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We explore the notion of representation of an affine extension of an abelianvariety -- such an extension is a faithfully flat affine morphism of$\Bbbk$-group schemes $q:G\to A$, where $A$ is an abelian variety. Wecharacterize the categories that arise as the category of representations of anaffine extension $q:G\to A$, generalizing the classical results of TannakaDuality established for affine $\Bbbk$-group schemes (that is, when$A=\operatorname{Spec}(\Bbbk)$). We also prove the existence of a contravariantequivalence between the category of affine extensions of a given $A$ and thecategory of faithful commutative Hopf sheaves on $A$, generalizing in thismanner the well-known op-equivalence between affine group schemes andcommutative Hopf algebras. If $\mathcal H_q$ is the Hopf sheaf on $A$associated to $q$, the category of representations of $q$ is equivalent to thecategory of $\mathcal H_q$-comodules.

Quasi-compact group schemes, Hopf sheaves, and their representations