Partial Hopf actions on generalized matrix algebras

TL;DR

This paper characterizes when a Hopf algebra can act partially on generalized matrix algebras, introducing the concept of opposite covariant pairs and unifying conditions for group algebra actions.

Contribution

It establishes necessary and sufficient conditions for partial Hopf actions on generalized matrix algebras, introducing opposite covariant pairs with a universal property.

Findings

Derived conditions for partial Hopf actions on matrix algebras.

Unified partial action conditions for group algebras.

Introduced the concept of opposite covariant pairs.

Abstract

Let $\Bbbk$ be a field, $H$ a Hopf algebra over $\Bbbk$, and $R = (_iM_j)_{1 \leq i,j \leq n}$ a generalized matrix algebra. In this work, we establish necessary and sufficient conditions for $H$ to act partially on $R$. To achieve this, we introduce the concept of an opposite covariant pair and demonstrate that it satisfies a universal property. In the special case where $H = \Bbbk G$ is the group algebra of a group $G$, we recover the conditions given in \cite{BP} for the existence of a unital partial action of $G$ on $R$.