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

Daniel Simson. "Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations." Colloquium Mathematicae 109.2 (2007): 307-343. <http://eudml.org/doc/283685>.

@article{DanielSimson2007,
abstract = {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.},
author = {Daniel Simson},
journal = {Colloquium Mathematicae},
keywords = {categories of left comodules; path coalgebras; categories of linear representations; equivalences; locally nilpotent locally finite representations; bound quivers; species; profinite algebras; linear topology; quivers with relations},
language = {eng},
number = {2},
pages = {307-343},
title = {Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations},
url = {http://eudml.org/doc/283685},
volume = {109},
year = {2007},
}

TY - JOUR
AU - Daniel Simson
TI - Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations
JO - Colloquium Mathematicae
PY - 2007
VL - 109
IS - 2
SP - 307
EP - 343
AB - 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.
LA - eng
KW - categories of left comodules; path coalgebras; categories of linear representations; equivalences; locally nilpotent locally finite representations; bound quivers; species; profinite algebras; linear topology; quivers with relations
UR - http://eudml.org/doc/283685
ER -