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\section*{Lie group}

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

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{lies_three_theorems}{Lie's three theorems}\dotfill \pageref*{lies_three_theorems} \linebreak
\noindent\hyperlink{classification}{Classification}\dotfill \pageref*{classification} \linebreak
\noindent\hyperlink{different_lie_group_structures_on_a_group}{Different Lie group structures on a group}\dotfill \pageref*{different_lie_group_structures_on_a_group} \linebreak
\noindent\hyperlink{different_topologies_on_a_lie_group}{Different topologies on a Lie group}\dotfill \pageref*{different_topologies_on_a_lie_group} \linebreak
\noindent\hyperlink{which_topological_groups_admit_lie_group_structure}{Which topological groups admit Lie group structure?}\dotfill \pageref*{which_topological_groups_admit_lie_group_structure} \linebreak
\noindent\hyperlink{homotopy_groups}{Homotopy groups}\dotfill \pageref*{homotopy_groups} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{in_differential_geometry}{In differential geometry}\dotfill \pageref*{in_differential_geometry} \linebreak
\noindent\hyperlink{in_gauge_theory}{In gauge theory}\dotfill \pageref*{in_gauge_theory} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{basic_examples}{Basic examples}\dotfill \pageref*{basic_examples} \linebreak
\noindent\hyperlink{classical_lie_groups}{Classical Lie groups}\dotfill \pageref*{classical_lie_groups} \linebreak
\noindent\hyperlink{exceptional_lie_groups}{Exceptional Lie groups}\dotfill \pageref*{exceptional_lie_groups} \linebreak
\noindent\hyperlink{infinitedimensional_examples}{Infinite-dimensional examples}\dotfill \pageref*{infinitedimensional_examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{homotopy_groups_2}{Homotopy groups}\dotfill \pageref*{homotopy_groups_2} \linebreak
\noindent\hyperlink{ReferencesOnInfiniteDimensionalLieGroups}{On infinite-dimensional Lie groups}\dotfill \pageref*{ReferencesOnInfiniteDimensionalLieGroups} \linebreak


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

A \textbf{Lie group} is a [[group]] with [[smooth structure]].

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

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{Lie group} is a [[smooth manifold]] whose [[forgetful functor|underlying]] [[set]] of [[elements]] is equipped with the [[structure]] of a [[group]] such that the [[magma|group multiplication]] and [[inverse]]-assigning [[functions]] are [[smooth functions]].

In other words, a Lie group is a [[group object]] [[internalization|internal to]] the [[category]] [[SmthMfd]] of [[smooth manifolds]].

\end{defn}
Usually the [[smooth manifold]] is assumed to be defined over the [[real numbers]] and to be of [[finite number|finite]] [[dimension]] (f.d.), but extensions of the definition to some other [[ground fields]] or to -[[infinite-dimensional manifolds]] are also relevant, sometimes under other names (such as [[Fréchet Lie group]] when the underlying manifold is an infinite-dimensional [[Fréchet manifold]]).

A real Lie group is called a \emph{[[compact Lie group]]} (or \emph{connected}, \emph{simply connected} Lie group, etc) if its underlying [[topological space]] is [[compact space|compact]] (or [[connected space|connected]], [[simply connected space|simply connected]], etc).

Every connected finite dimensional real Lie group is [[homeomorphism|homeomorphic]] to a [[product]] of a [[compact Lie group]] (its [[maximal compact subgroup]]) and a [[Euclidean space]].

Every [[abelian group|abelian]] connected compact finite dimensional real Lie group is a [[torus]] (a product of [[circles]] $T^n = S^1\times S^1 \times \ldots \times S^1$).

There is an [[infinitesimal space|infinitesimal]] version of a Lie group, a so-called [[local Lie group]], where the multiplication and the inverse are only partially defined, namely if the arguments of these operations are in a sufficiently small neighborhood of identity. There is a natural equivalence of local Lie groups by means of agreeing (topologically and algebraically) on a smaller neighborhood of the identity. The category of local Lie groups is equivalent to the category of connected and simply connected Lie groups.

The first order [[infinitesimal object|infinitesimal]] approximation to a Lie group is its [[Lie algebra]].

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

\hypertarget{lies_three_theorems}{}\subsubsection*{{Lie's three theorems}}\label{lies_three_theorems}

[[Sophus Lie]] has proved several theorems -- [[Lie's three theorems]] -- on the relationship between [[Lie algebra]]s and Lie groups. What is called [[Lie's third theorem]] is about the [[equivalence of categories]] of f.d. real Lie algebras and local Lie groups. [[Élie Cartan]] has extended this to a global integrability theorem called the Cartan-Lie theorem, nowadays after Serre also called Lie's third theorem.

