nLab Hall algebra




In terms of 2-Segal spaces

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

X 1×X 1( 2, 0)X 2 1X 1 X_1 \times X_1 \stackrel{(\partial_2, \partial_0)}{\leftarrow} X_2 \stackrel{\partial_1}{\rightarrow} X_1


ptX 0s 0X 1 pt \leftarrow X_0 \stackrel{s_0}{\to} X_1

admit pull-push integral transforms in some given cohomology theory hh. Then the Hall algebra of XX with coefficients in HH is the associative algebra structure on h(X 1)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 XX and with

XX (2)X×X X \leftarrow X^{(2)} \rightarrow X\times X

denoting the moduli stack of 2-flags of coherent sheaves on XX, then the corresponding pull-push multiplication on the motivic Grothendieck ring K(X)K(X) is called the motivic Hall algebra of XX (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.



The characterization via 2-Segal spaces/decomposition spaces is independently due to

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 A 2\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

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:

Last revised on November 14, 2023 at 22:39:44. See the history of this page for a list of all contributions to it.