## Actions of Hopf algebras.(English. Russian original)Zbl 0914.16019

Sb. Math. 189, No. 1, 149-159 (1998); translation from Mat. Sb. 189, No. 1, 147-157 (1998).
Let $$H$$ be a finite-dimensional Hopf algebra and $$A$$ a left noncommutative associative $$H$$-module algebra with center $$Z(A)$$. If $$H$$ is a pointed Hopf algebra and either $$\text{char }k=p>0$$ or $$\text{char }k=0$$, the algebra $$A$$ has no nilpotents and $$Z(A)$$ is an affine algebra. Then the extension $$Z(A)/(A^H\cap Z(A))$$ is integral.
Suppose that $$A$$ is integral over $$Z(A)$$ and one of the following conditions holds: the coradical of $$H$$ is cocommutative and $$\text{char }k=p>0$$; $$H$$ is a pointed Hopf algebra, $$A$$ has no nilpotents, $$Z(A)$$ is an affine algebra and $$\text{char }k=0$$; $$H$$ is cocommutative. Then $$A$$ is integral over $$Z(A)\cap A^H$$.
Suppose that the coradical $$H_0$$ of $$H$$ is cocommutative and either $$\text{char }k=0$$ or $$\text{char }k>\dim H$$. Then $$A$$ is fully integral over $$A^H$$ and if $$H$$ is a pointed Hopf algebra, then $$A^H=A^{G(H)}$$, where $$G(H)$$ is the set of all group-like elements of $$H$$. The previous result has the following corollary. Suppose that the coradical $$H_0$$ of a commutative Hopf algebra $$H$$ is a sub-Hopf algebra and either $$\text{char }k=0$$ or $$\text{char }k>\dim H$$. Then $$H$$ is cosemisimple. In particular, if $$H$$ is pointed, then $$H=kG(H)$$. Moreover, a commutative Hopf algebra $$H$$ with cocommutative coradical $$H_0$$ is cosemisimple and therefore cocommutative if either $$\text{char }k=0$$ or $$\text{char }k>\dim H$$. If in addition $$k$$ is algebraically closed then $$H$$ is a group algebra.

### MSC:

 16W30 Hopf algebras (associative rings and algebras) (MSC2000)
Full Text: