Inner Actions of Weak Hopf Algebras

TL;DR

This paper introduces a new concept of $(e,f)$-invertibility to define inner actions of weak Hopf algebras, providing conditions for their existence on algebras and characterizing when actions extend to smash products.

Contribution

It develops the notion of $(e,f)$-invertibility and applies it to establish criteria for inner actions of weak Hopf algebras on algebras.

Findings

Defined $(e,f)$-invertibility for ring elements

Established conditions for weak Hopf algebra actions on algebras

Characterized when weak Hopf algebra actions extend to smash products

Abstract

Let $R$ be an associative ring and $e,f$ idempotent elements of $R$. In this paper we introduce the notion of $(e,f)$-invertibility for an element of $R$ and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra $H$ and an algebra $A$, we present sufficient conditions for $A$ to admit an inner action of $H$. We also prove that if $A$ is a left $H$-module algebra then $H$ acts innerly on the smash product $A\#H$ if and only if $H$ is a quantum commutative weak Hopf algebra.