Maximal quadratic modules on *-rings

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to $\ast$-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime $\ast$-ideal, that every maximal proper quadratic module in a Noetherian $\ast$-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schm\" udgen's Strict Positivstellensatz for the Weyl algebra: Let $c$ be an element of the Weyl algebra $\mathcal{W}(d)$ which is not negative semidefinite in the Schr\" odinger representation. It is shown that under some conditions there exists an integer $k$ and elements $r_1,...,r_k \in \mathcal{W}(d)$ such that $\sum_{j=1}^k r_j c r_j^\ast$ is a finite sum of hermitian squares. This result is not a proper generalization however because we don't have the bound $k \le d$.Comment: 11 page

Prime ideals in nilpotent Iwasawa algebras

Let G be a nilpotent complete p-valued group of finite rank and let k be a field of characteristic p. We prove that every faithful prime ideal of the Iwasawa algebra kG is controlled by the centre of G, and use this to show that the prime spectrum of kG is a disjoint union of commutative strata. We also show that every prime ideal of kG is completely prime. The key ingredient in the proof is the construction of a non-commutative valuation on certain filtered simple Artinian rings

Rings of skew polynomials and Gel'fand-Kirillov conjecture for quantum groups

We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a ``q-deformation'' of the Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of automorphisms of certain non-commutaive rings of quotients coming from complex powers of quantum group generators; this is applied to explicit calculation of singular vectors in Verma modules over U_{q}(\gtsl_{n+1}). We finally give a definition of a $q-$connection with coefficients in a ring of skew polynomials and study the structure of quantum group modules twisted by a $q-$connection.Comment: 25 page

Modular Lie algebras and the Gelfand-Kirillov conjecture

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.Comment: 20 page

Magnesium in the atmosphere of the planet HD 209458 b: Observations of the thermosphere-exosphere transition region

Copyright © ESO, 2013The planet HD 209458 b is one of the most well studied hot-Jupiter exoplanets. The upper atmosphere of this planet has been observed through ultraviolet/optical transit observations with H i observation of the exosphere revealing atmospheric escape. At lower altitudes just below the thermosphere, detailed observations of the Na i absorption line has revealed an atmospheric thermal inversion. This thermal structure is rising toward high temperatures at high altitudes, as predicted by models of the thermosphere, and could reach ~ 10 000 K at the exobase level. Here, we report new near ultraviolet Hubble Space Telescope/Space Telescope Imaging Spectrograph (HST/STIS) observations of atmospheric absorptions during the planetary transit of HD 209458 b. We report absorption in atomic magnesium (Mg i), while no signal has been detected in the lines of singly ionized magnesium (Mg ii). We measure the Mg i atmospheric absorption to be 6.2   ±   2.9% in the velocity range from − 62 to − 19 km s-1. The detection of atomic magnesium in the planetary upper atmosphere at a distance of several planetary radii gives a first view into the transition region between the thermosphere and the exobase, where atmospheric escape takes place. We estimate the electronic densities needed to compensate for the photo-ionization by dielectronic recombination of Mg+ to be in the range of 108−109 cm-3. Our finding is in excellent agreement with model predictions at altitudes of several planetary radii. We observe Mg i atoms escaping the planet, with a maximum radial velocity (in the stellar rest frame) of −60 km s-1. Because magnesium is much heavier than hydrogen, the escape of this species confirms previous studies that the planet’s atmosphere is undergoing hydrodynamic escape. We compare our observations to a numerical model that takes the stellar radiation pressure on the Mg i atoms into account. We find that the Mg i atoms must be present at up to ~7.5 planetari radii altitude and estimate an Mg i escape rate of ~3   ×   107 g s-1. Compared to previous evaluations of the escape rate of H i atoms, this evaluation is compatible with a magnesium abundance roughly solar. A hint of absorption, detected at low level of significance, during the post-transit observations, could be interpreted as a Mg i cometary-like tail. If true, the estimate of the absorption by Mg i would be increased to a higher value of about 8.8   ±   2.1%.Centre National d’Études Spatiales (CNES)French Agence Nationale de la Recherche (ANR)NASA Exoplanet Science Institute (NExScI) - Sagan Exoplanet Fellowship programSTFC (Science & Technology Facilities Council)European Commissions Seventh Framework Programme - Marie Curie Intra-European Fellowshi

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

Modelling spectral and timing properties of accreting black holes: the hybrid hot flow paradigm

The general picture that emerged by the end of 1990s from a large set of optical and X-ray, spectral and timing data was that the X-rays are produced in the innermost hot part of the accretion flow, while the optical/infrared (OIR) emission is mainly produced by the irradiated outer thin accretion disc. Recent multiwavelength observations of Galactic black hole transients show that the situation is not so simple. Fast variability in the OIR band, OIR excesses above the thermal emission and a complicated interplay between the X-ray and the OIR light curves imply that the OIR emitting region is much more compact. One of the popular hypotheses is that the jet contributes to the OIR emission and even is responsible for the bulk of the X-rays. However, this scenario is largely ad hoc and is in contradiction with many previously established facts. Alternatively, the hot accretion flow, known to be consistent with the X-ray spectral and timing data, is also a viable candidate to produce the OIR radiation. The hot-flow scenario naturally explains the power-law like OIR spectra, fast OIR variability and its complex relation to the X-rays if the hot flow contains non-thermal electrons (even in energetically negligible quantities), which are required by the presence of the MeV tail in Cyg X-1. The presence of non-thermal electrons also lowers the equilibrium electron temperature in the hot flow model to <100 keV, making it more consistent with observations. Here we argue that any viable model should simultaneously explain a large set of spectral and timing data and show that the hybrid (thermal/non-thermal) hot flow model satisfies most of the constraints.Comment: 26 pages, 13 figures. To be published in the Space Science Reviews and as hard cover in the Space Sciences Series of ISSI - The Physics of Accretion on to Black Holes (Springer Publisher

X-Ray Polarization of Solar Flares Measured with Rhessi

The degree of linear polarization in solar flares has not yet been precisely determined despite multiple attempts to measure it with different missions. The high energy range in particular has very rarely been explored, due to its greater instrumental difficulties. We approached the subject using the Reuven Ramaty High Energy Spectroscopic Imager (RHESSI) satellite to study 6 X-class and 1 M-class flares in the energy range between 100 keV and 350 keV. Using RHESSI as a polarimeter requires the application of strict cuts to the event list in order to extract those photons that are Compton scattered between two detectors. Our measurements show polarization values between 2% and 54%, with errors ranging from 10% to 26% in 1 sigma level. In view of the large uncertainties in both the magnitude and direction of the polarization vector, the results can only reject source models with extreme properties.Comment: 26 pages, 11 figures, accepted for publication by Solar Physic

A constructive study of the module structure of rings of partial differential operators

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. © 2014 Springer Science+Business Media