On Semisimple Proto-Abelian Categories Associated to Inverse Monoids

Let $G$ be a finite abelian group written multiplicatively, with $\widehat{G}=G\bigsqcup \{0\}$ the pointed abelian group formed by adjoining an absorbing element $0$. There is an associated finitary, proto-abelian category ${\mathrm{Vect}}_{\widehat{G}}$, whose objects can be thought of as finite-dimensional vector spaces over $\widehat{G}$. The class of $\widehat{G}$-linear monoids are then defined in terms of this category. In this paper, we study the finitary, proto-abelian category $\mathrm{Rep}(M,\widehat{G})$ of finite-dimensional $\widehat{G}$-linear representations of a $\widehat{G}$-linear monoid $M$. Although this category is only a slight modification of the usual category of $M$-modules, it exhibits significantly different behavior for interesting classes of monoids. Assuming that the regular principal factors of $M$ are objects of $\mathrm{Rep}(M,\widehat{G})$, we develop a version of the Clifford-Munn-Ponizovski\u i Theorem and classify the $M$ for which each non-zero object of $\mathrm{Rep}(M,\widehat{G})$ is a direct sum of simple objects. When $M$ is the endomorphism monoid of an object in ${\mathrm{Vect}}_{\widehat{G}}$, we discuss alternate frameworks for studying its $\widehat{G}$-linear representations and contrast the various approaches.