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general linear group

Contents

Definition

Given a field kk, the general linear group GL(n,k)GL(n,k) (or GL n(k)GL_n(k)) is the group of invertible linear maps from the vector space k nk^n to itself. It may canonically be identified with the group of n×nn\times n matrices with entries in kk having nonzero determinant.

As a topological group

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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

Definition

Definition

(general linear group as a topological group)

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

The underlying group is the group of real or complex n×nn \times n matrices whose determinant is non-vanishing

GL(n,k)(AMat n×n(k)|det(A)0) GL(n,k) \;\coloneqq\; \left( A \in Mat_{n \times n}(k) \; \vert \; det(A) \neq 0 \right)

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

Mat n×n(k)k (n 2) Mat_{n \times n}(k) \simeq k^{(n^2)}

with its metric topology.

Lemma

(group operations are continuous)

Definition 1 is indeed well defined in that the group operations on GL(n,k)GL(n,k) are indeed continuous functions with respect to the given topology.

Proof

Observe that under the identification Mat n×n(k)k (n 2)Mat_{n \times n}(k) \simeq k^{(n^2)} matrix multiplication is a polynomial function

k (n 2)×k (n 2)k 2n 2k (n 2)Mat n×n(k). k^{(n^2)} \times k^{(n^2)} \simeq k^{ 2 n^2 } \longrightarrow k^{(n^2)} \simeq Mat_{n \times n}(k) \,.

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)Mat n×n(k)GL(n,k) \subset Mat_{n \times n}(k).

Definition

(stable general linear group)

The evident tower of embeddings

kk 2k 3 k \hookrightarrow k^2 \hookrightarrow k^3 \hookrightarrow \cdots

induces a corresponding tower diagram of embedding of the general linear groups (def. 1)

GL(1,k)GL(2,k)GL(3,k). GL(1,k) \hookrightarrow GL(2,k) \hookrightarrow GL(3,k) \hookrightarrow \cdots \,.

The colimit over this diagram in the category of topological group is called the stable general linear group denoted

GL(k)lim nGL(n,k). GL(k) \;\coloneqq\; \underset{\longrightarrow}{\lim}_n GL(n,k) \,.

Properties

Proposition

(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

GL(n,)Mat n×n() (n 2) GL(n,\mathbb{R}) \subset Mat_{n \times n}(\mathbb{R}) \simeq \mathbb{R}^{(n^2)}

(as in def. 1) coincides with the topology induced by regarding the general linear group as a subspace of the mapping space Maps(k n,k n)Maps(k^n, k^n),

GL(n,)Maps(k n,k n) GL(n,\mathbb{R}) \subset Maps(k^n, k^n)

i.e. the set of all continuous functions k nk nk^n \to k^n equipped with the compact-open topology.

Proof

On the one had, the universal property of the mapping space (this prop.) gives that the inclusion

GL(n,)Maps( n, n) GL(n, \mathbb{R}) \to Maps(\mathbb{R}^n, \mathbb{R}^n)

is a continuous function for GL(n,)GL(n,\mathbb{R}) equipped with the Euclidean metric topology, because this is the adjunct of the defining continuous action map

GL(n,)× n n. GL(n, \mathbb{R}) \times \mathbb{R}^n \to \mathbb{R}^n \,.

This implies that the Euclidean metric topology on GL(n,)GL(n,\mathbb{R}) is equal to or finer than the subspace topology coming from Map( n, n)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×n() (n 2)Mat_{n \times n}(\mathbb{R}) \simeq \mathbb{R}^{(n^2)}. Observe that a neighborhood base of a linear map or matrix AA consists of sets of the form

U A ϵ{BMat n×n()|1in|Ae iBe i|<ϵ} 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\}

for ϵ(0,)\epsilon \in (0,\infty).

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

U A ϵ= i=1 nV i K i, U_A^\epsilon \;=\; \bigcap_{i = 1}^n V_i^{K_i} \,,

where K i{e i}K_i \coloneqq \{e_i\} is a singleton and V iB Ae i (ϵ)V_i \coloneqq B^\circ_{A e^i}(\epsilon) is the open ball of radius ϵ\epsilon around Ae iA e^i.

Proposition

(connectedness properties of the general linear group)

For all nn \in \mathbb{N}

  1. the complex general linear group GL(n,)GL(n,\mathbb{C}) is path-connected;

  2. the real general linear group GL(n,)GL(n,\mathbb{R}) is not path-connected.

Proof

First observe that GL(1,k)=k{0}GL(1,k) = k \setminus \{0\} has this property:

  1. {0}\mathbb{C} \setminus \{0\} is path-connected,

  2. {0}=(,0)(0,)\mathbb{R} \setminus \{0\} = (-\infty,0) \sqcup (0,\infty) is not path connected.

Now for the general case:

  1. For k=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 {1,,n}({0})\underset{ \{1, \cdots, n\} }{\prod} (\mathbb{C} \setminus \{0\}) and hence connected (by this prop., since each factor space is).

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

Proposition

(compactness properties of the general linear group)

The topological general linear group GL(n,k)GL(n,k) (def. 1) is

  1. not compact;

  2. locally compact;

  3. paracompact Hausdorff.

Proof

Observe that

GL n(n,k)Mat n×n(k)k (n 2) GL_n(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}

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 1) of the of the open subspace k{0}kk \setminus \{0\} \subset k.

As an open subspace of Euclidean space, GL(n,k)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)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)GL(n,k) is locally compact.

From this it follows that GL(n,k)GL(n,k) is paracompact, since locally compact topological groups are paracompact (this prop.).

As a Lie group

Definition

Since the general linear group as a topological group (def. 1) is an open subspace of Euclidean space (proof of prop. 2) 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,)GL(n,\mathbb{R}) as a Lie group.

As an algebraic group

This group can be considered as a (quasi-affine) subvariety of the affine space M n×n(k)M_{n\times n}(k) of square matrices of size nn 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×n(k)×kM_{n \times n}(k) \times k defined by the equation det(M)t=1\det(M)t = 1 (where MM varies over the factor M n×n(k)M_{n \times n}(k) and tt over the factor kk).

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

One may in fact consider the set of invertible matrices over an arbitrary unital ring, not necessarily commutative. Thus GL n:RGL n(R)GL_n: R\mapsto GL_n(R) becomes a presheaf of groups on Aff=Ring opAff=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=[x 11,,x nn,t]/(det(X)t1)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?.

References

  • 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.

Revised on June 19, 2017 08:16:20 by Urs Schreiber (46.183.103.8)