The Albert algebra

(octonions) Definition

\(\Oc\) is the real vector space with basis \(e_0=1,e_1,\dots,e_7\) and the bilinear product with \(e_0\) the unit, \(e_k^2=-1\) (\(k\ge1\)), and

\[e_ie_{i+1}=e_{i+3},\quad e_{i+1}e_{i+3}=e_i,\quad e_{i+3}e_i=e_{i+1},\qquad e_ae_b=-e_be_a\ (a\ne b,\ a,b\ge1),\]

indices of \(e_1,\dots,e_7\) read mod 7. Conjugation \(\bar x\) negates \(e_1,\dots,e_7\). Put

\[\Nm:\Oc\to\R,\quad \Nm(x)=x\bar x=\textstyle\sum_k x_k^2,\qquad \Tr:\Oc\to\R,\quad \Tr(x)=x+\bar x=2x_0 .\]

Over \(\C\): \(\Oc_\C=\Oc\otimes_\R\C\), with \(\Nm\) and \(\Tr\) extended \(\C\)-bilinearly (so \(\Nm(x)=\sum x_k^2\) is no longer a sum of absolute values; see octonions over \(\C\)). The multiplication rule is drawn as the Fano plane.

(what survives) [Baez, §1.1, Thm 1], [SV, Ch. 1] Proved (cited)

For \(x,y,z\in\Oc_\C\):

\[\Nm(xy)=\Nm(x)\Nm(y),\qquad \Tr(xy)=\Tr(yx),\qquad \Tr((xy)z)=\Tr(x(yz))=\Tr((yz)x),\]

and \(\Oc\) is alternative (any two elements generate an associative subalgebra), but \((xy)z\ne x(yz)\) in general: \((e_1e_2)e_3=e_4e_3=-e_6\), \(e_1(e_2e_3)=e_1e_5=e_6\).

(Albert algebra) Definition

The Albert algebra \(\Jalg\) is the complex vector space of matrices

\[X=\begin{pmatrix}a&z&\bar y\\ \bar z&b&x\\ y&\bar x&c\end{pmatrix},\qquad a,b,c\in\C,\ \ x,y,z\in\Oc_\C,\]

so \(\dim_\C\Jalg=3+3\cdot8=27\), with the Jordan product

\[\jo:\Jalg\times\Jalg\to\Jalg,\qquad X\jo Y=\tfrac12(XY+YX),\]

identity \(\one=\diag(1,1,1)\) and trace \(\tr X=a+b+c\). Write \(\Jo=\{X:\tr X=0\}\), of dimension 26.

(cubic norm) [Baez, §3.4] Definition

(three special elements) Proved here

1. Real entries \(x,y,z\in\C\cdot1\): \(\Nm(x)=x^2\), \(\Tr((xy)z)=2xyz\), and \(\det X\) is the usual determinant of a symmetric matrix, e.g. \(\det\begin{pmatrix}1&.7&.3\\ .7&2&.5\\ .3&.5&3\end{pmatrix}=4.31\) by both formulas.
2. Diagonal: \(\det\diag(a,b,c)=abc\); in particular \(\det\one=1\).
3. One off-diagonal entry \(x\), \(y=z=0\): \(\det X=a\,(bc-\Nm(x))\), a scalar times a \(2\times2\) determinant.

(\(\Esix\), \(\Ffour\)) Definition

\[\Esix=\{g\in\GL(\Jalg):\det(gX)=\det X\ \text{for all }X\},\qquad \Ffour=\Aut(\Jalg)=\{g\in\GL(\Jalg):g(X\jo Y)=gX\jo gY\}.\]

(identification) Proved (cited)

1. \(\Esix\) is a simple algebraic group of type \(E_6\) and \(\Ffour\) one of type \(F_4\) [SV, Ch. 7]; \(\dim\esix=52+26=78\) [Yokota, Thm 3.2.1], and \(\Esix\) is connected, being diffeomorphic to (compact \(E_6\))\(\times\R^{78}\) [Yokota, Thm 3.14.1].
2. \(\dim\Ffour=\dim\Der(\Jalg)=52\) [CS; Baez, §4.2].
3. \(\Ffour=\{g\in\GL(\Jalg):\det(gX)=\det X,\ g\one=\one\}=\{g\in\Esix:g\one=\one\}\) [Yokota, §2.13, Definition, p. 65].