\hypertarget{classification}{}\subsubsection*{{Classification}}\label{classification}

\begin{prop}
\label{}\hypertarget{}{}
Every connected finite-dimensional real Lie group is [[homeomorphism|homeomorphic]] to a [[product]] of a compact Lie group and a [[Cartesian space|Euclidean space]]. Every abelian connected compact f.d. real Lie group is a [[torus]] (a product of circles $T^n = S^1\times S^1 \times \ldots \times S^1$).

\end{prop}
The [[simple Lie group]]s have a classification into infinite series of

\begin{itemize}%
\item [[classical Lie group]]s

\end{itemize}
and a finite snumber o

\begin{itemize}%
\item [[exceptional Lie group]]s

\end{itemize}
\hypertarget{different_lie_group_structures_on_a_group}{}\subsubsection*{{Different Lie group structures on a group}}\label{different_lie_group_structures_on_a_group}

For $G$ a bare [[group]] (without smooth structure) there may be more than one way to equip it with the [[stuff, structure, property|structure]] of a Lie group.

\begin{example}
\label{}\hypertarget{}{}
As bare [[abelian group]]s, the [[Cartesian space]]s $\mathbb{R}^n$ are, for all $n$, [[vector space]]s over the [[rational number]]s $\mathbb{Q}$ whose [[dimension]] is the [[cardinality]] of the [[continuum]], $2^{\aleph_0}$.

Therefore these are all [[isomorphic]] as bare group. But equipped with their canonical Lie group structure (as in the \hyperlink{Examples}{Examples}) they are of course not isomorphic.

\end{example}
\hypertarget{different_topologies_on_a_lie_group}{}\subsubsection*{{Different topologies on a Lie group}}\label{different_topologies_on_a_lie_group}

\begin{itemize}%
\item Linus Kramer, \emph{The topology of a simple Lie group is essentially unique}, (\href{http://arxiv.org/abs/1009.5457}{arXiv})

\end{itemize}
\begin{quote}%
Abstract: We study locally compact group topologies on simple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every `abstract' isomorphism between $S$ and a locally compact and $\sigma$-compact group $\Gamma$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.


\end{quote}
\hypertarget{which_topological_groups_admit_lie_group_structure}{}\subsubsection*{{Which topological groups admit Lie group structure?}}\label{which_topological_groups_admit_lie_group_structure}

\begin{itemize}%
\item \emph{[[Hilbert's fifth problem]]}

\end{itemize}
\hypertarget{homotopy_groups}{}\subsubsection*{{Homotopy groups}}\label{homotopy_groups}

List of [[homotopy groups]] of the manifolds underlying the classical [[Lie groups]] are for instance in (\hyperlink{Abanov09}{Abanov 09}).

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{in_differential_geometry}{}\subsubsection*{{In differential geometry}}\label{in_differential_geometry}

A central concept of [[differential geometry]] is that of a $G$-[[principal bundle]] $P \to X$ over a [[smooth manifold]] $X$ for $G$ a Lie group.

\hypertarget{in_gauge_theory}{}\subsubsection*{{In gauge theory}}\label{in_gauge_theory}

In the [[physics]] of [[gauge field]]s -- [[gauge theory]] -- Lie groups appear as local [[gauge group]]s parameterizing [[gauge transformation]]s: notably the [[Yang-Mills field]] is modeled by a $G$-[[principal bundle]] with [[connection on a bundle|connection]] for some Lie group $G$. For models that describe experimental observations the group $G$ in question is a quotient of a product of [[special unitary group]]s and the [[circle group]]. For details see [[standard model of particle physics]]

\hypertarget{in_higher_category_theory}{}\subsection*{{In higher category theory}}\label{in_higher_category_theory}

The notion of [[group]] generalizes in [[higher category theory]] to that of [[2-group]], \ldots{} [[∞-group]].

Accordingly, so does the notion of Lie group generalize to [[Lie 2-group]], \ldots{} [[∞-Lie group]]. For details see [[∞-Lie groupoid]].

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

\hypertarget{basic_examples}{}\subsubsection*{{Basic examples}}\label{basic_examples}

\begin{itemize}%
\item The [[real line]] $\mathbb{R}$ with its standard [[smooth structure]] and the group operation being addition is a Lie group. So is every [[Cartesian space]] $\mathbb{R}^n$ with the componentwise addition of real numbers.


\item The [[quotient]] of $\mathbb{R}$ by the subgroup of [[integer]]s $\mathbb{Z} \hookrightarrow \mathbb{R}$ is the [[circle group]] $S^1 = \mathbb{R}/\mathbb{Z}$. The quotient $\mathbb{R}^n/\mathbb{Z}^n$ is the $n$-[[dimension|dimensional]] [[torus]].


\item The [[automorphism group]] of any Lie group is canonically itself a Lie group: the \emph{[[automorphism Lie group]]}.



