On path coalgebras of quivers with relations

Abstract

Abstract. The notion of the path coalgebra of a quiver with relations introduced in [11] and [12] is studied. In particular, developing this topic in the context of the weak * topology, we give a criterion that allows us to verify whether or not a relation subcoalgebra of a path coalgebra is the path coalgebra of a quiver with relations.

Localization in tame and wild coalgebras

“…It is proved in [9,Example 4.11] that H is not the path coalgebra of a quiver with relations. Nevertheless, H is of wild comodule type because it contains the path coalgebra of the quiver…”

Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

“…In this case H = a,b∈Q 0 H(a, b), where H(a, b) = H ∩KQ(a, b) (see [13] and [14]). A quiver Q is said to be intervally finite if, for each pair a, b of vertices of Q, the set Q(a, b) of all paths from a to b in Q is finite.…”

“…(a) Statement (d) of Theorem 4.5 is proved in [31, Theorem 3.14(c), (d)] under the assumption that Q is locally finite. Unfortunately, this assumption is not sufficient (see [13]). Under the assumption that Q is intervally finite, made in Theorem 4.5, the proof given in [31, pp.…”

“…(b) A proof of (d) can also be found in [13], where a criterion is given to decide whether or not a relation subcoalgebra of K Q is of the form K (Q, Ω). It is shown in [14] that if Q is acyclic, then any tame relation subcoalgebra of K Q is of the form K (Q, Ω).…”

Serial coalgebras and their valued Gabriel quivers

“…We recall that, given an idempotent e ∈ C * , for each right C-comodule M, the vector space eM is endowed with a structure of right eCe-comodule given by (1) ⊗ x (0) using the sigma-notation of [31]. We refer the reader to [13,14] and [15] for basic definitions, notations and properties about quivers and path coalgebras. The localization in categories of comodules over path coalgebras is described in detail in [15].…”