Bipartite coalgebras and a reduction functor for coradical square complete coalgebras
Volume 112 / 2008
Abstract
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.
