Galois theory for bialgebroids, depth two and normal Hopf subalgebras
Abstract
Description
We reduce certain proofs in math.RA/0108067, math.RA/0408155, and math.QA/0409589 to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a left $T$-Galois extension for some right finite projective left bialgebroid over some algebra $R$ if and only if it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.
19 pp., to appear in Proceedings of the "Ferrara Algebra Workshop" jointly with the "Workshop on Hopf Algebras,Swansea" (to be published as a special issue of the "Annali dell'Universita' di Ferrara, sez. VII, Scienze Matematiche)
19 pp., to appear in Proceedings of the "Ferrara Algebra Workshop" jointly with the "Workshop on Hopf Algebras,Swansea" (to be published as a special issue of the "Annali dell'Universita' di Ferrara, sez. VII, Scienze Matematiche)