Coactions of Hopf Algebras on Algebras in Positive Characteristic

Let K be a field of positive characteristic $p > 0$. We study the coactions of the Hopf algebra of the multiplicative group $H_m$ with underlying algebra $H = K[X_1, \dots, X_m](X_1^{s_1}, \dots, X_n^{s_n})$, $n \geq 1, s_1 \geq \dots \geq s_n \geq 1$ on a K-algebra A. We give the rule for the set of additive endomorphisms of A, that define a coactions of $H_m$ on A commutative. For $s_1 = \dots = s_n = 1$ we obtain the explicit expression of such coactions in terms of n derivations of A.