We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.

Hepatic myelopathy (HM) is a rare complication of chronic liver diseases usually associated with a portosystemic shunt, causing a progressive spastic paraparesis, and is likely to be overlooked. Thirteen patients with liver cirrhosis associated with surgical or spontaneous portosystemic shunts were studied to determine the frequency and gravity of HM. Six patients exhibited clear-cut signs of spinal cord involvement and four of them exhibited varying degrees of disability. Neurological examination did not reveal any abnormalities in the other patients. Motor evoked potentials (MEPs) were measured in all patients; in five of them the examinations were done before and after orthotopic liver transplantation (OLT). The patients with clinical signs of spinal cord involvement exhibited severe neurophysiological abnormalities, whereas milder but unequivocal MEP abnormalities were found in four of the seven patients with normal clinical examination. The clinical and neurophysiological features of patients with slight MEP abnormalities improved after OLT, whereas the patients with a more advanced stage of disease (severe MEPs abnormalities) did not. Our findings indicate that MEP studies may disclose an impairment of the corticospinal pathways even before HM is clinically manifest and provide evidence that early diagnosis of HM and subsequent immediate liver transplantation have to be recommended.

Nichols algebras of group type with many cubic relations are classified under a technical assumption on the structure of Hurwitz orbits of the third power of the underlying indecomposable rack. All such Nichols algebras are finite-dimensional and their Hilbert series have a factorization into quantum integers. Also, all known finite-dimensional elementary Nichols algebras of group type turn out to have many cubic relations. The technical assumption of our theorem can be removed if a conjecture in the theory of cellular automata can be proven.Comment: 46 pages, 20 figures. We improved significantly the text by adding more definitions and more explanations. Accepted for publication in Transactions of the American Mathematical Societ

Abstract:A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin-Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize.

We analyse the so-called small-world network model (originally devised by Strogatz and Watts), treating it, among other things, as a case study of non-linear coupled difference or differential equations. We derive a system of evolution equations containing more of the previously neglected (possibly relevant) non-linear terms. As an exact solution of this entangled system of equations is out of question we develop a (as we think, promising) method of enclosing the "exact" solutions for the expected quantities by upper and lower bounds, which represent solutions of a slightly simpler system of differential equation. Furthermore we discuss the relation between difference and differential equations and scrutinize the limits of the spreading idea for random graphs. We then show that there exists in fact a "broad" (with respect to scaling exponents) crossover zone, smoothly interpolating between linear and logarithmic scaling of the diameter or average distance. We are able to corroborate earlier findings in certain regions of phase or parameter space (as e.g. the finite size scaling ansatz) but find also deviations for other choices of the parameters. Our analysis is supplemented by a variety of numerical calculations, which, among other things, quantify the effect of various approximations being made. With the help of our analytical results we manage to calculate another important network characteristic, the (fractal) dimension, and provide numerical values for the case of the small-world network.