Coverings of laura algebras: The standard case

Assem, Ibrahim; Bustamante, Juan Carlos; Meur, Patrick Le

doi:10.1016/j.jalgebra.2009.08.013

Cited by 9 publications

(10 citation statements)

References 37 publications

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“…In [51], Skowroński asked for which triangular algebras A we have that A is simply connected if and only if HH 1 (A) = 0. This problem has motivated several results: see [4,18,40] for example. The following theorem has a corollary that constrains which Lie algebras can be obtained as HH 1 (A) of a simply connected algebra A. Theorem 5.9.…”

Section: Simply Connected Algebrasmentioning

confidence: 99%

Maximal tori in $HH^1$ and the fundamental group

Briggs¹,

Degrassi²

2021

Preprint

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We investigate maximal tori in the Hochschild cohomology Lie algebra HH 1 (A) of a finite dimensional algebra A, and their connection with the fundamental groups associated to presentations of A. We prove that every maximal torus in HH 1 (A) arises as the dual of some fundamental group of A, extending work of Farkas, Green and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of A is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.

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Section: Simply Connected Algebrasmentioning

confidence: 99%

Maximal tori in $HH^1$ and the fundamental group

Briggs¹,

Degrassi²

2021

Preprint

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“…-In light of 3.1, (d) we can assume that w lies in a component given by relations of type (2) or else by relations of type (3). Let x i be the point determined by the edgec i c i+1 of w, see 2.2, (c) and (d).…”

Section: 2mentioning

confidence: 99%

Special biserial algebras with no outer derivations

Assem¹,

Bustamante²,

Meur³

2011

Colloq. Math.

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Let $A$ be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of $A$ with coefficients in the bimodule $A$ vanishes if and only if $A$ is representation finite and simply connected (in the sense of Bongartz and Gabriel), if and only if the Euler characteristic of $Q$ equals the number of indecomposable non uniserial projective injective $A$-modules (up to isomorphism). Moreover, if this is the case, then all the higher Hochschild cohomology groups of $A$ vanish.Comment: 13 pages, submitte

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“…It is called a Galois covering with group G if, moreover, the group G acts on in such a way that: (d) G acts freely on vertices; (e) p g = p for every g ∈ G; (f) the translation quiver morphism /G → induced by p is an isomorphism; (g) is connected. Given a connected translation quiver , there exists a group 1 (called the fundamental group of ) and a Galois covering → with group 1 called the universal cover of , which factors through any covering → . If p → is a covering (or a Galois covering with group G), then it naturally induces a covering (or a Galois covering with group G, respectively) → between the associated orbit-graphs.…”

Section: Galois Coverings Of Translation Quivers ([11 28])mentioning

confidence: 99%

“…If p → is a covering (or a Galois covering with group G), then it naturally induces a covering (or a Galois covering with group G, respectively) → between the associated orbit-graphs. It is proved in [11, 4.2] that if has only finitely many -orbits and if p → is the universal cover of translation quiver, then → is the universal cover of graphs, that is, 1 is isomorphic to 1 (and therefore is free).…”

Section: Galois Coverings Of Translation Quivers ([11 28])mentioning

confidence: 99%

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Galois Coverings of Weakly Shod Algebras

Meur

2010

Communications in Algebra

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We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebra not quasi-tilted of canonical type, we establish a correspondence between its Galois coverings and the Galois coverings of its connecting component. As a consequence we show that a weakly shod algebra which is not quasi-tilted of canonical type is simply connected if and only if its first Hochschild cohomology group vanishes.

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Coverings of laura algebras: The standard case

Cited by 9 publications

References 37 publications

Maximal tori in $HH^1$ and the fundamental group

Maximal tori in $HH^1$ and the fundamental group

Special biserial algebras with no outer derivations

Galois Coverings of Weakly Shod Algebras

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