The automorphism group and the Picard group of a monomial algebra<sup>*</sup>

Guil-Asensio, Francisco; Saorı́n, Manuel

doi:10.1080/00927879908826466

Cited by 25 publications

(56 citation statements)

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“…Guil-Asensio and Saorín worked on the outer automorphisms of any finite dimensional algebra (see [8]). The case of the finite dimensional monomial algebras was treated by them in their paper [9]. We are going to follow their notations.…”

Section: Theorem 27mentioning

confidence: 99%

“…This paper is organized in the following way: in the first section we will use the minimal projective resolution of a monomial algebra Λ (as a Λ-bimodule) given by Bardzell in [1] to get a handy description of the Lie algebra H 1 (Λ, Λ) in terms of parallel paths. The purpose of the second section is to link this description to Guil-Asensio and Saorín's work on the algebraic group of outer automorphisms of monomial algebras (see [9]). In section three we carry out the study of the Lie algebra H 1 (Λ, Λ).…”

Section: Introductionmentioning

confidence: 99%

“…Note that in the finite dimensional case in characteristic 0 this Lie algebra can be regarded as the Lie algebra of the algebraic group of outer automorphisms Out (Λ) = Aut (Λ)/Inn (Λ) of the algebra Λ. It has been treated by Guil-Asensio and Saorín in [8]. Huisgen-Zimmermann and Saorín proved in [11] that the identity component Out (Λ)…”

Section: Introductionmentioning

confidence: 99%

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The Lie Algebra Structure on the First Hochschild Cohomology Group of a Monomial Algebra

Strametz

2006

J. Algebra Appl.

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RésuméNousétudions la structure d'algèbre de Lie du premier groupe de la cohomologie de Hochschild d'une algèbre monomiale de dimension finie Λ, en termes combinatoires de son carquois, en quelconque caractéristique. Cela nous permet aussi d'examiner la composante de l'identité du groupe algébrique des automorphismes extérieurs de Λ en caractéristique zéro. Nous donnons des critères pour la (semi-)simplicitét la résolubilité. AbstractWe study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra Λ, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of Λ in characteristic zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given.2000 Mathematics Subject Classification: 16E40 16W20

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Section: Theorem 27mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Lie Algebra Structure on the First Hochschild Cohomology Group of a Monomial Algebra

Strametz

2006

J. Algebra Appl.

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“…So it is given by a finite quiver with relations whose set of vertices is (in bijection with) the set I/G = {[i] : i ∈ I} of Gorbits of elements of I. From [23][Proposition 10 and Theorem 12] we get a mapλ :…”

Section: Some Important Auxiliary Resultsmentioning

confidence: 99%

“…Assertion 4 follows directly from [23][Theorem 12], taking into account that the only inner graded automorphism induced by an element 1 − x, with x ∈ J, is the identity (see the proof of Lemma 4.3).…”

Section: Inner and Stably Inner Automorphismsmentioning

confidence: 99%

The symmetry, period and Calabi–Yau dimension of finite dimensional mesh algebras

Juan

Saorı́n

2015

Journal of Algebra

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On automorphism groups induced by bimodules

Guil-Asenio

Saorı́n

2001

Archiv der Mathematik

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Let R, S be rings and R M S a faithfully balanced bimodule. It is proved that the groups Out M (R) and Out M (S) of outer automorphisms of R and S which fix (up to isomorphism) the underlying modular structure of M are isomorphic. In case A, B are finite dimensional algebras and A M B is finite dimensional, it is proved that the isomorphism Out M (A) Out M (B) is algebraic, by showing in the way that every Outorbit of a finite dimensional module has a canonical structure of algebraic variety. The result is then applied to prove that the identity components of Out(A) and Out(B) are isomorphic when the Out-orbits of A M and M B are finite. That generalizes a result of Brauer and allows us to stablish the same conclusion for algebras which are iterated tilted from one another. The paper finishes with an identification of the iterated tilted algebras which have finite Picard group.For any given mathematical object, its automorphism group is always an important object of study, even to understand the object itself. In the case of a finite-dimensional algebra A over an algebraically field K, as shown in [10], the appropriate context to study its group of automorphisms Aut(A), or the group Out(A) = Aut(A) Inn(A) of outer automorphisms of A, is that of algebraic groups. That scheme has been pursued by the authors in [6], [7] leading to a very accurate knowledge of Aut(A) and Out(A) when A is, for instance, monomial or, more generally, gradable by the radical. For the study of the category A Mod of all A-modules or the category A mod of all finitedimensional modules, the group Out(A) is very interesting since it can be embedded in the Picard group Pic(A), which can be thought of as the group of (natural isoclasses of) Morita self-equivalences A Mod 3 A Mod. Then, a natural question arises: what is the relation between Out(A) and Out(B) when A and B are well-related? A result attributed to Brauer ([10]) states that the identity components of Out(A) and Out(B) are isomorphic when A and B are Morita equivalent.The goal of this work is to study the relationship between Out(A) and Out(B) when there exists a finite dimensional faithfully balanced bimodule A M B and to use that in order to extend the above mentioned Brauers result. In order to get that extension, we use our two main results, Theorems 1.4 and 2.5, which state that the stabilizers of A M and M B under the

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The automorphism group and the Picard group of a monomial algebra^*

Cited by 25 publications

References 6 publications

The Lie Algebra Structure on the First Hochschild Cohomology Group of a Monomial Algebra

The Lie Algebra Structure on the First Hochschild Cohomology Group of a Monomial Algebra

The symmetry, period and Calabi–Yau dimension of finite dimensional mesh algebras

On automorphism groups induced by bimodules

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The automorphism group and the Picard group of a monomial algebra*

Cited by 25 publications

References 6 publications

The Lie Algebra Structure on the First Hochschild Cohomology Group of a Monomial Algebra

The Lie Algebra Structure on the First Hochschild Cohomology Group of a Monomial Algebra

The symmetry, period and Calabi–Yau dimension of finite dimensional mesh algebras

On automorphism groups induced by bimodules

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The automorphism group and the Picard group of a monomial algebra^*