I'm a math student and I'm taking an algebra course. The professor introduced us to the field of quaternions ($\mathbb{Q}$), I became very curious about the topic and I saw that in addition to quaternions there are also two other fields called "octonions" ($\mathbb{O}$) and "sedenions" ($\mathbb{S}$).
The peculiarity of these fields lies in the fact that as the size increases, the product loses properties.
For octonions I saw that it is possible to define 480 different types of products for sedenions (as I understand) only 1. For convenience I therefore consider the product of the octets as the subcase in which it is a subspace of the sedenions
I wanted to know if there was a way to determine, given the indices of two sessions, the result. I noticed this:
- $e_1 e_k=\begin{cases}\pm e_{k+1}&k=2N\\ \pm e_{k-1}&k=2N+1\end{cases}$
- $e_2 e_k=\begin{cases}\pm e_{k+2}&k=2N+1\\ \pm e_{k-2}&k=2N\end{cases}$
I made this picture where you could clearly see this pattern:
In general they have a regular algorithmic trend that resembles a kind of "bouncing ball" (pass me the term)
And I wanted to know if it was possible to obtain the result of this product a priori, perhaps also considering algebras with larger dimensions (such as 32, 64, 128 etc...). For octonions the "antisymmetric tensor" is used, but I have not found the algorithm in which it is created.