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\begin{document}

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\section*{general linear group}

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

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

[[!include group theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{AsAtopologicalGroup}{As a topological group}\dotfill \pageref*{AsAtopologicalGroup} \linebreak
\noindent\hyperlink{definition_2}{Definition}\dotfill \pageref*{definition_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AsALieGroup}{As a Lie group}\dotfill \pageref*{AsALieGroup} \linebreak
\noindent\hyperlink{as_an_algebraic_group}{As an algebraic group}\dotfill \pageref*{as_an_algebraic_group} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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 [[field]] $k$, the \textbf{general linear group} $GL(n,k)$ (or $GL_n(k)$) is the [[group]] of [[invertible morphism|invertible]] [[linear maps]] from the [[vector space]] $k^n$ to itself. It may canonically be identified with the group of $n\times n$ [[matrix|matrices]] with entries in $k$ having nonzero [[determinant]].

\hypertarget{AsAtopologicalGroup}{}\subsubsection*{{As a topological group}}\label{AsAtopologicalGroup}

\hypertarget{context_2}{}\subsubsection*{{Context}}\label{context_2}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

Let $k = \mathbb{R}$ or $= \mathbb{C}$ be the [[real numbers]] or the [[complex numbers]] equipped with their [[Euclidean topology]].

\hypertarget{definition_2}{}\paragraph*{{Definition}}\label{definition_2}

\begin{defn}
\label{GLnAsTopologicalGroup}\hypertarget{GLnAsTopologicalGroup}{}
\textbf{(general linear group as a topological group)}

For $n \in \mathbb{N}$, as a [[topological group]] the \emph{general linear group} $GL(n, k)$ is defined as follows.

The underlying group is the [[group]] of [[real number|real]] or [[complex numbers|complex]] $n \times n$ [[matrices]] whose [[determinant]] is non-vanishing

\begin{displaymath}
GL(n,k)
  \;\coloneqq\;
  \left(
    A \in Mat_{n \times n}(k)
    \; \vert \;
    det(A) \neq 0
  \right)
\end{displaymath}
with group operation given by [[matrix multiplication]].

The [[topological space|topology]] on this set is the [[subspace topology]] as a subset of the [[Euclidean space]] of [[matrices]]

\begin{displaymath}
Mat_{n \times n}(k) \simeq k^{(n^2)}
\end{displaymath}
with its [[metric topology]].

\end{defn}
\begin{lemma}
\label{GLnTopologicalWellDefined}\hypertarget{GLnTopologicalWellDefined}{}
\textbf{(group operations are continuous)}

Definition \ref{GLnAsTopologicalGroup} is indeed well defined in that the group operations on $GL(n,k)$ are indeed [[continuous functions]] with respect to the given topology.

\end{lemma}
\begin{proof}
Observe that under the identification $Mat_{n \times n}(k) \simeq k^{(n^2)}$ [[matrix multiplication]] is a [[polynomial function]]

\begin{displaymath}
k^{(n^2)}
   \times
  k^{(n^2)}
   \simeq
  k^{ 2 n^2 }
    \longrightarrow
  k^{(n^2)}
    \simeq
  Mat_{n \times n}(k)
  \,.
\end{displaymath}
Similarly [[inverse matrix|matrix inversion]] is a [[rational function]]. Now [[rational functions are continuous]] on their [[domain]] of definition, and since a real matrix is invertible previsely if its determinant is non-vanishing, the domain of definition for matrix inversion is precisely $GL(n,k) \subset Mat_{n \times n}(k)$.

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{(stable general linear group)}

The evident [[tower]] of embeddings

\begin{displaymath}
k \hookrightarrow k^2 \hookrightarrow k^3 \hookrightarrow \cdots
\end{displaymath}
induces a corresponding [[tower diagram]] of embedding of the general linear groups (def. \ref{GLnAsTopologicalGroup})

\begin{displaymath}
GL(1,k)
    \hookrightarrow
  GL(2,k)
    \hookrightarrow
  GL(3,k)
    \hookrightarrow
  \cdots
  \,.
\end{displaymath}
The [[colimit]] over this [[diagram]] in the [[category]] of [[topological group]] is called the \emph{stable general linear group} denoted

\begin{displaymath}
GL(k) \;\coloneqq\; \underset{\longrightarrow}{\lim}_n GL(n,k)
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\paragraph*{{Properties}}\label{properties}

\begin{example}
\label{AsSubspaceOfTheMappingSpace}\hypertarget{AsSubspaceOfTheMappingSpace}{}
\textbf{(as a [[subspace]] of the [[compact-open topology|mapping space]])}

