Duality Principle and Braided Geometry

We give an overview of a new kind symmetry in physics which exists between observables and states and which is made possible by the language of Hopf algebras and quantum geometry. It has been proposed by the author as a feature of Planck scale physics. More recent work includes corresponding results at the semiclassical level of Poisson-Lie groups and at the level of braided groups and braided geometry.Comment: 24 page

Almost commutative Riemannian geometry: wave operators

Associated to any (pseudo)-Riemannian manifold $M$ of dimension $n$ is an $n+1$-dimensional noncommutative differential structure (\Omega^1,\extd) on the manifold, with the extra dimension encoding the classical Laplacian as a noncommutative `vector field'. We use the classical connection, Ricci tensor and Hodge Laplacian to construct (\Omega^2,\extd) and a natural noncommutative torsion free connection $(\nabla,\sigma)$ on $\Omega^1$. We show that its generalised braiding \sigma:\Omega^1\tens\Omega^1\to \Omega^1\tens\Omega^1 obeys the quantum Yang-Baxter or braid relations only when the original $M$ is flat, i.e their failure is governed by the Riemann curvature, and that \sigma^2=\id only when $M$ is Einstein. We show that if $M$ has a conformal Killing vector field $\tau$ then the cross product algebra $C(M)\rtimes_\tau\R$ viewed as a noncommutative analogue of $M\times\R$ has a natural $n+2$-dimensional calculus extending $\Omega^1$ and a natural spacetime Laplacian now directly defined by the extra dimension. The case $M=\R^3$ recovers the Majid-Ruegg bicrossproduct flat spacetime model and the wave-operator used in its variable speed of light preduction, but now as an example of a general construction. As an application we construct the wave operator on a noncommutative Schwarzschild black hole and take a first look at its features. It appears that the infinite classical redshift/time dilation factor at the event horizon is made finite.Comment: 39 pages, 4 pdf images. Removed previous Sections 5.1-5.2 to a separate paper (now ArXived) to meet referee length requirements. Corresponding slight restructure but no change to remaining conten

Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups $U_q(g)$. They have the same FRT generators $l^\pm$ but a matrix braided-coproduct \und\Delta L=L\und\tens L where $L=l^+Sl^-$, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices $BM_q(2)$; it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum double D(\usl) (also known as the `quantum Lorentz group') is the semidirect product as an algebra of two copies of \usl, and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.Comment: 45 pages. Revised (= much expanded introduction

Projective module description of the q-monopole

The Dirac q-monopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The Chern-Connes pairing of cyclic cohomology and K-theory is computed for the winding number -1. The non-triviality of this pairing is used to conclude that the quantum principal Hopf fibration is non-cleft. Among general results, we provide a left-right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (Hopf-Galois extensions) their associated covariant derivatives on projective modules.Comment: AMS-LaTeX 18 pages, no figures, correction of the Chern-number-sign-change Comments, 6 pages of new contents adde

Remarks on twisted noncommutative quantum field theory

We review recent results on twisted noncommutative quantum field theory by embedding it into a general framework for the quantization of systems with a twisted symmetry. We discuss commutation relations in this setting and show that the twisted structure is so rigid that it is hard to derive any predictions, unless one gives up general principles of quantum theory. It is also shown that the twisted structure is not responsible for the presence or absence of UV/IR-mixing, as claimed in the literature.Comment: 13 pages, v2: minor correction

Braided Hopf Algebras and Differential Calculus

We show that the algebra of the bicovariant differential calculus on a quantum group can be understood as a projection of the cross product between a braided Hopf algebra and the quantum double of the quantum group. The resulting super-Hopf algebra can be reproduced by extending the exterior derivative to tensor products.Comment: 8 page

Dopamine receptors gene expression in male rat hippocampus after administration of MDMA (Ecstasy) [La Expresión Génica de Receptores de Dopamina en el Hipocampo de Ratas Macho Después de la Administración de MDMA (Éxtasis)]

Ecstasy is one of the most popular amusing drugs among young people. Documents indicate some effects of Ecstasy on hippocampus and close relations between dopaminergic functions with reward learning. Therefore, the aim of this study was evaluation of the chronic effects of Ecstasy on memory in male Wistar rats and determination of dopamine receptors' gene expression in hippocampus. Forty adult male Wistar rats randomly distributed in five groups: Control, sham (received 1 ml/kg 0.9 saline) and three experimental groups were: Exp. 1 (2.5 mg/kg), Exp. 2 (5 mg/kg), and Exp. 3 (10 mg/kg) received MDMA intraperitoneally once every 7 days (3 times a day, 3 hours apart) for 4 weeks. Before the first injection animals trained in Shuttle Box memory and tested after the last injection. 24 hours after the final testing, brains of rats were dissected and hippocampus was removed and homogenized. After total RNA extraction and cDNA synthesis, expression of dopamine receptor genes in the hippocampus determined with Real-Time PCR. Our results showed that 2.5 and 5 mg/kg MDMA-treated groups had memory impairment. Also we found that MDMA increased the mRNA expression of dopamine receptors in hippocampus and the highest increase found in dopamine D1 receptors in the 5 mg/kg experimental group. We concluded that low doses of Ecstasy could increase Dopamine takers gene expression in hippocampus and disorder avoidance memory. But in high doses the increase in Dopamine takers gene expression was not as much as that in low doses and avoidance memory disorder was not observed. © 2015, Universidad de la Frontera. All rights reserved