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2007 | 109 | 2 | 307-343
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Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

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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.
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  • Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-12
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