Brauer Graph Algebras

Schroll, Sibylle

doi:10.1007/978-3-319-74585-5_6

Cited by 30 publications

(32 citation statements)

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“…The grading is chosen so that dim Ext 1 (S i , S i+1 ) = 1 for 1 ≤ i < n and dim Ext 1 (S n , S 1 (d)) = 1. For more information on Brauer tree algebras, we refer the reader to Schroll [8].…”

Section: A Combinatorial Model For a -Dgstabmentioning

confidence: 99%

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Perverse Equivalences and Dg-stable Combinatorics

Brightbill

2020

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Chuang and Rouquier [3] describe an action by perverse equivalences on the set of bases of a triangulated category of Calabi-Yau dimension −1. We develop an analogue of their theory for Calabi-Yau categories of dimension w < 0 and show it is equivalent to the mutation theory of w-simple-minded systems.Given a non-positively graded, finite-dimensional symmetric algebra A, we show that the differential graded stable category of A has negative Calabi-Yau dimension. When A is a Brauer tree algebra, we construct a combinatorial model of the dg-stable category and show that perverse equivalences act transitively on the set of |w|-bases.

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Section: A Combinatorial Model For a -Dgstabmentioning

confidence: 99%

“…Then B 1 (−1) has a Type α left collision with B 2 for some α ∈ {I, II, III} if and only if dim Hom A -dgstab (Φ(B 1 ), Φ(B 2 (1))) = 1. In this case, any nonzero morphism fits into a triangle (8) Φ(B 2 ) → Φ(M α (B 1 (−1)))(1) → Φ(B 1 ) → Φ(B 2 (1))…”

Section: Lemma 619 Two Colored Bead Arrangements Have the Same Class ...mentioning

confidence: 99%

Perverse Equivalences and Dg-stable Combinatorics

Brightbill

2020

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“…A Brauer tree consists of the data Γ = (T, e, v, m), where T is a tree, e is the number of edges of T , v is a vertex of T , called the exceptional vertex, and m is a positive integer, called the multiplicity of v. To any Brauer tree Γ, we can associate a basic finite-dimensional symmetric algebra A Γ . For the details of this process, we refer to [15].…”

Section: Brauer Tree Algebrasmentioning

confidence: 99%

“…The data of a Brauer tree determines, up to Morita equivalence, an algebra whose composition factors reflect the combinatorial data of the tree. We refer to Schroll [15] for a detailed introduction to the theory of Brauer tree algebras and their appearance in group theory, geometry, and homological algebra, but we mention here one application which is of particular relevance. Khovanov and Seidel [10] link the category D b dg (A), where A is a graded Brauer tree algebra on the the line with n vertices, to the triangulated subcategory of the Fukaya category generated by a chain of knotted Lagrangian spheres.…”

Section: Introductionmentioning

confidence: 99%

The Differential Graded Stable Category of a Self-Injective Algebra

Brightbill¹

2018

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Let A be a finite-dimensional, self-injective algebra, graded in nonpositive degree. We define A -dgstab, the differential graded stable category of A, to be the quotient of the bounded derived category of dg-modules by the thick subcategory of perfect dg-modules. We express A -dgstab as the triangulated hull of the orbit category A -grstab /Ω(1). This result allows computations in the dg-stable category to be performed by reducing to the graded stable category. We provide a sufficient condition for the orbit category to be equivalent to A -dgstab and show this condition is satisfied by Nakayama algebras and Brauer tree algebras. We also provide a detailed description of the dg-stable category of the Brauer tree algebra corresponding to the star with n edges.

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“…We show that the generalizations satisfy properties that are analogous to the properties of the algebras they generalize. Biserial, special biserial, and Brauer graph algebras have been extensively studied, see, for example, [1,2,5,7,10,11,13,14,19,20,23,25,27,28,32,31,34] and also the survey articles [29,30] and the references therein.…”

Section: Introductionmentioning

confidence: 99%