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\begin{document}

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\section*{algebraic group}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{linear_algebraic_groups_and_abelian_varieties}{Linear algebraic groups and abelian varieties}\dotfill \pageref*{linear_algebraic_groups_and_abelian_varieties} \linebreak
\noindent\hyperlink{linear_algebraic_group}{Linear algebraic group}\dotfill \pageref*{linear_algebraic_group} \linebreak
\noindent\hyperlink{abelian_variety}{Abelian variety}\dotfill \pageref*{abelian_variety} \linebreak
\noindent\hyperlink{elliptic_curve}{Elliptic curve}\dotfill \pageref*{elliptic_curve} \linebreak
\noindent\hyperlink{other_prominent_classes_of_algebraic_groups}{Other prominent classes of algebraic groups}\dotfill \pageref*{other_prominent_classes_of_algebraic_groups} \linebreak
\noindent\hyperlink{jacobian}{Jacobian}\dotfill \pageref*{jacobian} \linebreak
\noindent\hyperlink{unipotent_algebraic_groups}{Unipotent algebraic groups}\dotfill \pageref*{unipotent_algebraic_groups} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Given a (typically [[algebraically closed field|algebraically closed]]) field $k$, an \textbf{algebraic $k$-group} is a [[group object]] in the category of $k$-[[algebraic variety|varieties]].

\hypertarget{linear_algebraic_groups_and_abelian_varieties}{}\subsection*{{Linear algebraic groups and abelian varieties}}\label{linear_algebraic_groups_and_abelian_varieties}

There are two important classes of algebraic groups whose intersection is trivial (the identity group): Linear algebraic groups and abelian varieties.

Any algebraic group contains a unique normal linear algebraic subgroup $H$ such that their quotient $G/H$ is an abelian variety.

\hypertarget{linear_algebraic_group}{}\subsubsection*{{Linear algebraic group}}\label{linear_algebraic_group}

An algebraic $k$-group is [[linear algebraic group|linear]] if it is a [[Zariski topology|Zariski]]-closed subgroup of the [[general linear group]] $GL(n,k)$ for some $n$.

An algebraic group is linear iff it is affine.

An algebraic group scheme is \emph{affine} if the underlying scheme is [[affine scheme|affine]].

The category of affine group schemes is the [[opposite category|opposite]] of the category of commutative [[Hopf algebras]].

\hypertarget{abelian_variety}{}\subsubsection*{{Abelian variety}}\label{abelian_variety}

Another important class are connected algebraic $k$-groups whose underlying variety is [[projective variety|projective]]; these are automatically commutative so they are called \emph{abelian varieties}. In dimension $1$ these are precisely the [[elliptic curve]]s. If $k$ is a [[perfect field]] and $G$ an algebraic $k$-group, the theorem of Chevalley says that there is a unique linear subgroup $H\subset G$ such that $G/H$ is an abelian variety.

\hypertarget{elliptic_curve}{}\paragraph*{{Elliptic curve}}\label{elliptic_curve}

An abelian variety of dimension $1$ is called an \emph{[[elliptic curve]]}.

\hypertarget{other_prominent_classes_of_algebraic_groups}{}\subsection*{{Other prominent classes of algebraic groups}}\label{other_prominent_classes_of_algebraic_groups}

Some of the definitions of the following classes exist more generally for [[group schemes]].

\hypertarget{jacobian}{}\subsubsection*{{Jacobian}}\label{jacobian}

(\ldots{})

\hypertarget{unipotent_algebraic_groups}{}\subsubsection*{{Unipotent algebraic groups}}\label{unipotent_algebraic_groups}

(See also more generally [[unipotent group scheme]].)

\begin{defn}
\label{}\hypertarget{}{}
An element $x$ of an affine algebraic group is called \emph{unipotent} if its associated right translation operator $r_x$ on the affine [[coordinate ring]] $A[G]$ of $G$ is locally unipotent as an element of the ring of linear endomorphism of $A[G]$ where `'locally unipotent'` means that its restriction to any finite dimensional stable subspace of $A[G]$ is unipotent as a ring object.

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
([[Jordan-Chevalley decomposition]]) Any commutative linear algebraic group over a perfect field is the product of a unipotent and a [[semisimple object|semisimple algebraic group]].

\end{theorem}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The group objects in the category of [[algebraic schemes]] and [[formal scheme]]s are called (algebraic) [[group schemes]] and [[formal groups]], respectively.

Among group schemes are `the infinite-dimensional algebraic groups' of Shafarevich.

Algebraic analogues of [[loop group]]s are in the category of [[ind-scheme]]s. All linear algebraic $k$-groups are affine.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The [[affine line]] $\mathbb{A}^1$ comes canonically with the structure of a group under addition: the [[additive group]] $\mathbb{G}_a$.

The affine line without its origin, $\mathbb{A}^1 - \{0\}$ comes canonically with the structure of a group under multiplication: the [[multiplicative group]] $\mathbb{G}_m$.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

\begin{itemize}%
\item [[group scheme]]

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[isogeny]]


\item [[form of an algebraic group]]



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

The standard references are

\begin{itemize}%
\item M. Demazure, P. Gabriel, \emph{Groupes algebriques}, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970; English edition is \emph{Introduction to algebraic geometry and algebraic groups}, North-Holland, Amsterdam 1980 (North-Holland)


\item [[SGA3]] \emph{Sch\'e{}mas en groupes, 1962--1964, Lecture Notes in Mathematics 151, 152 and 153, 1970}


\item M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, \emph{Schemas en groupes}, i.e. SGA III-1, III-2, III-3


\item A. Borel, \emph{Linear algebraic groups}, Springer (2nd edition much expanded)


\item W. Waterhouse, \emph{Introduction to affine group schemes}, GTM 66, Springer 1979.


\item S. Lang, \emph{Abelian varieties}, Springer 1983.


\item D. Mumford, \emph{Abelian varieties}, 1970, 1985.


\item J. C. Jantzen, \emph{Representations of algebraic groups}, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)


\item T. Springer, \emph{Linear algebraic groups}, Progress in Mathematics 9, Birkh\"a{}user Boston (2nd ed. 1998, reprinted 2008)


\item Roe Goodman, Nolan R. Wallach, Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255. Springer, Dordrecht, 2009.


\item J. Milne, \emph{Algebraic groups, Lie Groups, and their arithmetic subgroups}, \href{http://www.jmilne.org/math/CourseNotes/ALA.pdf}{pdf}


\item [[!redirects algebraic groups]]



\end{itemize}


\end{document}