Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska ; Daniel Simson

Colloquium Mathematicae, Tome 111 (2008), p. 89-129 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2008-01-01

Zbl 1173.16008

EUDML-ID : urn:eudml:doc:286114