matrix lie groups - university of toronto · 2014-01-25 · 1 matrix lie groups 2 matrix...

Matrix Lie groupsand their Lie algebras

Mahmood Alaghmandan

A project in fulfillment of the requirement for the Lie algebra course

Department of Mathematics and StatisticsUniversity of Saskatchewan

March 2012

Mahmood Alaghmandan (U of S) Matrix Lie groups and their Lie algebras March 2012 1 / 36

Special orthogonal groupAn example of Lie groups

A matrix A is orthogonal if its transpose is equal to its inverse:

Atr = A−1.

EquivalentlyAAtr = I

If for orthogonal matrix A, det(A) = 1, A is call special orthogonalmatrix.

Orthogonal and special orthogonal groups

O(n) = {A ∈ Mn(R) : Atr = A−1}

andSO(n) = {A ∈ Mn(R) : Atr = A−1, det(A) = 1}.

Special orthogonal groupAn example of Lie groups

A matrix A is orthogonal if its transpose is equal to its inverse:

Atr = A−1.

EquivalentlyAAtr = I

If for orthogonal matrix A, det(A) = 1, A is call special orthogonalmatrix.

Orthogonal and special orthogonal groups

O(n) = {A ∈ Mn(R) : Atr = A−1}

andSO(n) = {A ∈ Mn(R) : Atr = A−1, det(A) = 1}.

Example.SO(2) is the set of all matrices[

cos θ − sin θsin θ cos θ

]for different angels θ.

The rotation with angle θ.

Example.

SO(3) forms the set all of rotations in R3 which preserves the lengthand orientation.Every rotation in SO(3) fixes a unique 1-dimensional linear subspaceof R3, due to Euler’s rotation theorem. cos θ − sin θ 0

sin θ cos θ 00 0 1

is the rotation with angel θ with the axis of rotation z-axis.

Example.

Applying complex matrices!

[0 11 0

], σ2 =

[0 −ii 0

], σ3 =

[1 00 −1

]called Pauli matrices are linearly independent. So for eachx = (xi)

3i=1 ∈ R3,

x1x2x3

= x1σ1 + x2σ2 + x3σ3.

A new interpretationNote that[

eiθ/2 00 e−iθ/2

](x1σ1 + x2σ2 + x3σ3)

[eiθ/2 0

0 e−iθ/2

(x1 cos θ + x2 sin θ)σ1 + (−x1 sin θ + x2 cos θ)σ2 + x3σ3.

As clearly we can see this is the rotation that appeared in a matrix inSO(3), (0.1).

Special unitary group SU(2)Let SU(2) be the set of all matrices

[a + di −b − cib − ci a− di

]where det(A) = 1 and a,b, c,d ∈ R.

We can see that these matrices form a group, we call them specialunitary matrices. Although this is a complex counterpart of SO(2), ithas a strong relation with SO(3).

A new interpretationNote that[

eiθ/2 00 e−iθ/2

](x1σ1 + x2σ2 + x3σ3)

[eiθ/2 0

0 e−iθ/2

(x1 cos θ + x2 sin θ)σ1 + (−x1 sin θ + x2 cos θ)σ2 + x3σ3.

As clearly we can see this is the rotation that appeared in a matrix inSO(3), (0.1).

Special unitary group SU(2)Let SU(2) be the set of all matrices

[a + di −b − cib − ci a− di

]where det(A) = 1 and a,b, c,d ∈ R.

We can see that these matrices form a group, we call them specialunitary matrices. Although this is a complex counterpart of SO(2), ithas a strong relation with SO(3).

