On Simply Connected Algebras

Bautista, R.; Larrión, F.; Salmerón, L.

doi:10.1112/jlms/s2-27.2.212

Cited by 51 publications

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“…Let Γ be a translation quiver with translation τ , that is, Γ is a quiver containing neither loops nor multiple arrows and τ is a bijection from a subset of Γ 0 to another one such that, for each a ∈ Γ 0 with τ (a) defined, there exists at least one arrow α : b → a and any such arrow determines a unique arrow σ(α) .2)]. Now we say that Γ is simply connected if Γ contains no oriented cycle and O(Γ ) is a tree; see [7,8]. By [9, (1.6), (4.1), (4.2)], this definition is equivalent to that in [9, (1.6)].…”

Section: Lemma Let R Be a Commutative Ring And Let Q Be A Finite Qumentioning

confidence: 99%

Hochschild Cohomology and Representation-finite Algebras

Buchweitz

Liu

2004

Proc. London Math. Soc.

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Section: Lemma Let R Be a Commutative Ring And Let Q Be A Finite Qumentioning

confidence: 99%

Hochschild Cohomology and Representation-finite Algebras

Buchweitz

Liu

2004

Proc. London Math. Soc.

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“…From [5], we recall that a spectroid S satisfies the separation condition if for each point s of S, two different direct summands of rad P s lie in different connected components of the spectroid S \ s , where s is the full subspectroid of S given by the start points of all paths ending at s.…”

Section: The Source-extension S (Denoted Bymentioning

confidence: 99%

Derived tubular strongly simply connected algebras

Barot¹,

Peña²

1999

Proc. Amer. Math. Soc.

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Abstract. Let A be a finite dimensional algebra over an algebraically closed field k. Assume A = kQ/I for a connected quiver Q and an admissible ideal I of kQ. We study algebras A which are derived equivalent to tubular algebras. If A is strongly simply connected and Q has more than six vertices, then A is derived tubular if and only if (i) the homological quadratic form χ A is a non-negative of corank two and (ii) no vector of χ Finite dimensional algebras over an algebraically closed field k are divided into two classes. On one hand, we have the tame algebras for which almost all indecomposable modules of each fixed dimension occur in a finite number of one-parametric families; on the other hand, the wild algebras for which the classification of indecomposable modules contains the classification of pairs of matrices under simultaneous conjugations. Among tame algebras, tubular algebras form a well-understood and important class [15]. The aim of this work is to study algebras which are derived equivalent to tubular algebras. We shall apply our results to obtain a new characterization of polynomial growth algebras.Two k-algebras A 1 and A 2 are said to be derived equivalent if their derived categories D b (A 1 -mod) and D b (A 2 -mod) are equivalent as triangulated categories; see [10]. If A 1 and A 2 are derived equivalent and A 1 is tubular, we say that A 2 is a derived tubular algebra. In [4], a characterization of derived tubular algebras of finite representation type was given; the characterization uses properties of the homological form and the Tits form of the algebra. In this work we extend those results to certain classes of tame algebras.If the algebra A is of finite global dimension, then the homological bilinear form of A is defined by dim, where dim M denotes the class of an A-module M in the Grothendieck group K • (A) of A. The corresponding quadratic form is denoted by χ A , whereas the Tits form q A is the "truncated version", defined for a semisimple module S as qThe radical rad q A is the subgroup of K • (A) of all vectors v such that q A (v+?) = q A (v) + q A (?). The corank of q A is the rank of rad q A .Let Q be a connected quiver and I an admissible ideal of kQ such that A = kQ/I is a finite dimensional k-algebra. We recall from [17] that A is said to be strongly

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“…We first recall that, by [15, (4.1)], a triangular algebra A is strongly simply connected if each full convex subcategory C of A satisfies the separation condition of [5]. Let thus A be a simply connected representation-finite algebra.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…For this purpose, we need the orbit quiver 6 A of A (see [5] or [14, p. 49]. Recall that, if A is representation-directed, then 6 A is a quiver defined as follows; a point in 0 A is the T-orbit 0(X) of an indecomposable A-module X, and there is an arrow 0(X)-»6(Y) if and only if there is an arrow t"X-*P in F(mod A), where m >0, and P is the unique indecomposable projective A-module in the T-orbit of Y.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7], [8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras.…”

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Strongly simply connected Auslander algebras

Assem

Brown

1997

Glasgow Math. J.

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Let k be an algebraically closed field. By an algebra is meant an associative finite dimensional ^-algebra A with an identity. We are interested in studying the representation theory of A, that is, in describing the category mod A of finitely generated right A-modules. Thus we may, without loss of generality, assume that A is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7], [8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.Let A be a representation-finite algebra, and {M u ...,M d } a complete set of representatives of the isomorphism classes of indecomposable A-modules. The algebra

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On Simply Connected Algebras

Cited by 51 publications

References 0 publications

Hochschild Cohomology and Representation-finite Algebras

Hochschild Cohomology and Representation-finite Algebras

Derived tubular strongly simply connected algebras

Strongly simply connected Auslander algebras

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