Derived Hall algebras

By Bertrand Toën. 2005

158 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/821f2fd92fafed7df464c9dca66d0857a8fa5b2a

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Citations in the corpus

From triangulated categories to cluster algebras (2005, background): “It allows us to bypass the ‘triangulated geometry’ of C, which unfortunately is even out of reach of the methods of [33,34], because the graded morphism spaces of the category C are not of finite total dimension.”
Higher Segal Spaces (2012, methodology): “Taking forX various incarnations of the Waldhausen S-construction, we recover “ classical” Hall algebras [Sch06] (C is the category of groupoids, h is the space of functions), derived Hall algebras of Toën [Toë06] (C is the category of spaces, h is the space of locally constant functions), motivic H”
On the Hall algebra of an elliptic curve, I (2005, unknown): “Drinfeld double) as the (yet inexistent) Hall algebra of the root category RX, where He+ Xcorresponds to the Hall algebra of Coh(X) and He− Xcorresponds to the Hall algebra of Coh(X)[1] (see, however [T] for a recent approach to Hall algebras for derived (or more precisely dg) categories). According”
The Hall algebra of a spherical object (2008, background): “It is shown in [22, 23] that it is an associative algebra with unit [0].”
From triangulated categories to Lie algebras: A theorem of Peng and Xiao (2005, background): “We remark that To¨en has recently shown how to construct an associative algebra from a dg-category [ 18 ] (under some finiteness assumptions), and it will clearly be of interest to investigate the connections between his derived Hall algebra and the quantised enveloping algebras.”
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].”
Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory (2015, methodology): “This was the case in the work of Toën [70] who dealt with the case where X is the Waldhausen S•-construction of a dg category, and in the work of Dyckerhoff and Kapranov [17] for exact ∞-categories.”
Moduli of objects in dg-categories (2005, background): “2 is most probably that such geometric derived Hall algebras exist, but also that they map to the one of [31] by means of a faisceaux-fonctions correspondence and trace formula for higher stacks.”
Hall algebras associated to triangulated categories (2006, methodology): “Abstract. By counting with triangles and the octohedral axiom, we find a direct way to prove the formula of To¨en in [ 13 ] for a triangulated category with (left) homological-finite condition.”