Special unitary group

A complex n × n matrix A is called unitary ifits columns are orthogonal. (Linear algebra)over Hilbert spaces H = Cn it forms a unitary operator i.e.A∗ = A−1 when A∗ is the adjoint operator of A. (FunctionalAnalysis)

Special Unitary group

SU(n) = {A ∈ Mn(C) : A∗A = AA∗ = I and det(A) = 1.}

Special unitary group

A complex n × n matrix A is called unitary ifits columns are orthogonal. (Linear algebra)over Hilbert spaces H = Cn it forms a unitary operator i.e.A∗ = A−1 when A∗ is the adjoint operator of A. (FunctionalAnalysis)

Special Unitary group

SU(n) = {A ∈ Mn(C) : A∗A = AA∗ = I and det(A) = 1.}

Outline

1 Matrix Lie groups

2 Matrix exponential

3 Lie algebra of a matrix Lie group

4 Lie algebra of some famous matrix Lie groups

5 Geometric interpretation

General linear groupGL(n,F)

General linear groupSet of all invertible matrices on field F also form a group with theregular multiplication of matrices denoted by GL(n,F).

Special linear groupAll invertible matrices A ∈ GL(n,F) s.t. det(A) = 1 denoted bySL(n,F).

All groups GL(n,R), SL(n,R), SL(n,C), SU(n), and SO(n) are allsubgroups of GL(n,C) for an appropriate n.

Topology on Mn(C)

Hilbert-Schmidt norm for matrix A = [ai,j ]i,j∈1,··· ,n is

‖A‖2 :=

n∑i,j=1

|ai,j |21/2

ConvergenceLet (Am)m be a sequence in Mn(C). Am converges to A if‖Am − A‖2 → 0.

Topology on Mn(C)

Hilbert-Schmidt norm for matrix A = [ai,j ]i,j∈1,··· ,n is

‖A‖2 :=

n∑i,j=1

|ai,j |21/2

ConvergenceLet (Am)m be a sequence in Mn(C). Am converges to A if‖Am − A‖2 → 0.

Matrix Lie group

Matrix (linear) Lie groupA matrix Lie group G is a subgroup of GL(n,C) such that if (Am)m isany convergent sequence in G, either A ∈ G or A is not invertible.

O(n),SO(n),SU(n),SL(n,R),SL(n,C),GL(n,R), and GL(n,C) are allmatrix Lie groups.

Matrix Lie group

Matrix (linear) Lie groupA matrix Lie group G is a subgroup of GL(n,C) such that if (Am)m isany convergent sequence in G, either A ∈ G or A is not invertible.

O(n),SO(n),SU(n),SL(n,R),SL(n,C),GL(n,R), and GL(n,C) are allmatrix Lie groups.

Outline

1 Matrix Lie groups

Matrix exponentialFor A ∈ Mn(C), we define

eA :=∞∑

where A0 = I.It is well defined:

‖eA‖2 ≤∞∑

1k !‖A‖k2 = e‖A‖2 <∞;

Main properties of eA

e0 = I,(

eA)∗

= e(A∗),(

= e(Atr ), and e(α+β)A = eαAeβA.

eA :=∞∑

‖eA‖2 ≤∞∑

1k !‖A‖k2 = e‖A‖2 <∞;

e0 = I,(

eA)∗

= e(A∗),(

eA :=∞∑

‖eA‖2 ≤∞∑

1k !‖A‖k2 = e‖A‖2 <∞;

e0 = I,(

eA)∗

= e(A∗),(

More properties

Matrix exponential essential properties

If AB = BA, then eA+B = eAeB = eBeA.eA is invertible and (eA)−1 = e−A.

If A is invertible, then eABA−1= AeBA−1.

Even if [A,B] 6= 0, we have

eA+B = limk→∞

1k B)k

called Lie product formula.

More properties

Matrix exponential essential properties

If AB = BA, then eA+B = eAeB = eBeA.eA is invertible and (eA)−1 = e−A.

If A is invertible, then eABA−1= AeBA−1.

Even if [A,B] 6= 0, we have

eA+B = limk→∞

1k B)k

called Lie product formula.

Calculation of the exponential of a matrix

Calculation of the exponential of nilpotent matrices is easy. Fordiagonalizable matrices also we have a good trick to simplify the

calculations. For an arbitrary matrix we can write as summation of onenilpotent matrix and one diagonalizable matrix which commute.

