Matrix groups and Lie groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 19 November 2010

Matrices

M_{n} (ℂ) = \{n \times n matrices with entries in ℂ\}

with product matrix multiplication is not a group because, if

g = (\begin{matrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})

then there does not exist

g^{- 1}

with

g g^{- 1} = g^{- 1} g = 1

The general linear group is ${G L}_{n} (ℂ) = \{g \in M_{n} (ℂ) ∣ there exists g^{- 1} with g g^{- 1} = g^{- 1} g = 1\} .$ The special linear group is is ${S L}_{n} (ℂ) = ker ϕ$ where $ϕ : {G L}_{n} (ℂ) \to {G L}_{1} (ℂ)$ is the homomorphism given by $ϕ (g) = det (g) .$

Orthogonal, Symplectic and Unitary groups

The orthogonal group is

O_{n} = \{g \in {G L}_{n} (ℂ) ∣ g \cdot 1 \cdot g^{t} = 1\} .

The symplectic group is

{S p}_{n} = \{g \in {G L}_{n} (ℂ) ∣ g \cdot J \cdot g^{t} = 1\} where J = (\begin{matrix} 0 & \begin{matrix} 1 & 0 \\ ⋱ \\ 0 & 1 \end{matrix} \\ \begin{matrix} - 1 & 0 \\ ⋱ \\ 0 & - 1 \end{matrix} & 0 \end{matrix})

The conjugation involution on

ℂ

- : ℂ \to ℂ

given by

\overline{x + y i} = x - y i .

The unitary group is

U_{n} (ℂ) = \{g \in {G L}_{n} (ℂ) ∣ g \cdot 1 \cdot {\overline{g}}^{t}\},

where

{\overline{g}}^{t} = \overline{{(\begin{matrix} g_{11} & \dots & g_{1 n} \\ ⋮ & ⋮ \\ g_{n 1} & \dots & g_{n n} \end{matrix})}^{t}} = \overline{(\begin{matrix} g_{11} & \dots & g_{n 1} \\ ⋮ & ⋮ \\ g_{1 n} & \dots & g_{n n} \end{matrix})} = (\begin{matrix} {\overline{g}}_{11} & \dots & {\overline{g}}_{n 1} \\ ⋮ & ⋮ \\ {\overline{g}}_{1 n} & \dots & {\overline{g}}_{n n} \end{matrix}) .

Linear transformations

Let

V

be a vector space.

End (V) = \{linear transformations g : V \to V\} .

b_{1}, \dots, b_{n}

is a basis of

V

then

\begin{matrix} End (V) & \tilde{\to} & M_{n} (ℂ) \\ g & \mapsto & (\begin{matrix} g_{11} & \dots & g_{1 n} \\ ⋮ & ⋮ \\ g_{n 1} & \dots & g_{n n} \end{matrix}) \end{matrix}

where

g b_{i} = g_{1 i} b_{1} + \dots + g_{n i} b_{n} .

A \{\begin{matrix} symmetric form \\ skew-symmetric form \\ Hermitian form \end{matrix} on V is a map \begin{matrix} ⟨, ⟩ : & V \otimes V & \to & ℂ \\ (v_{1}, v_{2}) & \mapsto & ⟨ v_{1}, v_{2} ⟩ \end{matrix}

such that

If $a_{1}, a_{2} \in ℂ$ and $v_{1}, v_{2}, v_{3} \in V$ then $⟨ a_{1} v_{1} + a_{2} v_{2}, v_{3} ⟩ = a_{1} ⟨ v_{1}, v_{3} ⟩ + a_{2} ⟨ v_{2}, v_{3} ⟩,$
If $v_{1}, v_{2} \in V$ then $\{\begin{array}{l} ⟨ v_{2}, v_{1} ⟩ = ⟨ v_{1}, v_{2} ⟩ . \\ ⟨ v_{2}, v_{1} ⟩ = - ⟨ v_{1}, v_{2} ⟩ . \\ ⟨ v_{2}, v_{1} ⟩ = \overline{⟨ v_{1}, v_{2} ⟩} . \end{array}$

Linear groups

The general linear group is

G L (V) = \{g \in End (V) ∣ there exists g^{- 1} with g^{- 1} g = g g^{- 1} = 1\} .

Let

⟨, ⟩

be a symmetric form on

V

. The orthogonal group is

O (V) \{g \in G L (V) ∣ ⟨ g v_{1}, g v_{2} ⟩= ⟨ v_{1}, v_{2} ⟩ for all v_{1}, v_{2} \in V\} .

Let

⟨, ⟩

be a skew-symmetric form on

V

. The symplectic group is

S p (V) \{g \in G L (V) ∣ ⟨ g v_{1}, g v_{2} ⟩= ⟨ v_{1}, v_{2} ⟩ for all v_{1}, v_{2} \in V\} .

