We introduce the principal representation category O ( G ) of reductive algebraic groups with Frobenius maps. When k is an algebraically closed field characteristic 0, we put forward a conjecture that this highest weight and provide some evidences conjecture. also study certain kind bound quiver algebras whose representations are related to .

Abstract Let ${\textbf{G}}$ be a connected reductive algebraic group defined over finite field $\mathbb{F}_q$ of $q$ elements and $\textbf{B}$ Borel subgroup $\mathbb{F}_q$. $\mathbb{k}$ we assume that $\mathbb{k}=\bar{\mathbb{F}}_q $ when $\textrm{char}\ \mathbb{k}=\textrm{char} \ \mathbb{F}_q$. We show the abstract-induced module $\mathbb{M}(\theta )=\mathbb{k}{\textbf{G}}\otimes _{\mathbb{k}...

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.

Manifolds with a commutative and associative multiplication on the tangent bundle are called F-manifolds if a unit field exists and the multiplication satisfies a natural integrability condition. They are studied here. They are closely related to discriminants and Lagrange maps. Frobenius manifolds are F-manifolds. As an application a conjecture of Dubrovin on Frobenius manifolds and Coxeter gr...