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Galois and biGalois objects over monomial non-semisimple Hopf algebras. (English) Zbl 1117.16025

Let \(A\) be a Hopf algebra over a field \(k\). A right \(A\)-Galois extension (of \(k\)) is a nonzero right \(A\)-comodule algebra \(Z\) such that a certain map from \(Z\otimes Z\) to \(Z\otimes A\) is bijective. The set of isomorphism classes of right \(A\)-Galois extensions is denoted \(\text{Gal}(A)\). Similarly one has left \(A\)-Galois extensions. An \(A\)-biGalois extension is a \(Z\) which is both a left \(A\)-Galois extension and a right \(A\)-Galois extension such that \(Z\) is an \(A\)-\(A\)-bicomodule. The set of isomorphism classes of \(A\)-biGalois extensions is denoted \(\text{BiGal}(A)\). It is a group under cotensor product.
The author determines \(\text{Gal}(A)\) and \(\text{BiGal}(A)\) for \(A\) a monomial non-semisimple Hopf algebra. A precise definition was given by X.-W. Chen, H.-L. Huang, Y. Ye, and P. Zhang, [J. Algebra 275, No. 2, 212-232 (2004; Zbl 1071.16030)]. There it was shown that the definition is equivalent to being a Hopf algebra constructed from a group datum \(D=(G,g,x,u)\), \(G\) a finite group, \(g\) a central element of \(G\), \(x\) a character of \(g\) with \(x(g)\neq 1\), and \(u\) in \(k\) satisfying certain conditions related to the orders of \(g\) and \(x(g)\). Various examples of monomial Hopf algebras are known, including Taft algebras and their generalizations. The author’s classification of \(\text{Gal}(A)\) and \(\text{BiGal}(A)\) involves identifying 6 types of group data, which involve whether or not \(u\) is \(0\) (\(u=0\) in 5 of them) and conditions involving the orders of \(g\) and \(x(g)\). The descriptions of \(\text{Gal}(A)\) involve some second cohomology groups, including \(H^2(G,k^*)\). \(\text{Gal}(A)\) need not be a group itself. The descriptions of \(\text{BiGal}(A)\) involve the subgroup of \(\operatorname{Aut}(G)\) of automorphisms \(s\) which fix \(g\), and a certain homomorphic image of \(H^2(G,k^*)\).

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)

Citations:

Zbl 1071.16030
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References:

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