general linear group (changes) in nLab

Context

Group Theory

Linear algebra

• Definition
  • As a topological group
    • Definition
    • Properties
  • As a Lie group
  • As an algebraic group
• Examples
• Properties
  • Representation theory
• Related concepts
• References

Context

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 2⟶k (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

k↪k 2↪k 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) \,.

Proof

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

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

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

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

This implies that the Euclidean metric topology on GL(n,ℝ)GL(n,\mathbb{R}) is equal to or finer than the subspace topology coming from Map(ℝ n,ℝ 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(ℝ)≃ℝ (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≔{B∈Mat n×n(ℝ)|∀1≤i≤n|Ae i−Be 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 i≔B 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 n∈ℕn \in \mathbb{N}

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

2. the real general linear group GL(n,ℝ)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. ℂ∖{0}\mathbb{C} \setminus \{0\} is path-connected,

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

Now for the general case:

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

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

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.).