Hereditary coalgebras and representations of species

Let C be a basic indecomposable hereditary K-coalgebra, where K is an arbitrary field. We investigate a technique for studying C and left C-comodules by means of the left valued Gabriel quiver of C, an associated Tits quadratic form and locally nilpotent representations of the Ext-species of C. One of the main aims of the paper is to prove the following result. Let {S(j)}j IC be a complete set of pairwise non-isomorphic simple left C-comodules and, given i,j IC, we set . Then every left C-comodule is a direct sum of finite dimensional C-comodules if and only if the following four conditions are satisfied: (a) for any j IC, the sum is finite, (b) the set is finite, (c) there is no infinite sequence j1,…,jm,… of elements of IC such that , (d) the integral Tits quadratic form of C defined by the formula is positive definite, where , and is the direct sum of IC copies of . In this case, we show that C is isomorphic to the (co)tensor coalgebra associated to the Ext-species of C, where F is the direct sum of the division algebras Fj=EndCS(j), , and . We also describe the Auslander–Reiten quiver Γ(C-comod) and the structure of the category C-comod of left C-comodules of finite dimension.