Hall algebras associated to triangulated categories

By Jie Xiao, Fan Xu. 2006

101 citations recorded by the source corpus.

By counting with triangles and the octohedral axiom, we find a direct way to prove the formula of To\"en in \cite{Toen2005} for a triangulated category with (left) homological-finite condition.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/792073ad8c95b9be36c37f8456d1e42a6fb58da6

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In Algebra and Number Theory

1. B. Keller
2. M. Kontsevich
3. Dong Ping Yang
4. Y. Soibelman
5. I. Burban
6. O. Schiffmann
7. M. Gorsky
8. Guodong Zhou

In Geometry and Topology

1. B. Keller
2. Dong Ping Yang
3. M. Kontsevich
4. Y. Soibelman
5. O. Schiffmann
6. M. Gorsky
7. T. Bridgeland
8. I. Burban

In Mathematical Physics

1. B. Keller
2. Dong Ping Yang
3. M. Kontsevich
4. Y. Soibelman
5. O. Schiffmann
6. M. Gorsky
7. I. Burban
8. Guodong Zhou

Citations in the corpus

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, background): “a from [70] is described in the following proposition. Proposition 12 There is a homomorphism of rings H(C) →HTo(C), where HTo(C) is the derived Hall algebra over Fq defined by To¨en in [70] (see also [76]), such that [π: Y →Ob(C)] → X y∈Y(Fq) [π(y)]#Aut(y)(Fq)q(y,y)<0 . 89 Moreover, for any stric”
• On the Hall algebra of an elliptic curve, I (2005, background): “ view DHX as the (yet inexistent) Hall algebra of the root category RX, where H+ X corresponds to the Hall algebra of Coh(X) and H− X corresponds to the Hall algebra of Coh(X)[1] (see, however [T] or [XX] for a recent approach to Hall algebras for derived categories). For F ∈ Coh(X) we put F[ǫ] = ( ”
• Semi-derived and derived Hall algebras for stable categories (2014, methodology): “T1], Toe¨n defined a version of Hall algebras for certain c lass of dg-enhanced triangulated categories. He called these objects derived Hall algebras. This work was further generalized by Xiao and Xu [XX1] for triangulated categories without assumptions on the existence of a dg-enhancement. The cons”
• Quantum groups via Hall algebras of complexes (2011, unknown): “Toën [11] solved this problem using a dg-enhancement, and it was later proved that the resulting formula defines an associative multiplication using only the axioms of a triangulated category [13].”
• The Hall algebra of a spherical object (2008, background): “It is shown in [22] [23] that it is an associative algebra with unit [0].”