Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska ¹   ; Daniel Simson ¹

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 112 (2008) no. 1, pp. 89-129

Résumé

Let $C$ be a coalgebra over an arbitrary field $K$. We show that the study of the category $C\hbox{-}{\rm Comod}$ 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, $C=C_1$, the second term of the coradical filtration of $C$. If $C=C_1$, we associate with $C$ a $K$-linear functor $\mathbb{H} _C : C\hbox{-}{\rm Comod}\rightarrow H_C\hbox{-}{\rm Comod}$ that restricts to a representation equivalence $\mathbb{H} _C : C\hbox{-}{\rm comod} \rightarrow H_C \hbox{-}{\rm comod}^\bullet_{\rm sp},$ where $H_C$ is a coradical square complete hereditary bipartite $K$-coalgebra such that every simple $H_C$-comodule is injective or projective. Here $H_C\hbox{-}{\rm comod}^\bullet_{\rm sp}$ is the full subcategory of $H_C\hbox{-}{\rm comod}$ whose objects are finite-dimensional $H_C$-comodules with projective socle having no injective summands of the form $\left[{\scriptstyle S(i')\atop \scriptstyle 0}\right]$ (see Theorem 5.11). Hence, we conclude that a coalgebra $C$ with $C=C_1$ 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.

Détail

DOI : 10.4064/cm112-1-5

Keywords: nbsp coalgebra nbsp arbitrary field study category hbox comod c comodules reduces study category representations nbsp certain bicomodule bipartite coalgebra coradical square complete coalgebra second term coradical filtration nbsp associate k linear functor mathbb hbox comod rightarrow hbox comod restricts representation equivalence mathbb hbox comod rightarrow hbox comod bullet where nbsp coradical square complete hereditary bipartite k coalgebra every simple c comodule injective projective here hbox comod bullet full subcategory hbox comod whose objects finite dimensional c comodules projective socle having injective summands form scriptstyle atop scriptstyle right see theorem nbsp hence conclude coalgebra pure semisimple only pure semisimple section nbsp get diagrammatic characterisation coradical square complete coalgebras pure semisimple tameness wildness coalgebras discussed