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

A good survey is given in

• Ben Webster, Hall algebras are Grothendieck groups (SBS)

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

Canonical references on Hall algebras include the following.

• M. Kapranov, Eisenstein series and quantum affine algebras, Journal Math. Sciences 84 (1997), 1311–1360.

• M. Kapranov, E. 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

• C. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591.

• O. Schiffmann, Lectures on Hall algebras, arXiv:math/0611617

• O. Schiffmann, E. Vasserot, The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, arXiv:0802.4001, (2008), to appear in Compositio Math.; The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A2, arXiv:0905.2555

• O. Schiffman, Drinfeld realization of the elliptic Hall algebra, arxiv/1004.2575

• O. Schiffmann, E. Vasserot, Hall algebras of curves, commuting varieties and Langlands duality, arxiv/1009.0678

• B. Toen, Derived Hall algebras, arxiv/0501343

• Alexander Efimov, Cohomological Hall algebra of a symmetric quiver, arxiv/1103.2736

• Description of seminar on stability conditions, Hall algebras and Stokes factors in Bonn 2009 (D. Huybrechts), pdf

• wikipedia: Hall algebra, Ringel-Hall algebra

• 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, 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

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