Aff(3) denotes the group of three-dimensional affine transformations. They are defined by twelf parameters \((a^i_\mu)\), with \(1 \le i \le 3\), \(0 \le \mu \le 3\) in the following way:

y^{1} = |
\(\mathbf{a^1_1}\)x^{1}+ |
\(\mathbf{a^1_2}\)x^{2}+ |
\(\mathbf{a^1_3}\)x^{3}+ |
\(\mathbf{a^1_0}\) |

y^{2} = |
\(\mathbf{a^2_1}\)x^{1}+ |
\(\mathbf{a^2_2}\)x^{2}+ |
\(\mathbf{a^2_3}\)x^{3}+ |
\(\mathbf{a^2_0}\) |

y^{3} = |
\(\mathbf{a^3_1}\)x^{1}+ |
\(\mathbf{a^3_2}\)x^{2}+ |
\(\mathbf{a^3_3}\)x^{3}+ |
\(\mathbf{a^3_0}\) |

The image of a line is a line, the image of parallel lines are parallel lines. But the image of a cube is not necessarily a cube: it may be deformed, the lengths of the edges may become different, and the angles between the edges may no longer be right angles. Thus, nor lengths, nor angles are preserved by affine transformations.

An affine transformation consists of two parts:

- A linear transformation defined by the \(a^i_j\) with \(j > 0\) which preserves the origin (in the picture the point O);
- A translation defined by the \(a^i_0\) (in the picture from O to O');

red quarks | green quarks | blue quarks | leptonic sector | |
---|---|---|---|---|

1. generation | \(\mathbf{a^1_1 \sim (d,u)_r}\) | \(\mathbf{a^1_2 \sim (d,u)_g}\) | \(\mathbf{a^1_3 \sim (d,u)_b}\) | \(\mathbf{a^1_0 \sim (e,\nu_e)}\) |

2. generation | \(\mathbf{a^2_1 \sim (s, c)_r}\) | \(\mathbf{a^2_2 \sim (s, c)_g}\) | \(\mathbf{a^2_3 \sim (s, c)_b}\) | \(\mathbf{a^2_0 \sim (\mu,\nu_\mu)}\) |

3. generation | \(\mathbf{a^3_1 \sim (b,t)_r}\) | \(\mathbf{a^3_1 \sim (b,t)_g}\) | \(\mathbf{a^3_1 \sim (b,t)_b}\) | \(\mathbf{a^3_1 \sim (\tau,\nu_\tau)}\) |

This table describes the association between the electroweak doublets of the SM and coefficients of the affine group Aff(3).

In Aff(3) there is almost no restriction for the coefficients \(a^i_\mu\). The only one is that the linear transformation should be non-degenerated, thus, for the determinant we have \(|(a^i_j)| \neq 0\). In our approach, this restriction plays no role and may be ignored.