On Mallios $\mathcal A$-connections as connections on principal sheaves

Abstract

Motivated by [5], we associate a vector sheaf $\varepsilon$ with a principal sheaf $\mathcal P\varepsilon)$, called sheaf of frames of $\varepsilon$. We show that $\mathcal A$-connections on $\varepsilon$ correspond to connections on $\mathcal P(\varepsilon)$. The latter are defined by an appropriate family of local matrices or, equivalently, by a morphism acting on $\mathcal P(\varepsilon)$, analogously to the operator of an \mathcal A$-collection.