Special Orthogonal Group and SU(2) Algebra in Three-Dimensional Rotations, Slides of Classical Mechanics

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Rotation

Group

Metric

Preserving

•^

A^

metric

is

used

to

measure

the

distance

in

a space.

-^

Euclidean

space

is

delta

•^

An

orthogonal

transformation

preserves

the

metric.

-^

Inverse

is

transpose

-^

Determinant

squared

is

1

•^

The

special

orthogonal

transformation

has

determinant

of

x^1

x^2

x^3 x^1

x^2

x^3^ ji ij j i ij^

u u

u u g

 

mn

j n i m ij^

g

S S g^

^

^

^

T^



^

j n

i^ m

S

S

SO(3)

Algebra

•^

The

Lie

algebra

comes

from

a

parameterized

curve.

-^

R (

)^

^

SO(3,

R )

-^

R (0)

=^

I

•^

The

elements

a

must

be

antisymmetric.^ –

Three

free

parameters

in

general

form

^

^

T

T

T^

^

dR d R R dR d

RR

d d

T^

^

a a

1

2

1

3

2

a

(^0) 



dR  d

a

Algebra

Basis

•^

The

elements

can

be

written

in

general

form.

-^

Use

three

parameters

as

coordinates

-^

Basis

of

three

matrices

a^1

^   

a^2

a^3

3 3 2 2 1 1 a

a

a

a

^

^

^

(^21) a

^

^

^

(^2)  1

(^41)

a

a^

^

^

1

(^31)

a

a^

Commutator

•^

The

structure

of

a

Lie

algebra

is

found

through

the

commutator.

-^

Basis

elements

squared

commute

•^

This

will

be

true

in

any

other

representation

of

the

Lie

group. k ijk j i^

a

a a

]

[

] , [ ] , [ ] , )

[(

2

i k ijk k i ijk

i j i j i i j i

a a

a a

a a a a a a a a

Special

Unitary

•^

If^

a^ space

is

complex

‐valued

metric

preservation

requires

Hermitian

matrices

-^

Inverse

is

complex

conjugate

-^

Determinant

squared

is

1

•^

The

special

unitary

transformation

has

determinant

of

•^

SU(2)

has

dimension

x^1

x^2

x^3 x^1

x^2

x^3^ ji ij j i ij^

u u

u u g

 

mn

j n i m ij^

g

S S g^

^

^

^

*^



^

j n

i^ m

S

S

Homomorphism

•^

The

SU(2)

and

SO(3)

groups

have

the

same

algebra.^ –

Isomorphic

Lie

algebras

•^

The

groups

themselves

are

not

isomorphic.

to

homomorphism

•^

SU(2)

is

simply

connected

and

is

the

universal

covering

group

for

the

Lie

algebra.

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