We provide an explicit construction for a class of commutative, non-associative algebras each the simple Chevalley groups simply laced type. Moreover, we equip these with associating bilinear form, which turns them into Frobenius algebras. This includes 3876-dimensional algebra on group type E8 acts by automorphisms. also prove that admit structure (axial) decomposition

We show that the theory of Frobenius fields is decidable. This is conjectured in [4], [8] and [13], and we prove it by solving a group theoretic problem to which this question is reduced there. To do this we present and develop the notion of embedding covers of finite and pro-finite groups. We also answer two other problems from [8], again by solving a corresponding group theoretic problem: A f...

If Q is a quasigroup that is free in the class of all quasigroups which are isotopic to an abelian group, then its multiplication group MltQ is a Frobenius group. Conversely, if MltQ is a Frobenius group, Q a quasigroup, then Q has to be isotopic to an abelian group. If Q is, in addition, finite, then it must be a central quasigroup. A quasigroup Q = Q(·) is usually defined as a binary system i...

The aim of the present paper is to lay foundation for a theory Ehresmann structures in positive characteristic, generalizing Frobenius-projective and Frobenius-affine defined previous work. This deals with atlases étale coordinate charts on varieties modeled homogeneous spaces algebraic groups, which we call Frobenius–Ehresmann structures. These are compared Cartan geometries as well higher-dim...

It is well known that Frobenius reciprocity is one of the central tools in the representation theory. In this paper, we discuss Frobenius reciprocity in the theory of automorphic functions. This Frobenius reciprocity was discovered by Gel’fand, Fomin, and PiatetskiShapiro in the 1960s as the basis of their interpretation of the classical theory of automorphic functions in terms of the represent...