Hopf algebroids and quantum groupoids

TL;DR

This paper introduces Hopf algebroids without commutative constraints, constructs quantum groupoids from quantum groups, and provides detailed examples including quantum sl(2), expanding the algebraic framework of quantum symmetries.

Contribution

It defines Hopf algebroids with non-commutative algebras and constructs quantum groupoids from quantum groups, including detailed examples like quantum sl(2).

Findings

Defined Hopf algebroids without commutativity constraints.

Constructed quantum groupoids from quantum groups with R-matrices.

Provided detailed example of quantum sl(2).

Abstract

We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras of the Drinfeld doubles of Hopf algebras when the $R$-matrices act properly. When this construction is applied to quantum groups, we get examples of quantum groupoids, which are semi-classical limits of Poisson groupoids. The example of quantum $sl(2)$ is worked out in details.