Pseudo-Galois Extensions and Hopf Algebroids

Abstract

Abstract. A pseudo-Galois extension is shown to be a depth two extension. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups, or more generally involutive Hopf algebras, and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observe that this Hopf algebroid is nontrivially isomorphic to a Connes-Moscovici Hopf algebroid, which raises interesting comparative questions.

Some Bialgebroids Constructed by Kadison and Connes–Moscovici are Isomorphic

“…The bialgebroid introduced by Kadison in [10] has as total algebra the following algebra structure on A e ⊗ H, which we denote by A e H:…”

“…Kadison found in [10] a universal property of A e H as algebra. We first establish an equivalent formulation, emphasizing the presence of the algebra A ⊗ A op : …”

Depth Two, Normality, and a Trace Ideal Condition for Frobenius Extensions

“…The second condition is called the D2 quasibase condition and is noted in [12]. It is based on condition (8) and the identifications Hom (…”

“…Suppose A | B is an H-separable right K-Galois extension for some Hopf algebra K. Then T op ∼ = R ⋊ K op as algebras, where K acts on the centralizer R via the Miyashita-Ulbrich action [8]. Then the module R T ∼ = R⋊K op R is a generator.…”

L-R-Smash biproducts, double biproducts and a braided category of Yetter-Drinfeld-Long bimodules

“…It was proved in [13] that, if H is moreover a Hopf algebra with bijective antipode, then D ♮ H is isomorphic to a diagonal crossed product D ⊲⊳ H as in [5], [7]; this result was used in [12] to give a very easy proof of the fact that two bialgebroids introduced independently in [6] and [8] are actually isomorphic.…”