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Defending the negated Kaplansky conjecture. (English) Zbl 0985.16026
Kaplansky’s 10-th conjecture at the end of his notes “Bialgebras” in 1975 was: For a given finite dimension \(n\), there are only finitely many isomorphism classes of Hopf algebras of dimension \(n\) over a field \(k\) whose characteristic does not divide \(n\).
In 1998 this conjecture was answered in the negative. Examples of infinite families of Hopf algebras of the same dimension \(n^4\) that are liftings of quantum linear spaces were discovered by Andruskiewitsch and Schneider, by Beattie, Dăscălescu and Grünenfelder, and by Gelaki and an infinite family of nonisomorphic Hopf algebras that are coordinate algebras of finite quantum subgroups of \(\text{GL}_q(2)\) was given by E. Müller.
In this paper, the author proves that in each infinite family in the examples quoted above, the Hopf algebras are cocycle deformations of each other. The key theorem for the proof of this result is:
Theorem: Let \(K\) be a Hopf subalgebra of a Hopf algebra \(H\), and let \(I\subset K\) be a Hopf ideal. If \(g\colon K\to k\) is an algebra map such that \(H/(gI)\neq 0\), then \(H/(gI)\) is an \((H/(I),H/(gIg^{-1}))\)-biGalois object and so the Hopf algebras \(H/(I)\) and \(H/(gIg^{-1})\) are monoidally Morita-Takeuchi equivalent.
In this theorem \(gI=\{\sum x_1g(x_2)\mid x\in I\}\), \(Ig^{-1}=\{\sum g^{-1}(x_1)x_2\mid x\in I\}\) and \((I)\) denotes the Hopf ideal of \(H\) generated by a Hopf ideal \(I\) of \(K\). Note that the condition \(H/(gI)\neq 0\) was recently noticed by A. Masuoka; this condition is satisfied for the situations considered in the paper.
Monoidally Morita-Takeuchi equivalent Hopf algebras that are finite dimensional are cocycle deformations of each other, i.e. they are quasi-isomorphic. These results lead to the new
Conjecture: There are finitely many Hopf algebras of a given finite dimension \(n\) up to cocycle deformation over a field \(k\) whose characteristic does not divide \(n\).
Note that recently A. Masuoka has improved results of Beattie, Dăscălescu and Raianu to show that all liftings of Nichols algebras of type \(A_2\) over an algebraically closed field are quasi-isomorphic and Beattie, Dăscălescu and Raianu have the same result for type \(B_2\), except if \(n=5\). This adds support to the conjecture above.

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
Full Text: DOI
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