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Cleft extensions for a Hopf algebra generated by a nearly primitive element. (English) Zbl 0809.16046
Fix an integer $$N \geq 2$$, and $$R$$ a commutative ring containing a root $$q$$ of the $$N$$-th cyclotomic polynomial over $$Z$$. Let $$A_ N$$ be the rank $$N^ 2$$ Hopf algebra over $$R$$ generated by a group-like element $$g$$ and a $$(1,g)$$-primitive element $$x$$, with relations $$g^ N = 1$$, $$x^ N = 0$$ and $$xg = qgx$$. If $$q$$ is a primitive $$N$$-th root of unity in a field $$R$$, $$A_ N$$ was constructed by the reviewer [Proc. Natl. Acad. Sci. USA 68, 2631-2633 (1971; Zbl 0222.16012)]. The author determines the isomorphism classes of $$A_ N$$-cleft extensions over a given algebra $$C$$. He also determines the set of crossed systems for $$A_ 2$$ over $$C$$.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
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