Normal Hopf subalgebras, depth two and Galois extensions

TL;DR

This paper explores the relationship between normal Hopf subalgebras, depth two extensions, and Galois extensions, establishing new characterizations and examples within Hopf algebra theory.

Contribution

It demonstrates that a Hopf subalgebra is normal if and only if it forms a Hopf-Galois extension, and characterizes weak Hopf-Galois extensions via an alternative Galois mapping.

Findings

Left endomorphism ring of depth two extension is a left S-Galois extension.

Normal Hopf subalgebras are precisely Hopf-Galois extensions.

Weak Hopf-Galois extensions are depth two and have bijective Galois mappings.

Abstract

Let $S$ be the left $R$-bialgebroid of a depth two extension with centralizer $R$ as defined in math.QA/0108067. We show that the left endomorphism ring of depth two extension, not necessarily balanced, is a left $S$-Galois extension of $A^{\rm op}$. 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 find a class of examples of the alternative Hopf algebroids in math.QA/0302325. We also 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.