A remark on nonlocal symmetries for the Calogero-Degasperis-Ibragimov-Shabat equation

We consider the Calogero-Degasperis-Ibragimov-Shabat (CDIS) equation and find the complete set of its nonlocal symmetries depending on the local variables and on the integral of the only local conserved density of the equation in question. The Lie algebra of these symmetries turns out to be a central extension of that of local generalized symmetries.Comment: arxiv version is already officia

Integrable Systems in n-dimensional Riemannian Geometry

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an (n)-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair in (\orth{n+1}) with the vector modified Korteweg-De Vries equation (vmKDV) \vk{t}= \vk{xxx}+\fr32 ||\vk{}||^2 \vk{x} as integrability condition. We indicate that other integrable vector evolution equations can be found by using a different Ansatz on the form of the Lax pair. We obtain these results by using the {\em natural} or {\em parallel} frame and we show how this can be gauged by a generalized Hasimoto transformation to the (usual) {\em Fren{\^e}t} frame. If one chooses the curvature to be zero, as is usual in the context of integrable systems, then one loses information unless one works in the natural frame

Higher dimensional Automorphic Lie Algebras

The paper presents the complete classification of Automorphic Lie Algebras based on $\mathfrak{sl}_n (\mathbb{C})$, where the symmetry group $G$ is finite and the orbit is any of the exceptional $G$-orbits in $\overline{\mathbb{C}}$. A key feature of the classification is the study of the algebras in the context of classical invariant theory. This provides on one hand a powerful tool from the computational point of view, on the other it opens new questions from an algebraic perspective, which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that Automorphic Lie Algebras associated to the $\mathbb{T}\mathbb{O}\mathbb{Y}$ groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only, thus they are group independent as Lie algebras. This can be established by defining a Chevalley normal form for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring.Comment: 43 pages, standard LaTeX2

Automorphic Lie Algebras and Cohomology of Root Systems

A cohomology theory of root systems emerges naturally in the context of Automorphic Lie Algebras, where it helps formulating some structure theory questions. In particular, one can find concrete models for an Automorphic Lie Algebra by integrating cocycles. In this paper we define this cohomology and show its connection with the theory of Automorphic Lie Algebras. Furthermore, we discuss its properties: we define the cup product, we show that it can be restricted to symmetric forms, that it is equivariant with respect to the automorphism group of the root system, and finally we show acyclicity at dimension two of the symmetric part, which is exactly what is needed to find concrete models for Automorphic Lie Algebras. Furthermore, we show how the cohomology of root systems finds application beyond the theory of Automorphic Lie Algebras by applying it to the theory of contractions and filtrations of Lie algebras. In particular, we show that contractions associated to Cartan $\mathbb{Z}$-filtrations of simple Lie algebras are classified by $2$-cocycles, due again to the vanishing of the symmetric part of the second cohomology group.Comment: 26 pages, standard LaTeX2

Equivariant decomposition of polynomial vector fields

To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the linear nilpotent. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the present paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch-Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D case

Versal Normal Form for Nonsemisimple Singularities

The theory of versal normal form has been playing a role in normal form since the introduction of the concept by V.I. Arnol'd. But there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields, that is, the linear part should be done completely first, before one computes the nonlinear terms. In this paper we address this issue by giving a complete description of a first order calculation in the case of the two- and three-dimensional irreducible nilpotent cases, which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics $L_4$ problem

Characterization of the human omega-oxidation pathway for omega-hydroxy-very-long-chain fatty acids

Very-long-chain fatty acids (VLCFAs) have long been known to be degraded exclusively in peroxisomes via beta-oxidation. A defect in peroxisomal beta-oxidation results in elevated levels of VLCFAs and is associated with the most frequent inherited disorder of the central nervous system white matter, X-linked adrenoleukodystrophy. Recently, we demonstrated that VLCFAs can also undergo omega-oxidation, which may provide an alternative route for the breakdown of VLCFAs. The omega-oxidation of VLCFA is initiated by CYP4F2 and CYP4F3B, which produce omega-hydroxy-VLCFAs. In this article, we characterized the enzymes involved in the formation of very-long-chain dicarboxylic acids from omega-hydroxy-VLCFAs. We demonstrate that very-long-chain dicarboxylic acids are produced via two independent pathways. The first is mediated by an as yet unidentified, microsomal NAD(+)-dependent alcohol dehydrogenase and fatty aldehyde dehydrogenase, which is encoded by the ALDH3A2 gene and is deficient in patients with Sjogren-Larsson syndrome. The second pathway involves the NADPH-dependent hydroxylation of omega-hydroxy-VLCFAs by CYP4F2, CYP4F3B, or CYP4F3A. Enzyme kinetic studies show that oxidation of omega-hydroxy-VLCFAs occurs predominantly via the NAD(+)-dependent route. Overall, our data demonstrate that in humans all enzymes are present for the complete conversion of VLCFAs to their corresponding very-long-chain dicarboxylic acids

Automorphic Lie Algebras with dihedral symmetry

The concept of Automorphic Lie Algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. Automorphic Lie Algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever-Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is $\mathfrak{sl}_2(\mathbb{C})$ and the poles of the Automorphic Lie Algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie algebra structure. In the present research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base Lie algebra. We find a uniform description of Automorphic Lie Algebras with dihedral symmetry, valid for poles at exceptional and generic orbits.Comment: 20 pages, 5 tables, standard LaTeX2