Scalar extension Hopf algebroids

TL;DR

This paper completes the proof that the antipode in certain Hopf algebroids is an antihomomorphism and introduces a generalized construction for symmetric Hopf algebroids that does not require the antipode to be invertible.

Contribution

It provides a complete proof of the antipode's antihomomorphism property and introduces a new generalized construction of symmetric Hopf algebroids without assuming invertibility of the antipode.

Findings

Complete proof of the antipode antihomomorphism property.

A new generalized construction of symmetric Hopf algebroids.

Construction does not require invertibility of the antipode.

Abstract

Given a Hopf algebra $H$, Brzezi\'nski and Militaru have shown that each braided commutative Yetter-Drinfeld $H$-module algebra $A$ gives rise to an associative $A$-bialgebroid structure on the smash product algebra $A \sharp H$. They also exhibited an antipode map making $A\sharp H$ the total algebra of a Lu's Hopf algebroid over $A$. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair $A$ and $A^{\mathrm{op}}$ of braided commutative Yetter-Drinfeld $H$-module algebras, and output is a symmetric Hopf algebroid $A\sharp H \cong H\sharp A^{\mathrm{op}}$ over $A$. This construction does not require that the antipode of $H$ is invertible.