Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

Daniel Simson¹

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 109 (2007) no. 2, pp. 307-343

Résumé

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.

Détail

DOI : 10.4064/cm109-2-12

Keywords: prove study category mbox comod comodules k coalgebra reduces study k linear representations quiver relations algebraically closed field study k linear representations k species relations perfect field given field quiver subcoalgebra path k coalgebra square containing square oplus square path coalgebra square mathfrak profinite bound quiver mathfrak category mbox comod c comodules equivalent category rep ell ell mathfrak locally nilpotent locally finite k linear representations bound profinite relation ideal mathfrak subset widehat given k species mathcal relation ideal mathfrak complete tensor k algebra widehat mathcal widehat mathcal bound species subcoalgebra square mathcal mathfrak cotensor k coalgebra square mathcal square mathcal defined subcoalgebra square mathcal containing square mathcal oplus square mathcal form square mathcal mathfrak category mbox comod equivalent category rep ell ell mathcal mathfrak locally nilpotent locally finite k linear representations mathcal bound profinite relation ideal mathfrak question basic k coalgebra form square mathfrak isomorphism discussed