The [[topological space|topology]] induced on the real general linear group when regarded as a [[topological subspace]] of [[Euclidean space]] with its [[metric topology]]

\begin{displaymath}
GL(n,\mathbb{R})
    \subset
  Mat_{n \times n}(\mathbb{R})
    \simeq
  \mathbb{R}^{(n^2)}
\end{displaymath}
(as in def. \ref{GLnAsTopologicalGroup}) coincides with the topology induced by regarding the general linear group as a [[subspace]] of the [[mapping space]] $Maps(k^n, k^n)$,

\begin{displaymath}
GL(n,\mathbb{R})
    \subset
  Maps(k^n, k^n)
\end{displaymath}
i.e. the set of all [[continuous functions]] $k^n \to k^n$ equipped with the [[compact-open topology]].

\end{example}
\begin{proof}
On the one had, the [[universal property]] of the [[mapping space]] (\href{Introduction+to+Topology+--+1#UniversalPropertyOfMappingSpace}{this prop.}) gives that the inclusion

\begin{displaymath}
GL(n, \mathbb{R}) \to Maps(\mathbb{R}^n, \mathbb{R}^n)
\end{displaymath}
is a [[continuous function]] for $GL(n,\mathbb{R})$ equipped with the [[Euclidean space|Euclidean]] [[metric topology]], because this is the [[adjunct]] of the defining continuous [[action]] map

\begin{displaymath}
GL(n, \mathbb{R}) \times \mathbb{R}^n \to \mathbb{R}^n
  \,.
\end{displaymath}
This implies that the [[Euclidean space|Euclidean]] [[metric topology]] on $GL(n,\mathbb{R})$ is equal to or [[finer topology|finer]] than the subspace topology coming from $Map(\mathbb{R}^n, \mathbb{R}^n)$.

We conclude by showing that it is also equal to or [[coarser topology|coarser]], together this then implies the claims.

Since we are speaking about a subspace topology, we may consider the open subsets of the ambient Euclidean space $Mat_{n \times n}(\mathbb{R}) \simeq \mathbb{R}^{(n^2)}$. Observe that a [[neighborhood base]] of a linear map or matrix $A$ consists of sets of the form

\begin{displaymath}
U_A^\epsilon
  \;\coloneqq\;
  \left\{B \in Mat_{n \times n}(\mathbb{R}) \,\vert\, \underset{{1 \leq i \leq n}}{\forall}\; |A e_i - B e_i| \lt \epsilon \right\}
\end{displaymath}
for $\epsilon \in (0,\infty)$.

But this is also a [[base for the topology|base]] element for the [[compact-open topology]], namely

\begin{displaymath}
U_A^\epsilon
   \;=\;
  \bigcap_{i = 1}^n V_i^{K_i}
  \,,
\end{displaymath}
where $K_i \coloneqq \{e_i\}$ is a [[singleton]] and $V_i \coloneqq B^\circ_{A e^i}(\epsilon)$ is the [[open ball]] of [[radius]] $\epsilon$ around $A e^i$.

\end{proof}
\begin{prop}
\label{ConnectednessOfGeneralLinearGroup}\hypertarget{ConnectednessOfGeneralLinearGroup}{}
\textbf{(connectedness properties of the general linear group)}

For all $n \in \mathbb{N}$

\begin{enumerate}%
\item the complex general linear group $GL(n,\mathbb{C})$ is [[path-connected topological space|path-connected]];


\item the real general linear group $GL(n,\mathbb{R})$ is not [[path-connected topological space|path-connected]].



\end{enumerate}
\end{prop}
\begin{proof}
First observe that $GL(1,k) = k \setminus \{0\}$ has this property:

\begin{enumerate}%
\item $\mathbb{C} \setminus \{0\}$ is path-connected,


\item $\mathbb{R} \setminus \{0\} = (-\infty,0) \sqcup (0,\infty)$ is not path connected.



\end{enumerate}
Now for the general case:

\begin{enumerate}%
\item For $k = \mathbb{C}$: every invertible complex matrix is diagonalizable by a sequence of elementary matrix operations (\href{inverse+matrix#FundamentalTheoremOfLinearAlgebra}{this prop.}). Each of these is clearly path-connected to the identity. Finally the subspace of invertible [[diagonal matrices]] is the [[product topological space]] $\underset{ \{1, \cdots, n\} }{\prod} (\mathbb{C} \setminus \{0\})$ and hence connected (by \href{ProductSpaceOfConnectedSpacesIsConnected}{this prop.}, since each factor space is).


\item For $k = \mathbb{R}$: the [[determinant]] function is a [[continuous function]] $GL(n,k) \to \mathbb{R} \setminus \{0\}$, and since the [[codomain]] is not path connected, the domain cannot be either.



