The universal cover of a quiver with relations

Martínez-Villa, Roberto; Peña, José Antonio de la

doi:10.1016/0022-4049(83)90062-2

Cited by 127 publications

(116 citation statements)

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“…It is well known that if G is torsion-free then the push-down functor F λ preserves indecomposability of modules and Auslander-Reiten sequences (see [15]). A locally bounded category R is called simply connected [1] if it is triangular (its quiver has no oriented cycles) and for any presentation R ∼ → KQ/I of R as a bound quiver category, the fundamental group Π 1 (Q, I) of (Q, I), defined in [17], [22], is trivial. It has been proved in [31] that a triangular locally bounded category R is simply connected if and only if each Galois covering of R is trivial.…”

Section: Self Injective Algebras With Simply Connected Galois Coveringsmentioning

confidence: 99%

Selfinjective algebras of tubular type

Białkowski

Skowroński

2002

Colloq. Math.

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Section: Self Injective Algebras With Simply Connected Galois Coveringsmentioning

confidence: 99%

Selfinjective algebras of tubular type

Białkowski

Skowroński

2002

Colloq. Math.

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“…Again if Q is connected then this group does not depend on the choice of x and we speak about the fundamental group of (Q, I) with respect to Ω and denote it by Π 1 ((Q, I), Ω) (cf. [7], [10], [1]). …”

Section: 3mentioning

confidence: 99%

Simply connected right multipeak algebras and the separation property

Kasjan¹

1999

Colloq. Math.

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Abstract. Let R = k(Q, I) be a finite-dimensional algebra over a field k determined by a bound quiver (Q, I). We show that if R is a simply connected right multipeak algebra which is chord-free and A-free in the sense defined below then R has the separation property and there exists a preprojective component of the Auslander-Reiten quiver of the category prin(R) of prinjective R-modules. As a consequence we get in 4.6 a criterion for finite representation type of prin(R) in terms of the prinjective Tits quadratic form of R.1. Introduction. Let k be a field. We consider triangular simply connected right multipeak algebras R = kQ/I, where Q is a finite quiver and I is an admissible ideal in the path algebra kQ. Triangularity means that the ordinary quiver Q of R has no oriented cycles. Following [13] we say that R is a right multipeak algebra if the right socle soc(R R ) of R is R-projective. The main objective of the paper is a criterion for R to have the separation property [2]. We prove in Section 4 that R has the separation property when R is chord-free (see 2.5) and A-free as a right multipeak algebra. Our main result, Theorem 4.5, is analogous to [21, Theorem 4.1] and [1, 1.2]; cf. [6].Recall from [1, 1.2] that if R is schurian, triangular, simply connected and does not contain any full subcategory (see 2.2) isomorphic to k A m , m ≥ 1, then R has the separation property. Our result is a version of this statement: algebras considered are right multipeak algebras and the requirement that R is A-free as a right multipeak algebra is a weaker version of A-freeness considered in [1], [3]. The condition that R is chord-free plays a similar role as the assumption that R is schurian. Note that the arguments used in [1] do not work in our situation: our assumptions on R do not imply that R is schurian. Moreover, R (viewed as a k-category) admits full subcategories isomorphic to k A m for some m ≥ 1, although R is A-free as a right multipeak algebra (see the Example in 2.5). Hence the arguments used in [3, 2.3] and [5, 2.9] do not apply here.

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“…Indeed, consider an orientation (s ,t ) of G such that π (s (e ) → t (e )) = s(π (e )) → t(π (e )) and fix any x 0 ∈ G 0 with π ( Remark 2.4. The construction of Galois coverings and the fundamental group of a hexagonal system follows similar arguments to those developed for finite dimensional algebras given as quotients of quiver algebras, see [1,5,9,11]. In particular, the proof of Theorem 1.1 is inspired in [11] where the concept of coverings in representation theory was introduced; the construction of the universal Galois covering follows [9].…”

Section: 6mentioning

confidence: 99%