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\begin{document}

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\section*{Hall algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_terms_of_2segal_spaces}{In terms of 2-Segal spaces}\dotfill \pageref*{in_terms_of_2segal_spaces} \linebreak
\noindent\hyperlink{MotivicHallAlgebra}{Motivic Hall algebra}\dotfill \pageref*{MotivicHallAlgebra} \linebreak
\noindent\hyperlink{in_terms_of_constructible_sheaves}{In terms of constructible sheaves}\dotfill \pageref*{in_terms_of_constructible_sheaves} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{in_terms_of_2segal_spaces}{}\subsubsection*{{In terms of 2-Segal spaces}}\label{in_terms_of_2segal_spaces}

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

\begin{displaymath}
X_1 \times X_1 \stackrel{(\partial_2, \partial_0)}{\leftarrow} X_2 \stackrel{\partial_1}{\rightarrow} X_1
\end{displaymath}
and

\begin{displaymath}
pt \leftarrow X_0 \stackrel{s_0}{\to} X_1
\end{displaymath}
admit pull-push [[integral transforms]] in some given [[cohomology theory]] $h$. Then the \textbf{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 \hyperlink{DyckerhoffKapranov12}{Dyckerhoff-Kapranov 12, def. 8.1.8}.

\hypertarget{MotivicHallAlgebra}{}\subsubsection*{{Motivic Hall algebra}}\label{MotivicHallAlgebra}

Specifically for a given [[algebraic stack]] $X$ and with

\begin{displaymath}
X \leftarrow X^{(2)} \rightarrow X\times X
\end{displaymath}
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 \emph{motivic Hall algebra} of $X$ (due to [[Dominic Joyce]] reviewed e.g. in \hyperlink{Bridgeland10}{Bridgeland 10, 4.2}). Discussion of motivic Hall algebras of [[Calabi-Yau 3-folds]] is in (\hyperlink{KontsevichSoibelman08}{Kontsevich-Soibelman 08}).

\hypertarget{in_terms_of_constructible_sheaves}{}\subsubsection*{{In terms of constructible sheaves}}\label{in_terms_of_constructible_sheaves}

The \emph{Hall algebra} of an [[abelian category]] is the [[Grothendieck group]] of [[constructible sheaves]]/[[perverse sheaves]] on the [[moduli stack]] of [[object]]s 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 category|monoidal]] in a canonical way.

This perspective is taken from (\hyperlink{Webster}{Webster11}). See there for more details.

\hypertarget{references}{}\subsection*{{References}}\label{references}

A good survey is given in

\begin{itemize}%
\item [[Ben Webster]], \emph{Hall algebras are Grothendieck groups} (\href{http://sbseminar.wordpress.com/2011/04/18/hall-algebras-are-grothendieck-groups/#more-3988}{SBS})

\end{itemize}
The characterization via [[2-Segal spaces]] is due to

\begin{itemize}%
\item Tobias Dyckerhoff, [[Mikhail Kapranov]], \emph{Higher Segal spaces I}, (\href{http://arxiv.org/abs/1212.3563}{arxiv:1212.3563})

\end{itemize}
Canonical references on Hall algebras include the following.

\begin{itemize}%
\item [[Kapranov|M. Kapranov]], \emph{Eisenstein series and quantum affine algebras}, Journal Math. Sciences \textbf{84} (1997), 1311--1360.


\item M. Kapranov, E. Vasserot, \emph{Kleinian singularities, derived categories and Hall algebras}, Math. Ann. \textbf{316} (2000), 565-576, \href{http://arxiv.org/abs/math/9812016}{arxiv/9812016}


\item [[Bernhard Keller]], Dong Yang, Guodong Zhou, \emph{The Hall algebra of a spherical object}, J. London Math Soc. (2) \textbf{80} (2009) 771--784, \href{http://dx.doi.org/10.1112/jlms/jdp054}{doi}, \href{www.math.jussieu.fr/~keller/publ/HallAlgSphericalObject.pdf}{pdf}


\item C. Ringel, \emph{Hall algebras and quantum groups}, Invent. Math. \textbf{101} (1990), no. 3, 583--591.


