linear algebra, higher linear algebra
(…)
Given a field $k$, the general linear group $GL(n,k)$ (or $GL_n(k)$) is the group of invertible linear maps from the vector space $k^n$ to itself. It may canonically be identified with the group of $n\times n$ matrices with entries in $k$ having nonzero determinant.
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Let $k = \mathbb{R}$ or $= \mathbb{C}$ be the real numbers or the complex numbers equipped with their Euclidean topology.
(general linear group as a topological group)
For $n \in \mathbb{N}$, as a topological group the general linear group $GL(n, k)$ is defined as follows.
The underlying group is the group of real or complex $n \times n$ matrices whose determinant is non-vanishing
with group operation given by matrix multiplication.
The topology on this set is the subspace topology as a subset of the Euclidean space of matrices
with its metric topology.
(group operations are continuous)
Definition is indeed well defined in that the group operations on $GL(n,k)$ are indeed continuous functions with respect to the given topology.
Observe that under the identification $Mat_{n \times n}(k) \simeq k^{(n^2)}$ matrix multiplication is a polynomial function
Similarly 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 $GL(n,k) \subset Mat_{n \times n}(k)$.
(stable general linear group)
The evident tower of embeddings
induces a corresponding tower diagram of embedding of the general linear groups (def. )
The colimit over this diagram in the category of topological group is called the stable general linear group denoted
(as a subspace of the mapping space)
The topology induced on the real general linear group when regarded as a topological subspace of Euclidean space with its metric topology
(as in def. ) coincides with the topology induced by regarding the general linear group as a subspace of the mapping space $Maps(k^n, k^n)$,
i.e. the set of all continuous functions $k^n \to k^n$ equipped with the compact-open topology.
On the one had, the universal property of the mapping space (this prop.) gives that the inclusion
is a continuous function for $GL(n,\mathbb{R})$ equipped with the Euclidean metric topology, because this is the adjunct of the defining continuous action map
This implies that the Euclidean metric topology on $GL(n,\mathbb{R})$ is equal to or 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, 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
for $\epsilon \in (0,\infty)$.
But this is also a base element for the compact-open topology, namely
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 $A e^i$.
(connectedness properties of the general linear group)
For all $n \in \mathbb{N}$
the complex general linear group $GL(n,\mathbb{C})$ is path-connected;
the real general linear group $GL(n,\mathbb{R})$ is not path-connected.
First observe that $GL(1,k) = k \setminus \{0\}$ has this property:
$\mathbb{C} \setminus \{0\}$ is path-connected,
$\mathbb{R} \setminus \{0\} = (-\infty,0) \sqcup (0,\infty)$ is not path connected.
Now for the general case:
For $k = \mathbb{C}$: every invertible complex matrix is diagonalizable by a sequence of elementary matrix operations (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 this prop., since each factor space is).
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.
(compactness properties of the general linear group)
The topological general linear group $GL(n,k)$ (def. ) is
Observe that
is an 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 ) 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 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 (this prop.).
Since the general linear group as a topological group (def. ) is an open subspace of Euclidean space (proof of prop. ) it inherits the structure of a smooth manifold (by 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 Lie group.
This group can be considered as a (quasi-affine) subvariety of the 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 groups 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 deformations called quantum linear groups?.
Over finite fields:
See at representation theory of the general linear group.
O.T. O’Meara, Lectures on Linear Groups, Amer. Math. Soc., Providence, RI, 1974.
B. Parshall, J.Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.
Last revised on October 19, 2022 at 07:24:45. See the history of this page for a list of all contributions to it.