Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska, and Daniel Simson. "Bipartite coalgebras and a reduction functor for coradical square complete coalgebras." Colloquium Mathematicae 112.1 (2008): 89-129. <http://eudml.org/doc/286114>.

@article{JustynaKosakowska2008,
abstract = {Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor $ℍ_\{C\}: C-Comod → H_\{C\}-Comod$ that restricts to a representation equivalence $ℍ_\{C\}: C-comod → H_\{C\}-comod^\{•\}_\{sp\}$, where $H_\{C\}$ is a coradical square complete hereditary bipartite K-coalgebra such that every simple $H_\{C\}$-comodule is injective or projective. Here $H_\{C\}-comod^\{∙\}_\{sp\}$ is the full subcategory of $H_\{C\}-comod$ whose objects are finite-dimensional $H_\{C\}$-comodules with projective socle having no injective summands of the form $[S(i^\{\prime \}) \atop 0]$ (see Theorem 5.11). Hence, we conclude that a coalgebra C with C = C₁ is left pure semisimple if and only if $H_\{C\}$ is left pure semisimple. In Section 6 we get a diagrammatic characterisation of coradical square complete coalgebras C that are left pure semisimple. Tameness and wildness of such coalgebras C is also discussed.},
author = {Justyna Kosakowska, Daniel Simson},
journal = {Colloquium Mathematicae},
keywords = {separated valued Gabriel quivers; wild coalgebras; irreducible morphisms; trivial extensions; stable categories; tame coalgebras; coradical filtrations},
language = {eng},
number = {1},
pages = {89-129},
title = {Bipartite coalgebras and a reduction functor for coradical square complete coalgebras},
url = {http://eudml.org/doc/286114},
volume = {112},
year = {2008},
}

TY - JOUR
AU - Justyna Kosakowska
AU - Daniel Simson
TI - Bipartite coalgebras and a reduction functor for coradical square complete coalgebras
JO - Colloquium Mathematicae
PY - 2008
VL - 112
IS - 1
SP - 89
EP - 129
AB - Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor $ℍ_{C}: C-Comod → H_{C}-Comod$ that restricts to a representation equivalence $ℍ_{C}: C-comod → H_{C}-comod^{•}_{sp}$, where $H_{C}$ is a coradical square complete hereditary bipartite K-coalgebra such that every simple $H_{C}$-comodule is injective or projective. Here $H_{C}-comod^{∙}_{sp}$ is the full subcategory of $H_{C}-comod$ whose objects are finite-dimensional $H_{C}$-comodules with projective socle having no injective summands of the form $[S(i^{\prime }) \atop 0]$ (see Theorem 5.11). Hence, we conclude that a coalgebra C with C = C₁ is left pure semisimple if and only if $H_{C}$ is left pure semisimple. In Section 6 we get a diagrammatic characterisation of coradical square complete coalgebras C that are left pure semisimple. Tameness and wildness of such coalgebras C is also discussed.
LA - eng
KW - separated valued Gabriel quivers; wild coalgebras; irreducible morphisms; trivial extensions; stable categories; tame coalgebras; coradical filtrations
UR - http://eudml.org/doc/286114
ER -