clifford algebra | The Hitchhiker's Guide to Computable Mathematics

The split-complex numbers are a special case of Clifford Algebra. This can be seen as follows. Let $X = \mathbb{R}^1$ . Let $e_1 = \langle a \rangle$ with $a \not = 0$ be a basis element of $\mathbb{R}^1$ . Then the Clifford Algebra $\mathcal{C}(\mathbb{R}^1)$ is defined as follows. The elements of the Clifford algebra are $\alpha + \beta e_1$ . Setting the rule for the Clifford product $e_1 \vee e_1$ as

(1) $\begin{equation*}e_1 \vee e_1 = 1\end{equation*}$

yields

$(\alpha_1 + \beta_1 e_1) \vee (\alpha_2 + \beta_2 e_1) = (\alpha_1 \alpha_2 + \beta_1 \beta_2) + (\alpha_1 \beta_2 + \beta_1 \alpha_2) e_1$

The isomorphism is $1 \mapsto 1$ , $e_1 \mapsto j$ . On the other hand, setting the rule to $e_1 \vee e_1 = 0$ or $e_1 \vee e_1 = -1$ in (1) yields a correspondence with dual-numbers or complex numbers respectively. I am planning to expand on this fact in one of the subsequent posts.

The Hitchhiker's Guide to Computable Mathematics

Konrad Burnik

Tag Archives: clifford algebra

Split-Complex Numbers as a Clifford Algebra