Given a 2-Segal space $X_\bullet$ such that the spans
and
admit pull-push integral transforms in some given cohomology theory $h$. Then the Hall algebra of $X$ with coefficients in $H$ is the associative algebra structure on $h(X_1)$ induced by these pull-push operations.
This is the perspective of Dyckerhoff-Kapranov 12, def. 8.1.8.
Specifically for a given algebraic stack $X$ and with
denoting the moduli stack of 2-flags of coherent sheaves on $X$, then the corresponding pull-push multiplication on the motivic Grothendieck ring $K(X)$ is called the motivic Hall algebra of $X$ (due to Dominic Joyce reviewed e.g. in Bridgeland 10, 4.2). Discussion of motivic Hall algebras of Calabi-Yau 3-folds is in (Kontsevich-Soibelman 08).
The Hall algebra of an abelian category is the Grothendieck group of constructible sheaves/perverse sheaves on the moduli stack of objects in the category. The Hall algebra is an algebra because the constructible derived category of the moduli stack of objects in an abelian category is monoidal in a canonical way.
This perspective is taken from (Webster11). See there for more details.
Survey:
Claus M. Ringel, The Hall multiplication], lecture notes (2006) [web]
Ben Webster, Hall algebras are Grothendieck groups (2011) [webpage]
Olivier Schiffmann, Lectures on Hall algebras [arXiv:math/0611617]
Matthew B. Young, Self-Dual Hall modules, PhD thesis, Stony Brook (2013) [pdf, pdf]
Wikipedia: Hall algebra, Ringel-Hall algebra
The characterization via 2-Segal spaces/decomposition spaces is independently due to
(now part of their book Higher Segal spaces, Springer Lec. Notes in Math. 2244 doi)
Anew light on the interplay between 2-Segal and Hall product:
Matthew B. Young, Relative 2-Segal spaces,
Algebraic & Geometric Topology 18&& (2018) 975–1039 [doi:10.2140/agt.2018.18.975]
We introduce a relative version of the 2–Segal simplicial spaces defined by Dyckerhoff and Kapranov, and Gálvez-Carrillo, Kock and Tonks. Examples of relative 2–Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the $\mathcal{R}_\bullet$-construction from Grothendieck–Witt theory. We show that a relative 2–Segal space defines a categorical representation of the Hall algebra associated to the base 2–Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative 2–Segal spaces.
Further references:
Mikhail Kapranov, Eisenstein series and quantum affine algebras, Journal Math. Sciences 84 (1997), 1311–1360.
Mikhail Kapranov, Eric Vasserot, Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000) 565-576, arXiv/9812016
Bernhard Keller, Dong Yang, Guodong Zhou, The Hall algebra of a spherical object, J. London Math Soc. (2) 80 (2009) 771–784, doi, pdf
Claus M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591
Green has introduced a coproduct on Hall algebras of a quiver which are related to quantum groups; Green’s theorem states that the Hall algebra of the category of representations of a quiver over a finite field is a twisted bialgebra. The coproduct than agrees with the one on quantum groups:
Elliptic Hall algebras
Toen’s derived Hall algebras
Bertrand Toen, Derived Hall algebras, Duke Math. J. 135(3): 587-615 (2006) doi arXiv:0501343
Alexander I. Efimov, Cohomological Hall algebra of a symmetric quiver, Compositio Mathematica 148:4 (2012) 1133-1146 doi arxiv/1103.2736
Description of seminar on stability conditions, Hall algebras and Stokes factors in Bonn 2009 (D. Huybrechts), pdf
sbseminar blog: Hall algebras and Donaldson-Thomas invariants-i
Bangming Deng, Jie Du, Brian Parshall, Jianpan Wang, Finite dimensional algebras and quantum groups, Mathematical Surveys and Monographs 150, Amer. Math. Soc. 2008. xxvi+759 pp. (chap. 10: Ringel-Hall algebras) MR2009i:17023)
David Hernandez, Bernard Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. Reine Angew. Math. 701 (2015) 77–126 doi arXiv/1109.0862
Parker E. Lowrey, The moduli stack and motivic Hall algebra for the bounded derived category, arxiv/1110.5117
Tobias Dyckerhoff, Higher categorical aspects of Hall Algebras, In: Herbera D., Pitsch W., Zarzuela S. (eds) Building Bridges Between Algebra and Topology. Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser 2015 arXiv:1505.06940 doi
Mauro Porta, Francesco Salla, Two-dimensional categorified Hall algebras, J. Eur. Math. Soc. 25:3 (2023) 1113–1205 doi (review in Zbl076835085)
J. Xiao, F. Xu, Hall algebras associated to triangulated categories, Duke Math. J. 143 (2) (2008) 357–373
A realization of (Drinfeld-Jimbo) quantum groups via a “Bridgeland” version of Hall algebra,
Motivic Hall algebras:
Tom Bridgeland, An introduction to motivic Hall algebra (arXiv:1002.4372)
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (arXiv:0811.2435)
M. Kontsevich, Y. Soibelman, Motivic Donaldson-Thomas invariants: summary of results, arXiv/0910.4315
Maxim Kontsevich, Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, arXiv/1006.2706
Richard Rimányi, Motivic characteristic classes in cohomological Hall algebras, Advances in Mathematics 360 (2020) 106888 doi arXiv:1808.05654
Alexei Latyntsev, Vertex algebras, moduli stacks, cohomological Hall algebras and quantum groups, PhD thesis, Oxford 2022 pdf
Last revised on November 14, 2023 at 22:39:44. See the history of this page for a list of all contributions to it.