Contents

group theory

# Contents

## Definition

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.

### As a topological group

Let $k = \mathbb{R}$ 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 $n \in \mathbb{N}$, 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

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

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

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

with its metric topology.

###### Lemma

(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.

###### Proof

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

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

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

###### Definition

(stable general linear group)

The evident tower of embeddings

$k \hookrightarrow k^2 \hookrightarrow k^3 \hookrightarrow \cdots$

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

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

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

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

#### 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

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

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

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

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

###### Proof

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

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

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

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

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

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

for $\epsilon \in (0,\infty)$.

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

$U_A^\epsilon \;=\; \bigcap_{i = 1}^n V_i^{K_i} \,,$

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 $A e^i$.

###### Proposition

(connectedness properties of the general linear group)

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

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

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

###### Proof

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

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

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

Now for the general case:

1. 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).

2. 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.

###### Proposition

(compactness properties of the general linear group)

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

###### Proof

Observe that

$GL_n(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}$

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 ) 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.).

### As a Lie group

###### Definition

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.

### As an algebraic 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?.

## Examples

Over finite fields:

## Properties

### Representation theory

• 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.