Quiver algebras, path coalgebras and coreflexivity. (Q1955981)

Let \(\Gamma\) be a quiver. Two algebraic objects can be attached to \(\Gamma\). First, the quiver algebra \(K[\Gamma]\), whose modules are the representations of the quiver. Secondly, the path coalgebra \(K\Gamma\), whose comodules are locally finite representations of the quiver. The links between these two notions are studied here. The notion of finite monomial dual of a quiver algebra \(K[\Gamma]\) is introduced. Namely, \(K[\Gamma]^o\) is the set of linear forms on \(K[\Gamma]\) which vanish on a cofinite monomial two-sided ideal of \(K[\Gamma]\). Generally, \(K\Gamma\) embeds in \(K[\Gamma]^o\). It is proved that this embedding is an isomorphism if, and only if, \(\Gamma\) has no oriented cycle and there is a finite number of arrows between any two vertices. Generally, \(K[\Gamma]\) embeds in the dual algebra \(K\Gamma^*\). It is proved that the image of this embedding is the rational part of \(K\Gamma^*\) if, and only if, for any vertex \(v\in V\) there is a finite number of paths starting or ending at \(v\). The coreflexivity of \(K\Gamma\) is studied, that is to say the question to know if \(K\Gamma\) is isomorphic to \((K\Gamma^*)^o\). -- Similar results are obtained for incidence coalgebras.