respectively, as well as for the construction of braided Hopf algebra cross products. We show that the notions of Hopf bimodules and crossed modules are equivalent. A generalization of the Radford-Majid criterion to the braided case is given and it is seen that bialgebra cross products over the Hopf algebra H are precisely described by H -crossed module bialgebras. We study the theory of (bicovariant) differential calculi in braided abelian categories and we construct $\NN_0$-graded bicovariant differential calculi out of first order bicovariant differential calculi. These objects are shown to be Hopf algebra differential calculi with universal bialgebra properties in the braided $\NN_0$-graded category.

A nonlinear change of basis allows to show that the non-standard quantum deformation of the (3+1) Poincare algebra has a bicrossproduct structure. Quantum universal R-matrix, Pauli-Lubanski and mass operators are presented in the new basis.

We describe a bilinear identity satisfied by certain multidimensional q-hypergeometric integrals. The identity can be considered as a deformation of the Riemann bilinear relation for the twisted de Rham (co)homologies. The identity also gives an explicit formula for scalar products of solutions of the qKZ and the dual qKZ equations, therefore providing a deformation of Gaudin-Korepin's formula for norms of the Bethe vectors.

Using realisations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner-Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner-Pollaczek polynomials. For the positive discrete series representations of the quantised universal enveloping algebra Uq(su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey-Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials.

We prove a binomial formula for Macdonald polynomials and consider applications of it.

Dunkl operators are differential-difference operators on $\b R^N$ which generalize partial derivatives. They lead to generalizations of Laplace operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on. In this paper we introduce two systems of biorthogonal polynomials with respect to Dunkl's Gaussian distributions in a quite canonical way. These systems, called Appell systems, admit many properties known from classical Hermite polynomials, and turn out to be useful for the analysis of Dunkl's Gaussian distributions. In particular, these polynomials lead to a new proof of a generalized formula of Macdonald due to Dunkl. The ideas for this paper are taken from recent works on non-Gaussian white noise analysis and from the umbral calculus.

We construct explicitly the quantum symplectic affine algebra U q ( sp ˆ 2n ) using bosonic fields. The Fock space decomposes into irreducible modules of level -1/2, quantizing the Feingold-Frenkel construction for q=1.

In this paper, we present a detailed analysis of the diagonalization of the higher spin Heisenberg model using its quantum affine symmetry U q ( sl(2) ^ ) . In particular, we describe the bosonizations of the latter algebra, its highest weight representations, vertex operators and screening operators. Finally, we use this bosonization method to compute the vacuum-to-vacuum expectation values and the form factors of any local operator.

We define a new class of unitary solutions to the classical Yang-Baxter equation (CYBE). These ``boundary solutions'' are those which lie in the closure of the space of unitary solutions to the modified classical Yang-Baxter equation (MCYBE). Using the Belavin-Drinfel'd classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer-Gervais solution to the MCYBE, we explicitly construct for all n > 2 a boundary solution based on the maximal parabolic subalgebra of sl(n) obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to n . We also give examples of non-boundary solutions for the classical simple Lie algebras.

Boundary solutions to the quantum Yang-Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the ``modified'' qYB equation, the latter being analogous to the modified classical Yang-Baxter (cYB) equation. We construct, for a large class of solutions r to the modified cYB equation, explicit ``boundary quantizations'', i.e., boundary solutions to the qYB equation of the form I+tr+ t 2 r 2 +... . In the last section we list and give quantizations for all classical r-matrices in sl(3)∧sl(3) .