Let $\mathcal{C}$ denote the category of unital associative finite-dimensional division $\mathbb{R}$-algebras. (As a full subcategory of that of unital $\mathbb{R}$-algebras.)
For $A\in \mathcal{C}$ of real dimension $0<d<\infty$, let us define the function $$ \begin{matrix} f_A &:&A & \rightarrow & \mathbb{R}_{\geq0} \\ && a & \mapsto & \bigl| \mathrm{det}\bigl( (x \mapsto ax)\in \mathrm{End}_{\mathbb{R}\textrm{-Mod}}(A) \bigr) \bigr|^{1/d} \end{matrix} $$ sending $a \in A$ to the $1/d$ power of the absolute value of the determinant of the left-$a$-multiplication self-map on $A$, viewed as an $\mathbb{R}$-linear self-map (so computed using an $\mathbb{R}$-basis for $A$).
Furthermore, $f_A$ is sort of "iso-natural" in that it's preserved under pre-composition by isomorphisms in $\mathcal{C}$; i.e. if $\psi : A' \xrightarrow{\simeq} A$ is a given isomorphism between objects in $\mathcal{C}$, then $f_A \circ \psi = f_{A'}$.
Some other immediate properties of $f_A$:
It is nonnegative-valued, and $f_A(a) = 0$ implies $a=0$ for $a\in A$. (The 2nd part uses that $A$ is a division alg.)
It satisfies $f_A(ta) = |t|f_A(a)$ for $a \in A, t \in \mathbb{R}$.
(Note these are the first two requirements for a norm; the remaining one is the triangle inequality.)
Next, by direct computation, we observe:
For $A \in \{ \mathbb{R} , \mathbb{C} , \mathbb{H}\}$ (using some fixed standard constructions for these), it seems the function $f$ does give a norm which actually agrees with the one induced by the ($\mathbb{R}$-bilinear) inner product having the following as orthonormal $\mathbb{R}$-bases:
- $\{ 1\}$ for $\mathbb{R}$,
- $\{ 1,i \}$ for $\mathbb{C}$,
- $\{ 1,i,j,k \}$ for $\mathbb{H}$.
Thus:
If we take it as already known that $\mathbb{R},\mathbb{C},\mathbb{H}$ exhausts all the isomorphism classes of $\mathcal{C}$ (which is a well-known result), then by the above (including "iso-naturality" of $f_A$ mentioned above), we can conclude $f_A$ is always a norm;
in particular $f_A$ always satisfies the triangle inequality.
My first question (assuming my reasoning above is sound) is:
- Is there a way to see more directly that $f_A$ satisfies the triangle inequality, without needing to know beforehand that any $A \in \mathcal{C}$ is isomorphic to one of $\mathbb{R},\mathbb{C},\mathbb{H}$?
Next:
If we relax the "associative" condition for $A \in \mathcal{C}$ to "alternative", then it is known that the isomorphism classes of $\mathcal{C}$ are then $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$. (Cf. Proposition 3.1 in https://ncatlab.org/nlab/show/normed+division+algebra; although that article is about normed division algebras, Prop 3.1 focuses on alternative div algebras but without a choice of norm.)
So my second question is:
- Is $f_{\mathbb{O}}$ also a norm, on the octonion space $\mathbb{O}$? For example, maybe it is the norm induced by the inner product having as orthonormal basis $e_0,e_1,\ldots,e_7$, using the notation at https://en.wikipedia.org/wiki/Octonion.
(I tried working through the computation for $\mathbb{O}$ but I found it got somewhat messy and I couldn't figure out how to make it more tractable.)
Edit: added the 2nd question about alternative division algebras, and the octonions $\mathbb{O}$.