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Rotation Group

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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Rotation Group

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  1. Rotation Group

  2. 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

  3. 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

  4. 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

  5. The elements can be written in general form. Use three parameters as coordinates Basis of three matrices Algebra Basis

  6. The one-parameter subgroups can be found through exponentiation. These are rotations about the coordinate axes. Subgroups

  7. 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

  8. 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

  9. The Lie algebra follows as it did in SO(3,R). The elements b must be Hermitian. Three free parameters in general form The basis elements commute as with SO(3). SU(2) Algebra

  10. Homomorphism • 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. next

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