Hopf algebroids with balancing subalgebra

TL;DR

This paper introduces a new axiomatic framework for Hopf algebroids with a balancing subalgebra, unifying various constructions including scalar extensions and weak Hopf algebras, and explores their behavior under twisting procedures.

Contribution

It proposes a set of abstract axioms for Hopf algebroids with a balancing subalgebra, unifying different existing constructions and analyzing their properties.

Findings

Scalar extension Hopf algebroids fit into the new axiomatic framework.

Weak Hopf algebra-based Hopf algebroids are encompassed by the framework.

The behavior of the balancing subalgebra under Drinfeld-Xu twisting is discussed.

Abstract

Recently, S. Meljanac proposed a construction of a class of examples of an algebraic structure with properties very close to the Hopf algebroids $H$ over a noncommutative base $A$ of other authors. His examples come along with a subalgebra $\mathcal{B}$ of $H\otimes H$, here called the balancing subalgebra, which contains the image of the coproduct and such that the intersection of $\mathcal{B}$ with the kernel of the projection $H\otimes H\to H\otimes_A H$ is a two-sided ideal in $\mathcal{B}$ which is moreover well behaved with respect to the antipode. We propose a set of abstract axioms covering this construction and make a detailed comparison to the Hopf algebroids of Lu. We prove that every scalar extension Hopf algebroid can be cast into this new set of axioms. We present an observation by G. B\"ohm that the Hopf algebroids constructed from weak Hopf algebras fit into our framework as well. At the end we discuss the change of balancing subalgebra under Drinfeld-Xu procedure of twisting of associative bialgebroids by invertible 2-cocycles.