nLab Hall algebra

Idea

In terms of 2-Segal spaces

Given a 2-Segal space $X_\bullet$ such that the spans

$$X_1 \times X_1 \stackrel{(\partial_2, \partial_0)}{\leftarrow} X_2 \stackrel{\partial_1}{\rightarrow} X_1$$

and

$$pt \leftarrow X_0 \stackrel{s_0}{\to} X_1$$

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.

Motivic Hall algebra

Specifically for a given algebraic stack $X$ and with

$$X \leftarrow X^{(2)} \rightarrow X\times X$$

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).

In terms of constructible sheaves

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.

References

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

• Tobias Dyckerhoff, Mikhail Kapranov, Higher Segal spaces I, (arxiv:1212.3563)

  (now part of their book Higher Segal spaces, Springer Lec. Notes in Math. 2244 doi)

• Imma Gálvez-Carrillo, Joachim Kock, Andrew Tonks, Decomposition spaces, incidence algebras and Möbius inversion, arXiv:1404.3202

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:

• J. A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (2), 361–377 (1995)
• J. Xiao, F. Xu, M. Zhao, Ringel-Hall algebras beyond their quantum groups I: Restriction functor and Green formula. Algebras & Represent. Theory 22 (2019) , 1299–1329 doi
• J. Xiao, Drinfeld double and Ringel-Green theory of Hall algebras, J Algebra 190 (1997) 100–144 Zbl0874.16026
• Matthew B. Young, Degenerate versions of Green’s theorem for Hall modules, J. Pure & Applied Algebra 225:4 (2021) 106557 doi

Elliptic Hall algebras

• Olivier Schiffmann, Eric Vasserot, The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Mathematica 147:1 (2011) 188-234 doi arXiv:0802.4001, (2008); The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbf{A}^2$, Duke Math. J. 162(2): 279-366 (2013) doi arXiv:0905.2555
• O. Schiffman, Drinfeld realization of the elliptic Hall algebra, J. Algebr. Comb. 35 (2012) 237–262 doi arxiv/1004.2575
• O. Schiffmann, Eric Vasserot, Hall algebras of curves, commuting varieties and Langlands duality, Math. Ann. 353 (2012) 1399-1451 doi arXiv:1009.0678

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,

• Tom Bridgeland, Quantum groups via Hall algebras of complexes, Ann of Math 177 (2) (2013) 739–759

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