The dual and the double of a Hopf algebroid are Hopf algebroids

TL;DR

This paper proves that the dual and double constructions of a certain class of Hopf algebroids preserve the Hopf algebroid structure under specific finiteness conditions.

Contribution

It establishes that the dual and Drinfeld double of a $ imes$-Hopf algebroid are also $ imes$-Hopf algebroids, extending the understanding of their structural properties.

Findings

Dual of a $ imes$-Hopf algebroid is $ imes$-Hopf under finiteness conditions.

The Drinfeld double of a $ imes$-Hopf algebroid is $ imes$-Hopf if the coopposite is $ imes$-Hopf.

Both the dual and the double retain the $ imes$-Hopf structure under certain conditions.

Abstract

Let $H$ be a $\times$-bialgebra in the sense of Takeuchi. We show that if $H$ is $\times$-Hopf, and if $H$ fulfills the finiteness condition necessary to define its skew dual $H^\vee$, then the coopposite of the latter is $\times$-Hopf as well. If in addition the coopposite $\times$-bialgebra of $H$ is $\times$-Hopf, then the coopposite of the Drinfeld double of $H$ is $\times$-Hopf, as is the Drinfeld double itself, under an additional finiteness condition.