Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations
Volume 109 / 2007
Abstract
We prove that the study of the category $C\mbox{-}{\rm Comod}$ of left comodules over a $K$-coalgebra $C$ reduces to the study of $K$-linear representations of a quiver with relations if $K$ is an algebraically closed field, and to the study of $K$-linear representations of a $K$-species with relations if $K$ is a perfect field. Given a field $K$ and a quiver $Q= (Q_0, Q_1)$, we show that any subcoalgebra $C$ of the path $K$-coalgebra $K\square Q$ containing $K\square Q_0 \oplus K\square Q_1$ is the path coalgebra $K\square (Q, \mathfrak{B})$ of a profinite bound quiver $(Q, \mathfrak{B})$, and the category $C\mbox{-}{\rm Comod}$ of left $C$-comodules is equivalent to the category ${\rm Rep}_K^{\ell n\ell f}( Q, \mathfrak{B})$ of locally nilpotent and locally finite $K$-linear representations of $Q$ bound by the profinite relation ideal $\mathfrak{B}\subset \widehat{KQ}$.
Given a $K$-species $\mathcal{M} = ( F_j, {}_iM_j)$ and a relation ideal $\mathfrak{B}$ of the complete tensor $K$-algebra $\widehat T(\mathcal{M})= \widehat {T_F(M)}$ of $\mathcal{M}$, the bound species subcoalgebra $T\square (\mathcal{M}, \mathfrak{B})$ of the cotensor $K$-coalgebra $T\square (\mathcal{M})=T\square _{F}(M)$ of $\mathcal{M}$ is defined. We show that any subcoalgebra $C$ of $T\square (\mathcal{M}) $ containing $T\square (\mathcal{M})_0 \oplus T\square (\mathcal{M})_1$ is of the form $T\square (\mathcal{M}, \mathfrak{B})$, and the category $C\mbox{-}{\rm Comod}$ is equivalent to the category ${\rm Rep}_K^{\ell n\ell f}( \mathcal{M}, \mathfrak{B})$ of locally nilpotent and locally finite $K$-linear representations of $\mathcal{M}$ bound by the profinite relation ideal $\mathfrak{B}$. The question when a basic $K$-coalgebra $C$ is of the form $T\square _{F}(M, \mathfrak{B})$, up to isomorphism, is also discussed.
