Mochizuki’s work on torally crys-stable bundles [9] has extensive implications for the theory of logarithmic connections on vector bundles of rank 2 on curves, once the language is translated appropriately. We describe how to carry out this translation, and give two classes of applications: first, one can conclude almost immediately certain results classifying Frobenius-unstable vector bundles ...

We give a new proof of the well known Wedderburn's little theorem (1905) that a finite‎ ‎division ring is commutative‎. ‎We apply the concept of Frobenius kernel in Frobenius representation theorem in finite group‎ ‎theory to build a proof‎.

Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. Given a semistable vector bundle E over X , we show that its direct image F∗E under the Frobenius map F of X is again semistable. We deduce a numerical characterization of the stable rank-p vector bundles F∗L, where L is a line bundle over X .

Different kinds of reduction for ordinary differential equations, such as λ –symmetry and σ reductions, are recovered particular cases Frobenius theorem distribution vector fields. This general approach provides some hints to tackle the reconstruction problem solve it under suitable assumptions on involved in process.

Tannaka duality describes the relationship between algebraic objects in a given category and functors into that category; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful “fibre functors” to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras...

We continue the investigation of Hopf-Galois extensions with central invariants started in [30]. Our objective is not to imitate algebraic geometry using Hopf-Galois extension but to understand their geometric properties. Let H be a finite-dimensional Hopf algebra over a ground field k. Our main object of study is an H-Galois extension U ⊇ O such that O is a central subalgebra of U . Let us bri...

A finite dimensional Lie algebra f is Frobenius if there is a linear Frobenius functional F : f→ C such that the skew bilinear form BF defined by BF (x, y) = F ([x, y]) is non-degenerate. The principal element of f is then the unique element F̂ such that F (x) = F ([F̂ , x]); it depends on the choice of functional. However, if f is a subalgebra of a simple Lie algebra g and not an ideal of any la...

In this note we characterize the compact weighted Frobenius-Perron operator $p$ on $L^1(Sigma)$ and determine their spectra. We also show that every weakly compact weighted Frobenius-Perron operator on $L^1(Sigma)$ is compact.