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

• Autorzy: Daniel Simson
• Języki publikacji: EN

Abstrakty

EN

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

Kategorie tematyczne

• 16G20: Representations of quivers and partially ordered sets
• 16W80: Topological and ordered rings and modules
• 16G60: Representation type (finite, tame, wild, etc.)
• 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2007
• Tom: 109
• Numer: 2
• Strony: 307-343

Daty

wydano

2007

Twórcy

autor

Daniel Simson

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

Identyfikatory

DOI

10.4064/cm109-2-12