Hopf algebroids and Galois extensions

Abstract

To a finite Hopf-Galois extension $A|B$ we associate dual bialgebroids $S := \End_BA_B$ and $T:=(A{ø}_{B}A{)}^B$ over the centralizer $R$ using the depth two theory in math.RA/0108067. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra \cite{KT,Doi} to depth two extensions using coring theory math.RA/0002105. Next we show that $T^{\mathrm{op}}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem 5.1 in math.QA/9505024 for smash products with special modules over the Drinfel'd double, the Miyashita-Ulbrich action, the fact that $R$ is a commutative algebra in the pre-braided category of Yetter-Drinfel'd modules \cite[Schauenburg]{Sch} and the equivalence of Yetter-Drinfel'd modules with modules over Drinfel'd double \cite[Majid]{Maj}. In our last section, an exposition of results of Sugano \cite{Su82,Su87} leads us to a Galois correspondence between sub-Hopf algebroids of $S$ over simple subalgebras of the centralizer with finite projective intermediate simple subrings of a finite projective H-separable extension of simple rings $A\supseteq B$.

19 pages, to appear in the Bulletin of the Belgian Mathematical Society - Simon Stevin in approx. the second issue of 2005

References in corpus (1)

• Hopf algebroids with bijective antipodes: axioms, integrals and duals