The relation of determinant and trace through matrix exponential

Let A ∈ Mn(C). Then det(eA) = etrace(A).

Calculation of the exponential of a matrix

Calculation of the exponential of nilpotent matrices is easy. Fordiagonalizable matrices also we have a good trick to simplify the

calculations. For an arbitrary matrix we can write as summation of onenilpotent matrix and one diagonalizable matrix which commute.

The relation of determinant and trace through matrix exponential

Let A ∈ Mn(C). Then det(eA) = etrace(A).

Derivative of matrix exponential function

Moreover A 7→ eA is a continuous map:If Am → A, then eAm → eA.So we can talk about the limit of the following matrices where ∆t → 0

e(t0+∆t)A − et0A

Therefore,

(etA)|t=t0 = lim∆t→0

∆t= Aet0A

Derivative of matrix exponential function

Moreover A 7→ eA is a continuous map:If Am → A, then eAm → eA.So we can talk about the limit of the following matrices where ∆t → 0

Therefore,

(etA)|t=t0 = lim∆t→0

∆t= Aet0A

Outline

1 Matrix Lie groups

Lie algebra of a matrix Lie group

Definition of Lie algebra of GFor G be a matrix Lie group.

g := {A ∈ Mn(C) : etA ∈ G for all t ∈ R}.

Note that:t should be real.For A ∈ g, etA ∈ G for all real t .

Definition of Lie algebra of GFor G be a matrix Lie group.

g := {A ∈ Mn(C) : etA ∈ G for all t ∈ R}.

Note that:t should be real.For A ∈ g, etA ∈ G for all real t .

Question. Is the Lie algebra of a matrix Lie group really a Lie algebra?

YES!It is really a (REAL) Lie algebra Lie bracket over matrices!

For A ∈ g, tA ∈ g for all t ∈ R.A + B ∈ g if A,B ∈ g.

Proof. If [A,B] = 0, then eA+B = eAeB.In general we apply Lie product formula, (2.1).

(1) For each k ,(

e1k Ae

1k B)k∈ G.

(2) Its limit (eA+B) must belong to G.

[A,B] ∈ g if A,B ∈ g.

Proof.G 3 etAeBe−tA = eetABe−tA

=⇒ etABe−tA ∈ g

(1) For each k ,(

e1k Ae

1k B)k∈ G.

[A,B] ∈ g if A,B ∈ g.

(1) For each k ,(

e1k Ae

1k B)k∈ G.

[A,B] ∈ g if A,B ∈ g.

Rest of the proof.But

(etABe−tA)|t=0 =ddt

(etA)|t=0Be−0A + e0A ddt

(Be−tA)|t=0 = AB − BA.

Why does it show that AB − BA ∈ g?Because:

(etABe−tA)|t=0 = lim∆t→0

e∆tABe−∆tA − B∆t

AND g is a complete vector space!

( g 3 Ak → A then limk→∞ eAk = eA. So eA ∈ G. Hence g is complete!)

[A,B] ∈ g if A,B ∈ g.

Anticommutativity and Jacobi identity are achievable because ofthe definition of the Lie bracket over matrices.

SummaryLet G be a matrix Lie group. Then g is a vector space over R.As a subspace of Mn(C), g is a complete topological space.Moreover, g is a Lie algebra with Lie bracket over matrices over field R.

In the following we will see that for a variety of matrix Lie groups, Liealgebra will actually form a complex Lie algebra. We call these groups,complex matrix Lie groups.

Anticommutativity and Jacobi identity are achievable because ofthe definition of the Lie bracket over matrices.

SummaryLet G be a matrix Lie group. Then g is a vector space over R.As a subspace of Mn(C), g is a complete topological space.Moreover, g is a Lie algebra with Lie bracket over matrices over field R.

In the following we will see that for a variety of matrix Lie groups, Liealgebra will actually form a complex Lie algebra. We call these groups,complex matrix Lie groups.

