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Lifting of quantum linear spaces and pointed Hopf algebras of order $$p^3$$. (English) Zbl 0919.16027
Let $$k$$ be an algebraically closed field of characteristic zero. Let $$A$$ be a noncosemisimple Hopf algebra over $$k$$ whose coradical $$A_0$$ is a Hopf subalgebra, e.g., if $$A$$ is pointed. Let $$\text{gr }A$$ be the graded Hopf algebra associated to the coradical filtration of $$A$$, $$\pi$$ the projection of $$\text{gr }A$$ on $$A_0$$. Then $$R=\{a$$ in $$\text{gr }A:(I\otimes\pi)\Delta a=a\otimes 1\}$$, the coinvariants of $$\pi$$, is a Hopf algebra in the braided category of Yetter-Drinfeld modules over $$A_0$$. $$\text{gr }A$$ can be recovered from $$R$$ by bosonization (biproduct with $$A_0$$). The authors propose to study $$A$$ by first studying $$R$$, transferring the information to $$\text{gr }A$$, and then lifting to $$A$$. In this article, they apply this plan when $$R$$ is a quantum linear space. This enables them to classify all pointed Hopf algebras of order $$p^3$$, $$p$$ an odd prime, and also to classify all pointed Hopf algebras whose coradical is Abelian and has index $$p$$ or $$p^2$$. Their lifting results then enable them to produce an infinite family of pointed, nonisomorphic Hopf algebras of dimension $$p^4$$. This answers in the negative a conjecture of Kaplansky.
Reviewer’s remark: This conjecture has also been answered recently by other authors using different constructions.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16W50 Graded rings and modules (associative rings and algebras)
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