Representation-tame incidence algebras of finite posets

Abstract

Abstract. Continuing the paper [Le], we give criteria for the incidence algebra of an arbitrary finite partially ordered set to be of tame representation type. This completes our result in [Le], concerning completely separating incidence algebras of posets.1. Introduction. Throughout, we assume that K is an algebraically closed field. We continue the study of representation-tame incidence Kalgebras of finite posets, that is, partially ordered sets, started in [Le]. We use the terminology and notation introduced in [Le]. In particular, given a finite-dimensional K-algebra A, we denote by mod A the category of all finite-dimensional right A-modules. The algebra A is said to be of tame representation type (or representation-tame) if, for each dimension d, the isomorphism classes of indecomposable modules in mod A of dimension d form at most finitely many one-parameter families. The reader is referred to [Dr] and [S; Sections 14.2-4] for precise definitions of representation-tame and representation-wild algebras.Throughout, we denote by Q = (Q 0 , Q 1 ) a finite connected quiver with Q 0 being the set of vertices and Q 1 the set of arrows. We assume that Q has no oriented cycles and no arrows having the same starting and ending vertex with another path. In particular Q has no multiple arrows. We view the quiver Q as a poset with respect to the partial order relation on Q 0 defined by the formula:

Incidence coalgebras of interval finite posets of tame comodule type

“…(b) Assume that J is G 3 , (N Z) * , or one of the three pg-critical posets in [20, (2.4a)-(2.4c)] (see also [21,22,28,54]). Then gl.dim KJ = 2 and the reduced Euler form q • J : Z J → Z coincides with its Euler form q J (see [47]).…”

“…In view of the equivalence (a1)⇔(a4), the implication (b1)⇒(b3) is a consequence of [35] and the description of finite A * m -free connected representation-tame posets in [20][21][22] (see also [53]).…”

On the Wildness of Cambrian Lattices

“…A general description of the representation-finite case has been made in [Lou75b,Lou75a]. The tame case has been considered in [Les03,Les02].…”

The representation theory of the incidence algebra of an inverse semigroup

“…The incidence algebra of a finite poset is in fact a basic finite dimensional algebra and so it is also very natural to study them from the point of view of representation theory. A very incomplete list of papers on this subject is [2,9,[15][16][17][18]24,28,29,47,48].…”