ABSTRACT
Throughout this paper we fix a field K. The notion of a path .RT-coalgebra C(Q, fJ) of a quiver with relations (Q,fi) introduced in [37, p. 135] and [38] is investigated in the paper. It is shown in Theorem 3.14 that the coalgebra C(Q,£l) is basic, the left Gabriel quiver of C(Q, fi) is Q and there exists a K-lmeax isomorphism comod-Cl(Q,fi) = nilrep/(<5,n) between the category of finite dimensional right C(Q, fi)-comodules and the category nilrep^(Q, fi) of nilpotent .ft'-linear representations of finite length of the quiver Q satisfying the relations in fi. Given any quiver Q, a correspondence between subcoalgebras H of the path coalgebra KQ and the relation ideals fi of the path .ftT-algebra KQ of Q is studied. As a consequence, we show in Theorem 4.9, that if K is algebraically closed and C is a basic /C-coalgebra such that its Gabriel's quiver Q is locally finite and has no oriented cycles, then there is a coalgebra isomorphism C S C(Q,f7), where fi is a relation ideal of the path /C-algebra KQ.
