Certain linear isomorphisms for hyperalgebras relative to a Chevalley group

Let $G$ be a simply connected and simple algebraic group defined and split over a finite prime field ${\mathbb{F}}_p$ of $p$ elements. In this paper, using an ${\mathbb{F}}_p$-linear map splitting Frobenius endomorphism on a hyperalgebra relative to $G$, we obtain some ${\mathbb{F}}_p$-linear isomorphisms induced by multiplication in the hyperalgebra.