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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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