Inversive Ring Geometry - Affine and Projective Groups

Affine and Projective Groups

The affine group on A is generated by the mappings x → x + c and x → x u, u ∈ U.

The group of projectivities on P(A) extends the affine group by including reciprocation x → x−1 as follows:

Represent translations by U(x, 1) = U(x + c, 1).

Represent "rotations" by U(x, 1) = U(x u, 1).

Include reciprocation with U(x, y) = U(y, x).

Note that if u ∈ U, then U(1, u) = U(u−1, 1) = U(u, 1).

Here the elements of P(A) present themselves as row vectors for matrix transformation; this way subsequent transformations appear on the right, consistent with reading order. Composition of mappings is represented by matrix multiplication where the matrices are of the 2 × 2 type exhibited with entries taken from the ring A. Call the set of them M(A, 2) so the group of projectivities G(A) ⊂ M(A, 2). For instance, in G(A) one finds the projectivity

Its action is U(x, 1) = U(xu, u) = U(u−1 xu, 1).

Thus the inner automorphism x → u−1 x u of the group of units U ⊂ A arises as a projectivity on P(A) by an element of G(A). For example, when A is the ring of quaternions then one obtains rotations of 3-space. In case A is the ring of biquaternions, which has two conjugations, projectivities include the mappings which provide a group representation for the Lorentz group of special relativity.