Why should we care about the Lie algebra of a matrixLie group?

They are somehow related to their matrix Lie groups!

PropositionLet G and H be two matrix Lie groups with Lie algebras g and hrespectively. Suppose that Φ : G→ H is a group homomorphism. Thethere exists an (Lie) algebra homomorphism φ : g→ h such thatΦ(eA) = eφ(A) for every A ∈ g.

The converse of This proposition is not correct in general. There aresome conditions if we impose we will see the converse. For example, ifboth groups are simply connected, every homomorphism between Liealgebras ends to a homomorphism between matrix Lie groups

They are somehow related to their matrix Lie groups!

PropositionLet G and H be two matrix Lie groups with Lie algebras g and hrespectively. Suppose that Φ : G→ H is a group homomorphism. Thethere exists an (Lie) algebra homomorphism φ : g→ h such thatΦ(eA) = eφ(A) for every A ∈ g.

The converse of This proposition is not correct in general. There aresome conditions if we impose we will see the converse. For example, ifboth groups are simply connected, every homomorphism between Liealgebras ends to a homomorphism between matrix Lie groups

Outline

1 Matrix Lie groups

The Lie algebra of general linear group GL(n,F)A is an arbitrary matrix in Mn(C), etA is invertible for all t ∈ R. SoetA ∈ GL(n,C). Hence, gl(n,C) = Mn(C).

If A is a real matrix, etA is real for all t ∈ R. On the other hand, if etA isa real matrix for every t ∈ R, we have that

A =ddt

(etA)|t=0 = lim∆t→0

e∆tA − I∆t

should be also real, because e∆tA−I∆t ∈ Mn(R) and Mn(R) is a closed

topological space. Therefore, gl(n,R) = Mn(R).

This argument will work for all real valued matrix Lie groups. It showsthat:

matrix Lie group is REAL =⇒ its Lie algebra is REAL.

A =ddt

e∆tA − I∆t

A =ddt

e∆tA − I∆t

The Lie algebra of special linear group SL(n,F)

det(etA) = et trace(A). So trace(A) = 0, if etA ∈ SL(n,C).On the other hand, if trace(A) = 0, det(etA) = et trace(A) = 1 and soetA ∈ SL(n,C).

sl(n,C) = {A ∈ Mn(C) : trace(A) = 0}.

Similarly for SL(n,R),

sl(n,R) = {A ∈ Mn(R) : trace(A) = 0}.

The Lie algebra of special linear group SL(n,F)

det(etA) = et trace(A). So trace(A) = 0, if etA ∈ SL(n,C).On the other hand, if trace(A) = 0, det(etA) = et trace(A) = 1 and soetA ∈ SL(n,C).

sl(n,C) = {A ∈ Mn(C) : trace(A) = 0}.

Similarly for SL(n,R),

sl(n,R) = {A ∈ Mn(R) : trace(A) = 0}.

Special Unitary groups.etA ∈ SU(n) if for each t ∈ R,

(etA)∗ = (etA)−1 = e−tA.

Also (etA)∗ = etA∗. Since etA∗

= e−tA, we have

A∗ =ddt

(etA∗)|t=0 =

(e−tA)|t=0 = −A.

On the other hand, if A∗ = −A

(etA)∗ = (etA∗) = e−tA = (etA)−1

which implies that A belong to the Lie algebra of SU(n). So

su(n) = {A ∈ Mn(C) : A∗ = −A and trace(A) = 0}.

Orthogonal and Special orthogonal groupWe can definitely predict that o(n) and so(n), the Lie algebras of O(n)and SO(n) respectively, are real matrices.

If Atr = −A,(etA)tr = etAtr

= e−tA = (etA)−1

for all t ∈ R. So A ∈ o(n).

Conversely, if A ∈ o(n), for each t we havee−tA = (etA)−1 = (etA)tr = etAtr

. If we get derivative from this equationfor t = 0 we have

−A =ddt

(e−tA)|t=0 =ddt

(etAtr)|t=0 = Atr .

