It is common popular-math knowledge that as we extend the real numbers to complex numbers, quaternions, octonions, sedenions, $32$-nions, etc. using the Cayley-Dickson construction, we lose algebraic properties at each step such as commutativity, associativity, alternativity, etc.

We can express this phenomenon in a more illustrative way. Consider the following multilinear maps $F_k$ of degree $k$:

  • Number two: $F_0: [\:] = 1 - (-1) = 2$.
  • Imaginary part: $F_1: [x] = x - x^*$.
  • Commutator: $F_2: [x, y] = xy - yx$.
  • Associator: $F_3: [x, y, z] = (xy)z - x(yz)$.
  • $16$-inator: $F_4: [x, y, z, w] = (x(yz))w+(w(yz))x-(xy)(zw)-(wy)(zx)$.

They measure the failure of the algebra to be of characteristic two, Hermitian, commutative, associative and Moufang, respectively. All of them are well-known except for the last one, which I adapted from the quadrilinear map described in section 5 of this paper and essentially comes from a linearized Moufang identity. (Note that depending on the convention, imaginary part, commutator and associator are often defined with an extra factor of $1/2$, and in the context of complex numbers, imaginary part almost always refers to $F_1$ divided by $2i$).

Since the maps are multilinear, they are determined by their values at the basis elements $1=e_0, e_1, e_2, e_3, \ldots, e_{2^n-1}$ (under the standard labeling) in the $n$th Cayley-Dickson algebra over the reals. All the maps above with degree $k \le n$ can then be checked to satisfy the following properties:

  • (A1) $[e_a, e_b, e_c, \ldots] = m e_i$ for some $i$, where $m\in\{-2,0,2\}$.
  • (A2) $[e_a, e_b, e_c, \ldots] = 0$ whenever $a, b, c, \ldots <2^{k-1}$.
  • (A3) $[e_1, e_2, e_4, \ldots, e_{2^{k-1}}] = 2 e_{2^{k}-1}$.
  • (A4) If $[x, y, \ldots] = z$, then $[\sigma(x), \sigma(y), \ldots] = \sigma(z)$ for any algebra automorphism $\sigma$.

Properties (A2) and (A3) together imply (rather trivially) that given a Cayley-Dickson algebra $\mathbb{A}$, $F_k$ vanishes in the next algebra $CD(\mathbb{A})$ if and only if both $F_k$ and $F_{k-1}$ vanish in $\mathbb{A}$; for $k < 4$ this fact holds in more generality for any $*$-algebra (see e.g. here) and provides a more uniform way to explain why commutativity, associativity, etc. break in sequence as we iterate the Cayley-Dickson process.

But rather than talk about algebraic properties, for the purposes of this question I would like to treat the maps themselves as the objects of interest. So let these properties (and multilinearity) be the axioms that define a $2^n$-inator in a real Cayley-Dickson algebra.

The first four maps $F_0, \ldots, F_3$ are in fact uniquely determined by these axioms when $k=n$. For example, (A3) implies $[\:]=2e_0=2$; (A2) and (A3) together with linearity imply $[a e_0+b e_1] = a[e_0]+b[e_1] = a\cdot 0 + b\cdot 2e_1 = 2b e_1$; the map $F_2$ in the quaternions is completely fixed by (A2), (A3) and (A4) applied twice, with an automorphism that cyclically permutes the imaginary basis elements and with the involution $e_1 \mapsto e_2, e_2 \mapsto e_1, e_3 \mapsto -e_3$; and similarly for $F_3$ in the octonions. I haven't checked whether the $16$-inator in the sedenions is uniquely recovered from the axioms, but I suspect it's probably nonunique, since the automorphism groups of the sedenions and above are "small" relative to the size of the algebras (the groups are all isomorphic to the $14$-dimensional octonionic automorphism group ($G_2$) times a finite group, while the algebras themselves double their dimension at each step). Nevertheless, it is natural to ask if one can continue the sequence by finding nontrivial examples of $F_k$ for $k>4$; my main question deals with the smallest case $k=n=5$.

This paper implies that if a $F_5$ exists, it is necessarily not expressible in terms of multiplication and real constants alone, unlike $F_0, F_2, F_3$ and $F_4$. There could still be a closed-form expression if we allow the use of conjugation as with $F_1$, but it looks unlikely to me. The possibility of using non-real constants seems more promising, because the automorphism groups of sedenions and above do not act transitively on imaginary elements of norm $1$, so some basis elements are "more special" than others. In fact, one can define new "conjugations" $x^{(8)} = e_{8} x e_{8}$, $x^{(16)} = e_{16} x e_{16}$, etc. which can be shown to be invariant under all automorphisms of the algebra; this follows from Lemma 2.1 here. An obvious followup to my main question would be whether the $2^n$-inators can be expressed in terms of multiplication, real constants, and the conjugations $x^*, x^{(8)}, x^{(16)}, x^{(32)}, \ldots, x^{(2^n)}$. But in principle it's not guaranteed that $F_n$ will be algebraically expressible at all, if it exists.


My question is:

Is there a $32$-inator in the $32$-nions, i.e. a multilinear map $F_5: [x,y,z,w,v]$ satisfying axioms (A1)-(A4)? Followup: can it be expressed in terms of multiplication, real constants and conjugations?

Such a map would necessarily be nontrivial by (A3), and would be identically zero in any sedenion subalgebra by (A2) and (A4). Some remarks:

  • In page 12 of the paper I linked above, it is suggested that the possible existence of higher-order maps might be related to projective geometry over the field $\mathbb{F}_2$.

  • The first problem looks simple enough that it could be solved with a computer search, but the search space seems to be quite large at first glance. The obvious upper bound on possible cases to check, taking into account only multilinearity and (A1), would be $(2\cdot 32 + 1)^{32^5} = 65^{2^{25}} \lesssim 2^{2^{28}}$. Perhaps some clever argument could reduce this bound to a more manageable number.

  • 2
    $\begingroup$ I see how $[w,x,y,z]$ is a polarization of the Moufang identity $x(yz)x=(xy)(zx)$, but I don't see how it's related to the other two Moufang identities. Are the three identities equivalent in any alternative algebra? $\endgroup$
    – coiso
    Commented Oct 12, 2023 at 19:12
  • $\begingroup$ In regards to conjugation. One way to view it is as an invertible operator on a vector of reals, that can only change the signs and the order of elements but not the magnitudes of the elements themselves. For an $n$ vector there are $2^n n!$ such conjugation operators. That gives a lot of options to work with. $\endgroup$ Commented Nov 4, 2023 at 0:27


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