\end{itemize}
\hypertarget{classical_lie_groups}{}\subsubsection*{{Classical Lie groups}}\label{classical_lie_groups}

The [[classical Lie groups]] include

\begin{itemize}%
\item the [[general linear group]] $GL(n)$


\item the [[orthogonal group]] $O(n)$ and [[special orthogonal group]] $SO(n)$;


\item the [[unitary group]] $U(n)$ and [[special unitary group]] $SU(n)$;


\item the [[symplectic group]] $Sp(2n)$.



\end{itemize}
\hypertarget{exceptional_lie_groups}{}\subsubsection*{{Exceptional Lie groups}}\label{exceptional_lie_groups}

The [[exceptional Lie groups]] incude

\begin{itemize}%
\item [[G2]], [[F4]], [[E6]], [[E7]] [[E8]],

\end{itemize}
\hypertarget{infinitedimensional_examples}{}\subsubsection*{{Infinite-dimensional examples}}\label{infinitedimensional_examples}

\begin{itemize}%
\item [[stable unitary group]]


\item [[loop group]]


\item [[diffeomorphism group]]

\begin{itemize}%
\item [[symplectomorphism group]], [[quantomorphism group]]

\end{itemize}


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

\begin{itemize}%
\item [[semisimple Lie group]], [[simple Lie group]], [[exceptional Lie group]]


\item [[real form]]


\item [[compact Lie group]], [[maximal compact subgroup]]


\item [[maximal torus]]


\item [[rank of a Lie group]]


\item [[conjugacy class]], [[Cartan-Dirac structure on a Lie group]]


\item [[Weyl group]]


\item [[Inönü-Wigner group contraction]]


\item [[infinite-dimensional Lie group]]


\item [[orbit method]]


\item [[Hamiltonian dynamics on Lie groups]]


\item [[complex Lie group]]


\item [[Poisson Lie group]]


\item [[invariant differential form]], [[Maurer-Cartan form]]



\end{itemize}
[[!include infinitesimal and local - table]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item A. L. Onishchik (ed.) \emph{Lie Groups and Lie Algebras}

\begin{itemize}%
\item \emph{I.} A. L. Onishchik, E. B. Vinberg, \emph{Foundations of Lie Theory},


\item \emph{II.} V. V. Gorbatsevich, A. L. Onishchik, \emph{Lie Transformation Groups}



\end{itemize}
Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993


\item [[Hans Duistermaat]], J. A. C. Kolk, \emph{Lie groups}, 2000


\item Joachim Hilgert, [[Karl-Hermann Neeb]], \emph{Structure and Geometry of Lie Groups}, Springer Monographs in Mathematics, Springer-Verlag New York, 2012 (\href{https://link.springer.com/book/10.1007/978-0-387-84794-8}{doi:10.1007/978-0-387-84794-8})


\item Mark Haiman, lecture notes by [[Theo Johnson-Freyd]], \emph{Lie groups}, Berkeley 2009 (\href{http://math.berkeley.edu/~theojf/LieGroups.pdf}{pdf})


\item [[Eckhard Meinrenken]], \emph{Lie groups and Lie algebas}, Lecture notes 2010 (\href{http://www.math.toronto.edu/mein/teaching/LectureNotes/lie.pdf}{pdf})



\end{itemize}
\hypertarget{homotopy_groups_2}{}\subsubsection*{{Homotopy groups}}\label{homotopy_groups_2}

\begin{itemize}%
\item Alexander Abanov, Homotopy groups of Lie groups 2009 (\href{http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cpA1-upload.pdf}{pdf})

\end{itemize}
\hypertarget{ReferencesOnInfiniteDimensionalLieGroups}{}\subsubsection*{{On infinite-dimensional Lie groups}}\label{ReferencesOnInfiniteDimensionalLieGroups}

References on [[infinite-dimensional Lie groups]]

\begin{itemize}%
\item [[Andreas Kriegl]], [[Peter Michor]], \emph{Regular infinite dimensional Lie groups} Journal of Lie Theory Volume 7 (1997) 61-99 (\href{http://www.heldermann-verlag.de/jlt/jlt07/MICHPL.PDF}{pdf})


\item Rudolf Schmid, \emph{Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics} Advances in Mathematical Physics Volume 2010, (\href{http://www.emis.de/journals/HOA/AMP/Volume2010/280362.pdf}{pdf})


\item Josef Teichmann, \emph{Infinite dimensional Lie Theory from the point of view of Functional Analysis} (\href{http://www.math.ethz.ch/~jteichma/diss.pdf}{pdf})


\item [[Karl-Hermann Neeb]], \emph{Monastir summer school: Infinite-dimensional Lie groups} (\href{http://www.math.uni-hamburg.de/home/wockel/data/monastir.pdf}{pdf})



\end{itemize}
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