Hereditary coalgebras and representations of species

Abstract

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 Extspecies of C.

Ringel–Hall algebras of hereditary pure semisimple coalgebras

“…For coalgebra representations we use the notation and terminology of [11], [24] and [26]. The reader is referred to [1], [2], [9], [23], [29], and [30] for the terminology and notation of representation theory, and to [12] and [31] for background on coalgebras and comodules.…”

“…If an indecomposable coalgebra C is hereditary and left pure semisimple, then ( C Q, C d) is one of the valued quivers in Tables 1.1 and 1.2 below (see [7], [11,Theorem 4.14], [13]). Moreover, in this case the map…”

“…A representation X of M is said to be of finite length if X has a finite composition series (see [2], [11] and [23,Chapter 14]). We denote by Rep(M) the Grothendieck category of all linear representations of M, and by rep(M) ⊇ rep f (M) the full subcategories of Rep(M) formed by finitely generated objects and by finitely generated representations of finite length, respectively.…”

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

“…We denote by Rep K (M) the category of all Klinear representations of M (see [18] for a precise definition). Following [18] and the definition (3.2), we define the K-linear functor…”

“…If the index set I M is finite and each bimodule i M j is of finite K-dimension, M is called finite. Throughout this section, we freely use the terminology and notation introduced in [18]. In particular, we denote by (Q M , d M ) the valued quiver of the species M, where Q M 0 = I M is the set of vertices of the quiver Q M .…”