In this paper we consider some algebraic structures associated to a class of outer automorphisms of generalized Kac-Moody (GKM) algebras. These structures have recently been introduced in [2] for a smaller class of outer automorphisms in the case of ordinary Kac-Moody algebras with symmetrizable Cartan matrices. A GKM algebra G = G(A) is essentially described by its Cartan matrix, A = (aij)i,j∈...

Recent progress in diffusion imaging has lead to in-vivo acquisitions of fiber orientation data in the beating heart. Current methods are however limited in resolution to a few short-axis slices. For this particular application and others where the diffusion volume is subsampled, partial or even damaged, the reconstruction of a complete volume can be challenging. To address this problem, we pre...

One of the most appealing results metric-affine gauge theory gravity is a close parallel between Riemann curvature two-form and Cartan torsion two-form: While former field strength Lorentz-group connection one-form, latter can be understood as coframe one-form. This parallel, unfortunately, not fully established until one adopts Trautman's idea introducing an affine-vector-valued zero-from, mea...

The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms and their parallel transport, that ties together various approaches. We begin with detailed introduction to bundles. then introduce the concept of so-called relative supermanifolds as well bundles connections defined in these categories. Studying objects turns out be utmost importance orde...

In this paper we introduce a reference along a null curve in an n-dimensional Lorentzian space with the minimun number of curvatures. That reference generalizes the reference of Bonnor for null curves in Minkowski space-time and it is called the Cartan frame of the curve. The associated curvature functions are called the Cartan curvatures of the curve. We characterize the null helices (that is,...

We explore the ultra-relativistic limit of a class four dimensional gravity theories, known as Lovelock-Cartan gravities, in first order formalism. First, we review well Einstein-Hilbert action. A very useful scale symmetry involving vierbeins and boost connection is presented. Moreover, field equations to find formal solutions. Some remarkable results are obtained: Riemann Weitzenb\"ock like m...

Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to ...

We present an explicit numerical method to obtain the Cartan-KhanejaGlaser decomposition of a general element G ∈ SU(2 ) in terms of its ‘Cartan’ and ‘non-Cartan’ components. This effectively factors G in terms of group elements that belong in SU(2) with n < N , a procedure that an be iterated down to n = 2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained b...