Classification of real three-dimensional Lie bialgebras and their Poisson-Lie groups

Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using the Sklyanin bracket, the Poisson structures on the related Poisson-Lie groups are obtained.Comment: 17 page

On a suggestion relating topological and quantum mechanical entanglements

We analyze a recent suggestion \cite{kauffman1,kauffman2} on a possible relation between topological and quantum mechanical entanglements. We show that a one to one correspondence does not exist, neither between topologically linked diagrams and entangled states, nor between braid operators and quantum entanglers. We also add a new dimension to the question of entangling properties of unitary operators in general.Comment: RevTex, 7 eps figures, to be published in Phys. Lett. A (2004

From the braided to the usual Yang-Baxter relation

Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in order to satisfy the usual Yang-Baxter relation. As a consequence, a general connection between braided and {\it unbraided} (usual) Yang-Baxter algebras is derived and also analysed.Comment: 13 Latex page

A braided Yang-Baxter Algebra in a Theory of two coupled Lattice Quantum KdV: algebraic properties and ABA representations

A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of Algebraic Bethe Ansatz techniques. The conjecture that this monodromy matrix algebra leads, {\it in the cylinder continuum limit}, to a Perturbed Minimal Conformal Field Theory description is analysed and supported.Comment: Latex file, 46 page

Multiparametric quantum gl(2): Lie bialgebras, quantum R-matrices and non-relativistic limits

Multiparametric quantum deformations of $gl(2)$ are studied through a complete classification of $gl(2)$ Lie bialgebra structures. From them, the non-relativistic limit leading to harmonic oscillator Lie bialgebras is implemented by means of a contraction procedure. New quantum deformations of $gl(2)$ together with their associated quantum $R$-matrices are obtained and other known quantizations are recovered and classified. Several connections with integrable models are outlined.Comment: 21 pages, LaTeX. To appear in J. Phys. A. New contents adde

Spectral extension of the quantum group cotangent bundle

The structure of a cotangent bundle is investigated for quantum linear groups GLq(n) and SLq(n). Using a q-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on SLq(n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators -- the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SLq(n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. Relation between the two operators is given by a modular functional equation for Riemann theta function.Comment: 38 pages, no figure

The association between environmental exposures during childhood and the subsequent development of Crohn's Disease: A score analysis approach

Background Environmental factors during childhood are thought to play a role in the aetiology of Crohn's Disease (CD). In South Africa, recently published work based on an investigation of 14 childhood environmental exposures during 3 age intervals (0-5, 6-10 and 11-18 years) has provided insight into the role of timing of exposure in the future development of CD. The 'overlapping' contribution of the investigated variables however, remains unclear. The aim of this study was to perform a post hoc analysis using this data and investigate the extent to which each variable contributes to the subsequent development of CD relative to each aforementioned age interval, based on a score analysis approach. Methods Three methods were used for the score analysis. Two methods employed the subgrouping of one or more (similar) variables (methods A and B), with each subgroup assigned a score value weighting equal to one. For comparison, the third approach (method 0) involved no grouping of the 14 variables. Thus, each variable held a score value of one. Results Results of the score analysis (Method 0) for the environmental exposures during 3 age intervals (0-5, 6-10 and 11-18 years) revealed no significant difference between the case and control groups. By contrast, results from Method A and Method B revealed a significant difference during all 3 age intervals between the case and control groups, with cases having significantly lower exposure scores (approximately 30% and 40% lower, respectively). Conclusion Results from the score analysis provide insight into the 'compound' effects from multiple environmental exposures in the aetiology of CD.IS