We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i. e. for fusion categories), obtained recently in our joint work with D. Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols–Zoeller and Schauenburg, respectively. We also gi...

In a paper by Xu, some simple Lie algebras of generalized Cartan type were constructed, using the mixtures of grading operators and down-grading operators. Among them, are the simple Lie algebras of generalized Witt type, which are in general nongraded and have no torus. In this paper, some representations of these simple Lie algebras of generalized Witt type are presented. 1. INTRODUCTION The ...

In 1880 S. Lie (1842-1899) studied the groups of transformations depending on a finite number of parameters and now called Lie groups of transformations. Ten years later he discovered that these groups are only examples of groups of transformations solutions of linear or nonlinear systems of ordinary differential (OD) or partial differential (PD) equations which may even be of high order and ar...

In this thesis, we explore three alternatives to the current paradigm of the standard inflationary big bang scenario. The three alternative themes are spin torsion (or Einstein-Cartan-Kibble-Sciama) theories, extra dimensions (braneworld cosmology) and changing global symmetry. In the spin torsion theories, we found new cosmological solutions with a cosmological constant as alternative to the s...

The motion of spinless particles in Riemann-Cartan (RC) U4 and teleparallel spacetimes is revisited on the light of gravitational waves hitting on the spinless test particles. It is shown that in the case Cartan contortion is totally skew-symmetric the spinless particles follow geodesics but if this symmetry is dropped they are able to follow nongeodesic worldlines in torsionic background. The ...

Growth (resp. atrophy) describes the physical processes by which a material of solid body increases (resp. decreases) its size by addition (resp. removal) of mass. In the present contribution, we propose a sound mathematical analysis of growth, relying on the decomposition of the geometric deformation tensor into the product of a growth tensor describing the local addition of material and an el...

Abstract In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space $${\mathbb {R}}^3$$ R 3 to case Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., Riemannian 3-manifolds whose metr...

We show that if the cochain complex computing Ext groups (in category of modules over Hopf algebroids) admits a cocyclic structure, then noncommutative Cartan calculus structure on Tor dualises in cyclic sense to Coext Cotor. More precisely, duals chain resp. spaces two classical derived functors lead complexes compute more exotic ones, giving opposite module an operad with multiplication induc...

In the present work the properties of Cartan subalgebras and their connection with regular elements in finite dimensional Lie algebras are extended to the case of Leibniz algebras. It is shown that Cartan subalgebras and regular elements of a Leibniz algebra correspond to Cartan subalgebras and regular elements of a Lie algebra by a natural homomorphism. Conjugacy of Cartan subalgebras of Leibn...