An algebraic framework for the Drinfeld double based on infinite groupoids

TL;DR

This paper develops an algebraic framework for the Drinfeld double using infinite groupoids, demonstrating its structure as a weak multiplier Hopf algebra and establishing connections with quantum groupoids and module categories.

Contribution

It introduces a new algebraic framework for the Drinfeld double based on infinite groupoids, extending weak multiplier Hopf algebra theory.

Findings

The Drinfeld double is a weak multiplier Hopf algebra.

The double preserves algebraic quantum groupoid structures.

Modules over the double correspond to Yetter-Drinfeld modules.

Abstract

The Drinfeld double associated to the weak multiplier Hopf ($*$-) algebra pairing $\left\langle A, B\right\rangle$ is constructed. We show that the Drinfeld double is again a weak multiplier Hopf ($*$-) algebra. If $A$ and $B$ are algebraic quantum groupoids, then so does the double. We also prove the correspondence between modules over the Drinfeld double and Yetter-Drinfeld modules. Finally, we prove that the double is a quasitriangular weak multiplier Hopf algebra.