Let

⟨, ⟩

be a Hermitian form on

V

. The unitary group is

U (V) \{g \in G L (V) ∣ ⟨ g v_{1}, g v_{2} ⟩= ⟨ v_{1}, v_{2} ⟩ for all v_{1}, v_{2} \in V\} .

Exponential maps

satisfies

e^{x} e^{y} = e^{x + y} and {\frac{d e^{x}}{d x}|}_{x = 0} = 1 .

e^{x}

is a polynomial then

e^{x} = 1 + x + \frac{1}{2} x^{2} + \frac{1}{3!} x^{3} + \dots .

So we have an exponential map

\begin{matrix} e : & ℂ & \to & ℂ ∖ \{0\} \\ r + i θ & \mapsto & e^{r} e^{θ} \end{matrix}

which is really a map

\begin{matrix} e : & M_{1} (ℂ) & \to & {G L}_{1} (ℂ) \\ x & \mapsto & e^{x} \end{matrix}

which is a special case of

\begin{matrix} e : & M_{n} (ℂ) & \to & {G L}_{n} (ℂ) \\ x & \mapsto & e^{x} . \end{matrix}

Example: $e^{(\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix})} = (\begin{matrix} e & 0 \\ 0 & e^{2} \end{matrix}) and e^{(\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix})} = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}) .$

Lie groups

Conceptually, a Lie group is a group

G

with an exponential map.

$M_{n} (ℂ)$ is a vector space with basis $\{E_{i j} ∣ 1 \leq i \leq j \leq n\} where E_{i j} = \begin{matrix} \begin{matrix} j^{th} \end{matrix} \\ \begin{matrix} i^{th} \end{matrix} & (\begin{matrix} 0 \\ ⋱ & 1 \\ 0 & 0 \end{matrix}) \end{matrix}$

Example: $(\begin{matrix} 0 & 5 \\ 2 & 1 \end{matrix}) = 5 E_{12} + 2 E_{21} + E_{22} .$

Let $G$ be a group of matrices (so $G \subseteq {G L}_{n} (ℂ)$ ). If it turns out that $𝔤 = log (G) = \{x \in M_{n} (ℂ) ∣ e^{x} \in G\}$ is a subspace of $M_{n} (ℂ)$ , then $e : 𝔤 \to G$ is an exponential map for $G$ .

𝔤

is the Lie algebra of

G

One parameter subgroups

Let

G

be a Lie group. A one parameter subgroup of

G

is a map

γ : ℂ \to G such that γ (t_{1} + t_{2}) = γ (t_{1}) γ (t_{2}) .

Example: Let $x \in M_{n} (ℂ)$ . Then $\begin{matrix} γ_{x} : & ℂ & \to & {G L}_{n} (ℂ) \\ t & \mapsto & e^{t x} \end{matrix}$ is a one-parameter subgroup of ${G L}_{n} (ℂ)$ .

Example: Let $x \in 𝔤$ . Then $\begin{matrix} γ_{x} : & ℂ & \to & G \\ t & \mapsto & e^{t x} \end{matrix}$ is a one-parameter subgroup of $G$ . $\begin{matrix} 𝔤 & \tilde{\leftrightarrow} & \{one parameter subgroups of G\} \\ x & \mapsto & γ_{x} \end{matrix}$

A one parameter subgroup of ${G L}_{n} (ℂ)$ is a homomorphism $ℂ \to {G L}_{n} (ℂ) .$

${S L}_{2}$ embeddings

{S L}_{2} (ℂ) = \{g \in {G L}_{2} ∣ det (g) = 1\} .

The group

{S L}_{2} (ℂ)

is generated by

(\begin{matrix} 1 & t \\ 0 & 1 \end{matrix}) and (\begin{matrix} 1 & 0 \\ s & 1 \end{matrix}) where t, s \in ℂ .

For each

1 \leq i < j \leq n

\begin{matrix} ϕ_{i j} : & {S L}_{2} (ℂ) & \to & {G L}_{2} (ℂ) \\ (\begin{matrix} 1 & t \\ 0 & 1 \end{matrix}) & \mapsto & (\begin{matrix} 1 & 0 \\ ⋱ & t \\ 0 & 1 \end{matrix}) \\ (\begin{matrix} 1 & 0 \\ s & 1 \end{matrix}) & \mapsto & (\begin{matrix} 1 & 0 \\ s & ⋱ \\ 0 & 1 \end{matrix}) \end{matrix}

is an embedding of

S L_{2} (ℂ)

G L_{n} (ℂ)

. A maximal torus in

G L_{n} (ℂ)

is an embedding of the group

T = \{(\begin{matrix} * & 0 \\ ⋱ \\ 0 & * \end{matrix})\} into {G L}_{n} (ℂ) .

