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3 papers
Actions of Hopf algebras
A. A. Totok M. V. Lomonosov Moscow State University
Abstract:
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.
UDC:
512.667.7
MSC: 16W30 Received: 28.04.1997
DOI:
10.4213/sm299
Fulltext:
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