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Hopf bigalois extensions. (English) Zbl 0878.16020
The author constructs a very interesting example that shows that when $$A$$ is noncommutative and $$A/k$$ is some $$H$$-Galois extension for a noncommutative Hopf algebra $$H$$, then $$H$$ is not unique. This fact has been shown previously for a field $$A$$ by C. Greither and B. Pareigis [in J. Algebra 106, 239-258 (1987; Zbl 0615.12026)] and for commutative $$A$$ and $$H$$ by F. Van Oystaeyen and Y. Zhang [in $$K$$-Theory 8, No. 3, 257-269 (1994; Zbl 0814.16034)]. In the resulting construction, $$A/k$$ becomes an $$L$$-$$H$$ bigalois extension for a new Hopf algebra $$L=L(A,H)$$. The author discusses Hopf bigalois extensions and shows that equivalences of monoidal categories $$^B{\mathcal M}\cong{^H{\mathcal M}}$$ are classified by $$B$$-$$H$$ bigalois extensions.

MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16D90 Module categories in associative algebras 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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References:
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