Remarks on Hall algebras of triangulated categories

Xiao, Jie; Xu, Fan

doi:10.1215/21562261-2871803

Cited by 9 publications

(5 citation statements)

References 24 publications

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“…Similar to (2.3), the odd periodic derived Hall number F L X,Y also has the derived Riedtmann-Peng formula (cf. [21,16])…”

Section: And With the Multiplication Defined On Basis Elements Bymentioning

confidence: 99%

Periodic derived Hall algebras of hereditary abelian categories

Zhang¹

2023

Preprint

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Let m be a positive integer and D m (A) be the m-periodic derived category of a finitary hereditary abelian category A . Applying the derived Hall numbers of the bounded derived category D b (A), we define an m-periodic extended derived Hall algebra for D m (A), and use it to give a global, unified and explicit characterization for the algebra structure of Bridgeland's Hall algebra of periodic complexes. Moreover, we also provide an explicit characterization for the odd periodic derived Hall algebra of A defined by Xu-Chen [22].

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“…Similar to (2.3), the odd periodic derived Hall number F L X,Y also has the derived Riedtmann-Peng formula (cf. [21,16])…”

Section: And With the Multiplication Defined On Basis Elements Bymentioning

confidence: 99%

Periodic derived Hall algebras of hereditary abelian categories

Zhang¹

2023

Preprint

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“…Our choice of the structure constants actually defines the dual algebra of the Hall coalgebra. It is known that this dual algebra and the usual Hall algebra are naturally isomorphic, see [70] for a detailed discussion.…”

Section: Hall Algebrasmentioning

confidence: 99%

Exact structures and degeneration of Hall algebras

Fang¹,

Gorsky²

2020

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We study degenerations of the Hall algebras of exact categories induced by degree functions on the set of isomorphism classes of indecomposable objects. We prove that each such degeneration of the Hall algebra H(E) of an exact category E is the Hall algebra of a smaller exact structure E ′ < E on the same additive category A. When E is admissible in the sense of Enomoto, for any E ′ < E satisfying suitable finiteness conditions, we prove that H(E ′ ) is a degeneration of H(E) of this kind.In the additively finite case, all such degree functions form a simplicial cone whose face lattice reflects properties of the lattice of exact structures. For the categories of representations of Dynkin quivers, we recover degenerations of the negative part of the corresponding quantum group, as well as the associated polyhedral structure studied by Fourier, Reineke and the first author.Along the way, we give minor improvements to certain results of Enomoto and Brüstle-Langford-Hassoun-Roy concerning the classification of exact structures on an additive category. We prove that for each idempotent complete additive category A, there exists an abelian category whose lattice of Serre subcategories is isomorphic to the lattice of exact structures on A. We show that every Krull-Schmidt category admits a unique maximal admissible exact structure and that the lattice of smaller exact structures of an admissible exact structure is Boolean.

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“…where Ext 1 T (X, Y ) L is defined to be Hom T (X, Y [1]) L [1] which denotes the subset of Hom(X, Y [1]) consisting of morphisms l : X → Y [1] whose cone Cone(l) is isomorphic to L [1]. Here the definition we used is a version in [25] so-called the Drinfeld dual of the derived Hall algebra given by Toën in [22] and also by Xiao-Xu in [24].…”

Section: A New Proof Of Green's Formulamentioning

confidence: 99%

“…[X] for all [X] ∈ Iso(T ), and one can see [24,28] for more details. Obviously we can define the derived Hall algebra of D Z/t (A) in case t is an odd integer, and we denote it by DH Z/t (A).…”

Section: Hall Algebras Of Odd-periodic Relative Derived Categoriesmentioning

confidence: 99%

“…Green [5] found a famous formula on Ringel-Hall numbers, usually called Green's formula, to show the bialgebra structure of the Ringel-Hall algebra and then generalized the Ringel's result to any type case. These works have led to extensive research into Ringel-Hall algebras, and a great deal of progresses has been made, see for example [18,15,6,23,21,7,13,14,9,19,20,2,22,24,8,25,1,3,4].…”

Section: Introductionmentioning

confidence: 99%

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Modified Ringel–Hall algebras, Green's formula and derived Hall algebras

Lin

Peng

2019

Journal of Algebra

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In this paper we define the modified Ringel-Hall algebra MH(A) of a hereditary abelian category A from the category C b (A) of bounded Z-graded complexes. Two main results have been obtained. One is to give a new proof of Green's formula on Ringel-Hall numbers by using the associative multiplication of the modified Ringel-Hall algebra. The other is to show that in certain twisted cases the derived Hall algebra can be embedded in the modified Ringel-Hall algebra. As a consequence of the second result, we get that in certain twisted cases the modified Ringel-Hall algebra is isomorphic to the tensor algebra of the derived Hall algebra and the torus of acyclic complexes and so the modified Ringel-Hall algebra is invariant under derived equivalences.

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Remarks on Hall algebras of triangulated categories

Cited by 9 publications

References 24 publications

Periodic derived Hall algebras of hereditary abelian categories

Periodic derived Hall algebras of hereditary abelian categories

Exact structures and degeneration of Hall algebras

Modified Ringel–Hall algebras, Green's formula and derived Hall algebras

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