\end{enumerate}
\end{proof}
\begin{prop}
\label{TopologicalGeneralLinearGroup}\hypertarget{TopologicalGeneralLinearGroup}{}
\textbf{(compactness properties of the general linear group)}

The [[topological group|topological]] general linear group $GL(n,k)$ (def. \ref{GLnAsTopologicalGroup}) is

\begin{enumerate}%
\item not [[compact topological space|compact]];


\item [[locally compact topological space|locally compact]];


\item [[paracompact Hausdorff topological space|paracompact Hausdorff]].



\end{enumerate}
\end{prop}
\begin{proof}
Observe that

\begin{displaymath}
GL_n(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}
\end{displaymath}
is an [[open subset|open]] [[subspace]], since it is the [[pre-image]] under the [[determinant]] function (which is a [[polynomial]] and hence continuous, as in the proof of lemma \ref{GLnTopologicalWellDefined}) of the of the open subspace $k \setminus \{0\} \subset k$.

As an open subspace of Euclidean space, $GL(n,k)$ is not compact, by the [[Heine-Borel theorem]].

As Euclidean space is Hausdorff, and since every [[subspace]] of a Hausdorff space is again Hausdorff, so $Gl(n,k)$ is Hausdorff.

Similarly, as Euclidean space is [[locally compact topological space|locally compact]] and since an open subspace of a locally compact space is again locally compact, it follows that $GL(n,k)$ is locally compact.

From this it follows that $GL(n,k)$ is paracompact, since locally compact topological groups are paracompact (\href{topological+group#ConnectedLocallyCompactTopologicalGroupsAreSigmaCompact}{this prop.}).

\end{proof}
\hypertarget{AsALieGroup}{}\subsubsection*{{As a Lie group}}\label{AsALieGroup}

\begin{defn}
\label{}\hypertarget{}{}
Since the general linear group as a [[topological group]] (def. \ref{GLnAsTopologicalGroup}) is an [[open subspace]] of [[Euclidean space]] (proof of prop. \ref{TopologicalGeneralLinearGroup}) it inherits the structure of a [[smooth manifold]] (by \href{differentiable+manifold#OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds}{this prop.}). The group operations (being [[rational functions]]) are [[smooth functions]] with respect to this smooth structure. This is the general linear group $GL(n,\mathbb{R})$ as a \emph{[[Lie group]]}.

\end{defn}
\hypertarget{as_an_algebraic_group}{}\subsubsection*{{As an algebraic group}}\label{as_an_algebraic_group}

This group can be considered as a (quasi-affine) sub[[variety]] of the [[affine scheme|affine space]] $M_{n\times n}(k)$ of square matrices of size $n$ defined by the condition that the [[determinant]] of a matrix is nonzero. It can be also presented as an affine subvariety of the affine space $M_{n \times n}(k) \times k$ defined by the equation $\det(M)t = 1$ (where $M$ varies over the factor $M_{n \times n}(k)$ and $t$ over the factor $k$).

This [[variety]] is an algebraic $k$-group, and if $k$ is the field of real or complex numbers it is a [[Lie group]] over $k$.

One may in fact consider the set of invertible matrices over an arbitrary unital [[ring]], not necessarily commutative. Thus $GL_n: R\mapsto GL_n(R)$ becomes a [[presheaf]] of [[group]]s on $Aff=Ring^{op}$ where one can take rings either in commutative or in noncommutative sense. In the commutative case, this functor defines a [[group scheme]]; it is in fact the affine group scheme represented by the commutative ring $R = \mathbb{Z}[x_{11}, \ldots, x_{n n}, t]/(det(X)t - 1)$.

Coordinate rings of general linear groups and of special general linear groups have [[quantum group|quantum deformations]] called [[quantum linear groups]].

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

Over [[finite fields]]

\begin{itemize}%
\item [[GL(2,3)]]

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

\begin{itemize}%
\item [[translation group]], [[affine group]]


\item [[Gauss decomposition]],


\item [[special linear group]], [[projective linear group]]


\item [[orthogonal group]], [[unitary group]],


\item [[Borel subgroup]], [[flag variety]]


\item [[group of units]]


\item [[jet group]]


\item [[general linear supergroup]], [[quantum linear group]]



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

\begin{itemize}%
\item O.T. O'Meara, \emph{Lectures on Linear Groups}, Amer. Math. Soc., Providence, RI, 1974.


\item B. Parshall, J.Wang, \emph{Quantum linear groups}, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.



\end{itemize}
[[!redirects general linear group]] [[!redirects general linear groups]]



\end{document}