\item O. Schiffmann, \emph{Lectures on Hall algebras}, \href{http://arXiv.org/abs/math/0611617}{arXiv:math/0611617}


\item O. Schiffmann, E. Vasserot, \emph{The elliptic Hall algebra}, Cherednik Hecke algebras and Macdonald polynomials, \href{http://arxiv.org/abs/0802.4001}{arXiv:0802.4001}, (2008), to appear in Compositio Math.; \emph{The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A2, \href{http://arXiv.org/abs/0905.2555}{arXiv:0905.2555}}


\item O. Schiffman, \emph{Drinfeld realization of the elliptic Hall algebra}, \href{http://arxiv.org/abs/1004.2575}{arxiv/1004.2575}


\item O. Schiffmann, E. Vasserot, \emph{Hall algebras of curves, commuting varieties and Langlands duality}, \href{http://arxiv.org/abs/1009.0678}{arxiv/1009.0678}


\item [[Toen|B. Toen]], \emph{Derived Hall algebras}, \href{http://arxiv.org/abs/math/0501343}{arxiv/0501343}


\item [[Alexander I. Efimov|Alexander Efimov]], \emph{Cohomological Hall algebra of a symmetric quiver}, \href{http://arxiv.org/abs/1103.2736}{arxiv/1103.2736}


\item Description of seminar on stability conditions, Hall algebras and [[Stokes phenomenon|Stokes factors]] in Bonn 2009 ([[Daniel Huybrechts|D. Huybrechts]]), \href{http://www.math.uni-bonn.de/people/compgeo/Hall.pdf}{pdf}


\item wikipedia: \href{http://en.wikipedia.org/wiki/Hall_algebra}{Hall algebra}, \href{http://en.wikipedia.org/wiki/Ringel&#8211;Hall_algebra}{Ringel-Hall algebra}


\item sbseminar blog: \href{http://sbseminar.wordpress.com/2009/03/25/hall-algebras-and-donaldson-thomas-invariants-i}{Hall algebras and Donaldson-Thomas invariants-i}


\item Bangming Deng, Jie Du, [[Brian Parshall]], Jianpan Wang, \emph{Finite dimensional algebras and quantum groups}, Mathematical Surveys and Monographs \textbf{150}, Amer. Math. Soc. 2008. xxvi+759 pp. (chap. 10: Ringel-Hall algebras) \href{http://www.ams.org/mathscinet-getitem?mr=2457938}{MR2009i:17023)}


\item David Hernandez, Bernard Leclerc, \emph{Quantum Grothendieck rings and derived Hall algebras}, \href{http://arxiv.org/abs/1109.0862}{arxiv/1109.0862}


\item Parker E. Lowrey, \emph{The moduli stack and motivic Hall algebra for the bounded derived category}, \href{http://arxiv.org/abs/1110.5117}{arxiv/1110.5117}


\item [[Tobias Dyckerhoff]], \emph{Higher categorical aspects of Hall Algebras}, (\href{https://arxiv.org/abs/1505.06940}{arXiv:1505.06940})



\end{itemize}
Motivic Hall algebras:

\begin{itemize}%
\item [[Tom Bridgeland]], \emph{An introduction to motivic Hall algebra} (\href{http://arxiv.org/abs/1002.4372}{arXiv:1002.4372})


\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Stability structures, motivic Donaldson-Thomas invariants and cluster transformations} (\href{http://arxiv.org/abs/0811.2435}{arXiv:0811.2435})


\item [[Kontsevich|M. Kontsevich]], [[Yan Soibelman|Y. Soibelman]], \emph{Motivic Donaldson-Thomas invariants: summary of results}, \href{http://arxiv.org/abs/0910.4315}{arxiv/0910.4315}


\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants}, \href{http://arxiv.org/abs/1006.2706}{arxiv/1006.2706}



\end{itemize}
[[!redirects Hall algebras]]

[[!redirects motivic Hall algebra]] [[!redirects motivic Hall algebras]]



\end{document}