Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

• Autorzy: Justyna Kosakowska , Daniel Simson
• Języki publikacji: EN

Abstrakty

EN

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 $[S(i') \atop 0]$ (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.

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: 2008
• Tom: 112
• Numer: 1
• Strony: 89-129

Daty

wydano

2008

Twórcy

autor

Justyna Kosakowska

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

autor

Daniel Simson

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

Identyfikatory

DOI

10.4064/cm112-1-5