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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)
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