2D geometric algebra – Geometric Algebra

In two dimensions the geometric algebra, indicated as $\mathbb{G} _2$ or also $Cl_2$ as it is based on Clifford’s theory of algebras, is generated by the bases $e_1$ and $e_2$ which combine by contraction and extension, or by scalar product and external product.

$e_1 e_1 = e_1 \cdot e_1 + e_1 \wedge e_1 = 1 + 0 = 1$ (scalar)
$e_1 e_2 = e_1 \cdot e_2 + e_1 \wedge e_2 = 0 + e_1_2$ (bivector)

since geometric algebra is NOT commutative, it is also worth calculating:

$e_2 e_1 = e_2 \cdot e_1 + e_2 \wedge e_1 = 0 + e_2_1 = - e_1e_2$ (opposite bivector)
$e_1_2 e_1 = -e_2$
$e_1_2 e_2= e_1$

and the most interesting product of all:

$e_1_2 e_1_2 = e_1e_2e_1e_2 = -1$

We say interesting because it shows us two things:

the square of the bivector is -1 and therefore can be assimilated to the imaginary unit $i$ , even if for caution we will indicate it with a different symbol: $I$ uppercase
the geometric algebra $\mathbb {G} _2$ is closed, i.e. the attempt to extend the elements of degree 2 by means of an external product leads to the collapse in lower degrees

Summarizing, in two dimensions the geometric algebra is formed by the following bases:

which combine to form the multivector

$M = a + \underbrace{v_1e_1 + v_2e_2}_\text{vector} + b_1e_1e_2$

The even sub-algebra: generalized complex numbers

It is interesting to note that in $G_2$ we have an even subalgebra, formed by the elements of even degree (scalars and bivectors). It is subalgebra because the elements of even degree are a closed set with respect to addition and multiplication.

The well-known vectors and the even subalgebra, which we have called generalized complex numbers, are two subsets of 2D geometric algebra. Let’s find out what interactions exist between these two worlds.