Also note that Atr = −A forces the entries of the main diagonal to bezero, so automatically trace(A) = 0. Therefore,

o(n) = so(n) = {A ∈ Mn(R) : Atr = −A}.

Orthogonal and Special orthogonal groupWe can definitely predict that o(n) and so(n), the Lie algebras of O(n)and SO(n) respectively, are real matrices.

If Atr = −A,(etA)tr = etAtr

= e−tA = (etA)−1

for all t ∈ R. So A ∈ o(n).

Conversely, if A ∈ o(n), for each t we havee−tA = (etA)−1 = (etA)tr = etAtr

. If we get derivative from this equationfor t = 0 we have

−A =ddt

(e−tA)|t=0 =ddt

(etAtr)|t=0 = Atr .

Also note that Atr = −A forces the entries of the main diagonal to bezero, so automatically trace(A) = 0. Therefore,

o(n) = so(n) = {A ∈ Mn(R) : Atr = −A}.

Outline

1 Matrix Lie groups

Lie groups

ManifoldA manifold is a second countable Hausdorff topological space M that islocally homemorphic to an open subset of Rn.

What does it mean?It means that for each point x ∈ M there exists a neighborhood U of xand a homeomorphism φ from U onto some open set in Rn.

Definition of Lie groupA Lie group G is a group as well as a manifold so that the groupoperation i.e. (x , y) 7→ xy and the mapping inverse of the group i.e.x 7→ x−1 both are smooth functions.

Matrix Lie groups are Lie groups

GL(n,C) is a Lie groupWe know that det : Mn(C)→ C is a continuous map; therefore,GL(n,C) = det−1(C \ {0}) is an open subset of Cn (u R2n).Moreover, if we look at matrix product as a combination of coordinatewise multiplication, we can judge that this product is smooth.Consequently, the process of generating the inverse of a matrix is asmooth function. Hence, GL(n,C) is a Lie group.

Every closed subgroup of Lie group G is a Lie group. But the definitionof linear Lie group implies that every matrix Lie group indeed is a Liegroup, since it forms a closed subgroup of GL(n,C) for some n.

Matrix Lie groups are Lie groups

GL(n,C) is a Lie groupWe know that det : Mn(C)→ C is a continuous map; therefore,GL(n,C) = det−1(C \ {0}) is an open subset of Cn (u R2n).Moreover, if we look at matrix product as a combination of coordinatewise multiplication, we can judge that this product is smooth.Consequently, the process of generating the inverse of a matrix is asmooth function. Hence, GL(n,C) is a Lie group.

Every closed subgroup of Lie group G is a Lie group. But the definitionof linear Lie group implies that every matrix Lie group indeed is a Liegroup, since it forms a closed subgroup of GL(n,C) for some n.

The tangent space of Lie groupsSO(2) is nothing but the simplest manifold, a circle! For each matrixA ∈ so(2), we have that

[0 a−a 0

]for some a ∈ R.

So so(2) is bijective with R

The tangent line at the identity of the group SO(2) is bijective with so(2).

Indeed, general theory of Lie groups shows that for each Lie group G,the tangent space of G at the identity of the group is bijective with a Liealgebra.Specially for matrix Lie group G, the tangent space of G at the identityof the group is bijective with g.

References.

HALL, BRIAN C., Lie groups, Lie algebras, and representations. Anelementary introduction. Graduate Texts in Mathematics, 222.Springer-Verlag, New York, 2003.

SEPANSKI, MARK R., Compact Lie groups. Graduate Texts inMathematics, 235. Springer, New York, 2007.

STILLWELL, JOHN, Naive Lie theory. Undergraduate Texts inMathematics. Springer, New York, 2008.

WESTRA, D. B., SU(2) and SO(3),http://www.mat.univie.ac.at/ westra/so3su2.pdf.

Cartoons from http://thenoisychannel.com

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