Extensions of the Endomorphism Algebra of Weak Comodule Algebras

Let $H$ be a weak Hopf algebra, and let $A/B$ be a weak right $H$-Galois extension. In this paper, we mainly discuss the extension of the endomorphism algebra of a module over $A$. A necessary and sufficient condition for such an extension of the endomorphism algebra to be weak $H$-Galois is obtained by using Hopf–Galois theory and Morita theory.