Twisted traces of quantum intertwiners and quantum dynamical R-matrices corresponding to generalized Belavin-Drinfeld triples

This paper is a continuation of math.QA/9907181 and math.QA/9908115. We consider traces of intertwiners between certain representations of the quantized enveloping algebra associated to a semisimple complex Lie algebra g, which are twisted by a ``generalized Belavin-Drinfeld triple'', i.e a triple consisting of two subdiagrams of the Dynkin diagram of g together with an isomorphism between them. The generating functions F(lambda,mu) for such traces depend on two weights lambda and mu. We show that F(lambda,mu) satisfy two sets of difference equations in the variable lambda: the Macdonald-Ruijsenaars (MR) equations and the quantum Knizhnik-Zamolodchikov (qKZB) equations. These equations involve as a main ingredient the quantum dynamical R-matrices constructed in math.QA/9912009. When the generalized Belavin-Drinfeld triple is an automorphism, we show that F(lambda,mu) satisfy another two sets of difference equations with respect to the weight mu. These dual MR and dual qKZB equations involve the usual Felder's dynamical R-matrix. These results were first obtained by the first author and A. Varchenko in the special case of the trivial Belavin-Drinfeld triple. However, the symmetry between lambda and mu which exists in that case is destroyed in the twisted setting. At the end, we brielfly treat the (simialr) case of Kac-Moody algebras g and derive the classical limits of all the previous results.Comment: 30 pages, late

Spherical Hall algebra of Spec (Z)

We study an arithmetic analog of the Hall algebra of a curve, when the curve is replaced by the spectrum of the integers compactified at infinity. The role of vector bundles is played by lattices with quadratic forms. This algebra H consists of automorphic forms with respect to GL_n(Z), n>0, with multiplication given by the parabolic pseudo-Eisenstein series map. We concentrate on the subalgebra SH in H generated by functions on the Arakelov Picard group of Spec(Z). We identify H with a Feigin-Odesskii type shuffle algebra, with the function defining the shuffle algebra expressed through the Riemann zeta function. As an application we study relations in H. Quadratic relations express the functional equation for the Eisenstein-Maass series. We show that the space of additional cubic relations (lying an an appropriate completion of H and considered modulo rescaling), is identified with the space spanned by nontrivial zeroes of the zeta function

Variants Cause Spastic Paraplegia Associated with Cerebral Hypomyelination

Oculodentodigital dysplasia is an autosomal dominant disorder due to variants characterized by dysmorphic features. Neurologic symptoms have been described in some patients but without a clear neuroimaging pattern. To understand the pathophysiology underlying neurologic deficits in oculodentodigital dysplasia, we studied 8 consecutive patients presenting with hereditary spastic paraplegia due to variants. Clinical disease severity was highly variable. Cerebral MR imaging revealed variable white matter abnormalities, consistent with a hypomyelination pattern, and bilateral hypointense signal of the basal ganglia on T2-weighted images and/or magnetic susceptibility sequences, as seen in neurodegeneration with brain iron accumulation diseases. Patients with the more prominent basal ganglia abnormalities were the most disabled ones. This study suggests that -related hereditary spastic paraplegia is a complex neurodegenerative disease affecting both the myelin and the basal ganglia. variants should be considered in patients with hereditary spastic paraplegia presenting with brain hypomyelination, especially if associated with neurodegeneration and a brain iron accumulation pattern

Skin-impedance in Fabry Disease: A prospective, controlled, non-randomized clinical study

<p>Abstract</p> <p>Background</p> <p>We previously demonstrated improved sweating after enzyme replacement therapy (ERT) in Fabry disease using the thermo-regularity sweat and quantitative sudomotor axon reflex tests. Skin-impedance, a measure skin-moisture (sweating), has been used in the clinical evaluation of burns and pressure ulcers using the portable dynamic dermal impedance monitor (DDIM) system.</p> <p>Methods</p> <p>We compared skin impedance measurements in hemizygous patients with Fabry disease (22 post 3-years of bi-weekly ERT and 5 ERT naive) and 22 healthy controls. Force compensated skin-moisture values were used for statistical analysis. Outcome measures included 1) moisture reading of the 100^threpetitive reading, 2) rate of change, 3) average of 60–110^threading and 4) overall average of all readings.</p> <p>Results</p> <p>All outcome measures showed a significant difference in skin-moisture between Fabry patients and control subjects (p < 0.0001). There was no difference between Fabry patients on ERT and patients naïve to ERT. Increased skin-impedance values for the four skin-impedance outcome measures were found in a small number of dermatome test-sites two days post-enzyme infusions.</p> <p>Conclusion</p> <p>The instrument portability, ease of its use, a relatively short time required for the assessment, and the fact that DDIM system was able to detect the difference in skin-moisture renders the instrument a useful clinical tool.</p

The double Ringel-Hall algebra on a hereditary abelian finitary length category

In this paper, we study the category $\mathscr{H}^{(\rho)}$ of semi-stable coherent sheaves of a fixed slope $\rho$ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of $\mathscr{H}^{(\rho)}$ and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.Comment: 29 page