On noncommutative equivariant bundles

TL;DR

This paper explores a noncommutative generalization of equivariant vector bundles, highlighting differences from classical cases and providing examples where traditional equivalences fail in noncommutative settings.

Contribution

It introduces a noncommutative framework for equivariant bundles, analyzing conditions under which classical equivalences break down in noncommutative algebra.

Findings

Invertibility of $ heta$ when $H$ is commutative ensures equivalence.

Noncommutative Hopf algebras can produce non-invertible $ heta$.

Examples show the divergence between two notions of equivariance in noncommutative cases.

Abstract

We discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let $A$ be a $\mathbb{K}$-algebra, $M$ a left $A$-module, $H$ a Hopf $\mathbb{K}$-algebra, $\delta:A\to H\otimes A:=H\otimes_{\mathbb{K}} A$ an algebra coaction, and let $(H\otimes A)_\delta$ denote $H\otimes A$ with the right $A$-module structure induced by~$\delta$. The usual definitions of an equivariant vector bundle naturally lead, in the context of $\mathbb{K}$-algebras, to an $(H\otimes A)$-module homomorphism \[\Theta:H\otimes M\to (H\otimes A)_\delta\otimes_AM\] that fulfills some appropriate conditions. On the other hand, sometimes an $(A,H)$-Hopf module is considered instead, for the same purpose. When $\Theta$ is invertible, as is always the case when $H$ is commutative, the two descriptions are equivalent. We point out that the two notions differ in general, by giving an example of a noncommutative Hopf algebra $H$ for which there exists such a $\Theta$ that is not invertible and a left-right $(A,H)$-Hopf module whose corresponding homomorphism $M\otimes H\to (A\otimes H)_\delta\otimes_AM$ is not an isomorphism.