2D 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.