(four kinds of moves) Proved here

1. \(\sigma(a,b,c;x,y,z)=(b,c,a;y,z,x)\) lies in \(\Ffour\).
2. \(\varphi\), applying \(e_k\mapsto e_{k+1}\) (\(k=1,\dots,7\), mod 7) to each of \(x,y,z\), lies in \(\Ffour\).
3. For real \(g\in\SL_3\), the map \(X\mapsto gXg^T\) lies in \(\Esix\). For \(g=\diag(\lambda_1,\lambda_2,\lambda_3)\) with \(\lambda_1\lambda_2\lambda_3=1\) and not all \(\lambda_k^2=1\), it moves \(\one\) to \(\diag(\lambda_1^2,\lambda_2^2,\lambda_3^2)\ne\one\), so it is not in \(\Ffour\).
4. \(X\mapsto sX\) lies in \(\Esix\) only when \(s^3=1\).

Proof

Proof of (1), (2), (4)

(1) \(\sigma(X)=PXP^T\) for the permutation matrix \(P\) with \(P e_1=e_3, Pe_2=e_1,Pe_3=e_2\); real scalar matrices associate with octonion matrices, so \(\sigma(X\jo Y)=\sigma X\jo\sigma Y\). (2) The map \(e_k\mapsto e_{k+1}\) sends each relation \(e_ie_{i+1}=e_{i+3}\) to another one, so it is an automorphism of \(\Oc\); applied entrywise it preserves every matrix product. (4) \(\det(sX)=s^3\det X\).

Proof

Proof of (3): the computation where associativity is replaced

\(\SL_3(\R)\) is generated by diagonal matrices and shears \(I+tE_{ij}\), so it suffices to treat these.

Diagonal. \(a\mapsto\lambda_1^2a\), \(x\mapsto\lambda_2\lambda_3x\), \(y\mapsto\lambda_3\lambda_1y\), \(z\mapsto\lambda_1\lambda_2z\) (and cyclically). Every term of \(\det\) picks up \((\lambda_1\lambda_2\lambda_3)^2=1\); for instance \(\Tr((xy)z)\mapsto\lambda_2\lambda_3\cdot\lambda_3\lambda_1\cdot\lambda_1\lambda_2\,\Tr((xy)z)\).

Shear \(g=I+tE_{12}\) (the others follow by conjugating with \(\sigma\) and transposition). Row then column operation give

\[a'=a+t\Tr(z)+t^2b,\qquad z'=z+tb,\qquad y'=y+t\bar x,\qquad b'=b,\ c'=c,\ x'=x .\]

Using \(\Nm(p+q)=\Nm(p)+\Nm(q)+\Tr(p\bar q)\), \(\bar{\bar x}=x\) and \(x\bar x=\Nm(x)\):

\[\Nm(y')=\Nm(y)+t\Tr(yx)+t^2\Nm(x),\qquad \Nm(z')=\Nm(z)+tb\Tr(z)+t^2b^2,\] \[\Tr((xy')z')=\Tr((xy)z)+tb\Tr(xy)+t\Nm(x)\Tr(z)+2t^2b\Nm(x).\]

Collect the coefficient of \(t\) in \(\det X'\):

\[\Tr(z)(bc-\Nm(x))-b\Tr(yx)-bc\Tr(z)+b\Tr(xy)+\Nm(x)\Tr(z)=b\big(\Tr(xy)-\Tr(yx)\big)=0,\]

and of \(t^2\): \(b(bc-\Nm(x))-b\Nm(x)-b^2c+2b\Nm(x)=0\). So \(\det X'=\det X\).

The only octonionic input was \(\Tr(xy)=\Tr(yx)\) and expanding \((x\bar x)z=\Nm(x)z\), which needs only alternativity. On the \(\SL_3/\SO_3\) page the same step was “\(\det\) is multiplicative”.