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

Daniel Simson

Colloquium Mathematicae, Tome 107 (2007), p. 307-343 / Harvested from The Polish Digital Mathematics Library

Résumé

We prove that the study of the category C-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 = (Q₀,Q₁), we show that any subcoalgebra C of the path K-coalgebra K◻Q containing $K^{◻}Q₀ ⊕ K^{◻}Q₁$ is the path coalgebra $K^{◻}(Q,)$ of a profinite bound quiver (Q,), and the category C-Comod of left C-comodules is equivalent to the category $Rep_{K}^{ℓnℓf}(Q,)$ of locally nilpotent and locally finite K-linear representations of Q bound by the profinite relation ideal $ ⊂ \widehat{KQ}$. Given a K-species $ℳ = (F_{j},_{i}M_{j})$ and a relation ideal of the complete tensor K-algebra $T̂(ℳ ) = \widehat{T_F(M)}$ of ℳ, the bound species subcoalgebra $T^{◻}(ℳ,)$ of the cotensor K-coalgebra $T^{◻}(ℳ ) = T^{◻}_{F}(M)$ of ℳ is defined. We show that any subcoalgebra C of $T^{◻}(ℳ )$ containing $T^{◻}(ℳ )₀ ⊕ T^{◻}(ℳ )$₁ is of the form $T^{◻}(ℳ, )$, and the category C-Comod is equivalent to the category $Rep_{K}^{ℓnℓf}(ℳ, )$ of locally nilpotent and locally finite K-linear representations of ℳ bound by the profinite relation ideal . The question when a basic K-coalgebra C is of the form $T^{◻}_{F}(M, )$, up to isomorphism, is also discussed.

Détail

Publié le : 2007-01-01

Zbl 1142.16005

EUDML-ID : urn:eudml:doc:283685