A Borel subgroup in

G L_{n} (ℂ)

is an embedding of the group

B = \{(\begin{matrix} * & * \\ ⋱ \\ 0 & * \end{matrix})\} into {G L}_{n} (ℂ) .

Weyl groups

Let

G

be a Lie group. Let

T

be a maximul torus in

G

The normalizer of $T$ in $G$ is the

largest subgroup

N

G

such that

T

is normal in

N

. The Weyl group of

G

W = N / T .

Example: The normalizer of $T$ in ${G L}_{n} (ℂ)$ is $N = \{\begin{array}{l} n \times n matrices with \\ (a) exactly one nonzero entry in each row and each column \\ (b) the nonzero entries are in {G L}_{1} (ℂ) . \end{array}\}$ The map that changes nonzero entries to $1$ is a homomorphism $ϕ : N \to S_{n} with ker ϕ = T .$

Example:

So the symmetric group

S_{n}

is the Weyl group of

{G L}_{n} (ℂ)

Generators and relations for Weyl groups

Let

$S_{n}$ is generated by $s_{1}, s_{2}, \dots, s_{n - 1} .$

		Proof.
	Stretch $w$ . $□$

The symmetric group is given by generators

s_{1}, \dots, s_{n - 1}

and relations

Elementary matrices

\{E_{i j} ∣ 1 \leq i, j \leq n\}

is a basis of

M_{n} (ℂ)

and

\begin{matrix} x_{i j} : & ℂ & \to & {G L}_{n} (ℂ) \\ t & \mapsto & e^{t E_{i j}} \end{matrix}

is a one parameter subgroup.

\begin{array}{rcl} x_{i j} (t) & = & 1 + t E_{i j} + \frac{1}{2} t^{2} E_{i j}^{2} + \frac{1}{3!} t^{3} E_{i j}^{3} + \dots = 1 + t E_{i j} + 0 + 0 + \dots \\ = & \begin{matrix} \begin{matrix} j^{th} \end{matrix} \\ \begin{matrix} i^{th} \end{matrix} & (\begin{matrix} 1 \\ ⋱ & t \\ 0 & 1 \end{matrix}) . \end{matrix} \end{array}

The elementary matrices in

{G L}_{n} (ℂ)

are

x_{i j} (t), 1 \leq i, j \leq n, t \in ℂ .

g \in {G L}_{n} (ℂ)

then

\begin{array}{l} x_{i j} (t) \cdot g = g except 5 \cdot (j^{th} row) is added to the i^{th} row, and \\ g \cdot x_{i j} (t) = g except 5 \cdot (i^{th} column) is added to the j^{th} column. \end{array}

So elementary matrices in

{G L}_{n} (ℂ)

are row and column operations.

Generators and relations for Lie groups

${G L}_{n} (ℂ)$ is generated by $x_{i j} (t), 1 \leq i, j \leq n, t \in ℂ .$

		Proof.
	Let $g \in {G L}_{n} (ℂ)$ . Finding $g^{- 1}$ by row reduction is equivalent to finding $x_{i_{l} j_{l}} (t_{l}) \dots x_{i_{1} j_{1}} (t_{1}) g = 1 .$ Since $x_{i j} {(t)}^{- 1} = x_{i j} (- t),$ $g = x_{i_{1} j_{1}} (- t_{1}) \dots x_{i_{l} j_{l}} (- t_{l}) is a product of elementary matrices.$ $□$

The relations are

x_{i j} (t_{1}) x_{i j} (t_{2}) = x_{i j} (t_{1} + t_{2}), h_{i} (t_{1}) h_{i} (t_{2}) = h_{i} (t_{1} t_{2}),

x_{i j} (t) x_{k l} (s) = x_{k l} (s) x_{i j} (t), x_{i j} (t) x_{j l} (s) = x_{j l} (s) x_{i j} (t) x_{i l} (s t),

x_{i j} (t) x_{j i} (- t^{- 1}) x_{i j} (t) = s_{i j} h_{i} (t) h_{j} (- t^{- 1}),

w x_{i j} (t) w^{- 1} = x_{w (i) w (j)} (t), for w \in S_{n},

h x_{i j} (t) = x_{i j} (h_{i} t h_{j}^{- 1}), for h = (\begin{matrix} h_{1} & 0 \\ ⋱ \\ 0 & h_{n} \end{matrix}) \in T,

where

s_{i j} = \begin{matrix} \begin{matrix} i^{th} & j^{th} \end{matrix} \\ (\begin{matrix} 1 \\ ⋱ \\ 1 \\ 0 & 1 \\ 1 \\ ⋱ \\ 1 \\ 1 & 0 \\ 1 \\ ⋱ \\ 1 \end{matrix}) . \end{matrix}

References

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

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