Actions of Hopf algebras

We consider an action of a finite-dimensional Hopf algebra $H$ on a non-commutative associative algebra $A$. Properties of the invariant subalgebra $A^H$ in $A$ are studied. It is shown that if $A$ is integral over its centre $\mathrm Z(A)$ then in each of three cases $A$ will be integral over $\mathrm Z(A)^H$ (the invariant subalgebra in $\mathrm Z(A)$):

• 1) the coradical $H_0$ is cocommutative and char $\operatorname {char}k=p>0$,
• 2) $H$ is pointed, $A$ has no nilpotent elements, $\mathrm Z(A)$ is an affine algebra, and $\operatorname {char}k=0$,
• 3) $H$ is cocommutative.

We also consider an action of a commutative Hopf algebra $H$ on an arbitrary associative algebra, in particular, the canonical action of $H$ on the tensor algebra $T(H)$. A structure theorem on Hopf algebras is proved by application of the technique developed. Namely, every commutative finite-dimensional Hopf algebra $H$ whose coradical $H_0$ is a sub-Hopf algebra or cocommutative, where $\operatorname {char}k=0$ or $\operatorname {char}k>\dim H$, is cosemisimple, that is, $H=H_0$. In particular, a commutative pointed Hopf algebra with $\operatorname {char}k=0$ or $\operatorname {char}k>\dim H$ will be a group Hopf algebra. An example is also constructed showing that the restrictions on $\operatorname {char}k$ are essential.