Push-forward of Hopf--Galois extensions: the non central case

TL;DR

We address the problem of extending Hopf--Galois theory to noncentral settings by introducing a push-forward construction for Hopf--Galois extensions using twisted tensor products. The main method shows that, given an $H$-Galois extension $B\subset A$ and an algebra map $\mathsf{F}: B\to C$ with a twisting map $\psi:A\otimes C\to C\otimes A$ compatible with $\mathsf{F}$, the twisted push-forward $C\otimes^\psi_B A$ is a faithfully flat $H$-Galois extension of $C$; the translation map is explicitly described as $\tau^\psi(h)=(1_C\otimes_B h^{<1>})\otimes (1_C\otimes_B h^{<2>})$. The paper develops the theory of when the twist descends to quotients, analyzes special cases such as Galois objects and total-space quotients, and studies concrete examples including deformed SU(2) bundles, showing how push-forwards behave under quotienting. It also compares Ehresmann--Schauenburg bialgebroids before and after push-forward, establishing morphisms that connect the original and pushed-forward ES structures under base-change. Overall, the work extends noncentral Hopf--Galois theory and provides algebraic tools for pullback-like constructions in noncommutative geometry with potential applications to quantum principal bundles.

Abstract

We study the push-forward of Hopf--Galois extensions as the algebraic counterpart of the pullback of principal bundles. We apply the theory of twisted tensor product algebras to endow covariant extensions of modules along a map $\mathsf{F}$ with an algebra structure, under compatibility conditions between $\mathsf{F}$ and the twisting map. The push-forward of an $H$-Galois extension $B \subset A$ along a map $\mathsf{F} : B \to C$ is an $H$-Galois extension of $C$. The corresponding Ehresmann--Schauenburg algebroids are compared.

Theorems & Definitions (61)

• Theorem 2.1
• Example 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 3.1
• proof