nLab
Hall algebra

Idea

In terms of 2-Segal spaces

Given a 2-Segal space $X_{•}$ such that the spans

$$X_1\times X_1\stackrel{({\partial }_2,{\partial }_0)}{\leftarrow }{X}_2\stackrel{{\partial }_1}{\to }{X}_1$$X_1 \times X_1 \stackrel{(\partial_2, \partial_0)}{\leftarrow} X_2 \stackrel{\partial_1}{\rightarrow} X_1

and

$$\mathrm{pt}\leftarrow X_0\stackrel{s_0}{\to }{X}_1$$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.

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 is due to

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

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
• 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
• 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