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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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