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Rotation Group. 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 +1. Metric Preserving. x 3. x 2. x 1.

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Metric preserving l.jpg

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 +1.

Metric Preserving

x3

x2

x1


Special orthogonal group l.jpg

Group definitions: A, B G

Closure: AB G

Associative: A(BC) = (AB)C

Identity: 1A = A1 = A

Inverse: A-1 = AA-1 = 1

Rotation matrices form a group.

Inverse is the transpose

Identity is d or I

Associativity from matrix multiplication

Closure from orthogonality

For three dimensional rotations the group is SO(3,R).

Special Orthogonal Group


So 3 algebra l.jpg

The Lie algebra comes from a parameterized curve.

R(e)  SO(3,R)

R(0) = I

The elements a must be antisymmetric.

Three free parameters in general form

SO(3) Algebra


Algebra basis l.jpg

The elements can be written in general form.

Use three parameters as coordinates

Basis of three matrices

Algebra Basis


Subgroups l.jpg

The one-parameter subgroups can be found through exponentiation.

These are rotations about the coordinate axes.

Subgroups


Commutator l.jpg

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.

Commutator


Special unitary l.jpg

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 +1.

SU(2) has dimension 3

Special Unitary

x3

x2

x1


Su 2 algebra l.jpg

The Lie algebra follows as it did in SO(3, Hermitian matricesR).

The elements b must be Hermitian.

Three free parameters in general form

The basis elements commute as with SO(3).

SU(2) Algebra


Homomorphism l.jpg
Homomorphism Hermitian matrices

  • The SU(2) and SO(3) groups have the same algebra.

    • Isomorphic Lie algebras

  • The groups themselves are not isomorphic.

    • 2 to 1 homomorphism

  • SU(2) is simply connected and is the universal covering group for the Lie algebra.

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