Determinants in Jordan algebras of Euclidean type

Question

As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type. Each (some?) of such algebras admits a cone of positive definite elements, and there is a version of determinant in terms of which one can formulate a version of the Sylvester criterion of positive definiteness. E.g. basic examples are real symmetric and complex Hermitian matrices, and all the relevant notions are standard. A less standard example is quaternionic Hermitian matrices, where positivity is defined as in the real and complex cases, but the determinant is not so well known - it is called the Moore determinant.

I am interested in the exceptional example of such an algebra - Hermitian 3x3 matrices with octonionic entries. I would like to have a reference to the explicit formula for the determinant.

Answer 1

We denote by $J_{3}(\mathbb{O})$ be the space: $$ J_{3}(\mathbb{O}) = \left\{ \begin{pmatrix} \lambda_1 & a_1 & \overline{a_2} \\ \overline{a_1} & \lambda_2 & a_3 \\ a_2 & \overline{a_3} & \lambda_3 \end{pmatrix}, \ a_i \in \mathbb{O}, \ \lambda_i \in \mathbb{C} \right\}.$$ For any $A \in J_3(\mathbb{O})$, we denote by $\det_{\mathbb{O}}(A)$ the number:

$${\det}_{\mathbb{O}}(A) = \lambda_1 \lambda_2 \lambda_3 + a_1(a_3a_2) + \big((\overline{a_2})( \overline{a_3}) \big)(\overline{a_1}) - \lambda_2 a_2 \overline{a_2} - \lambda_1 a_3 \overline{a_3} - \lambda_3 a_1 \overline{a_1}.$$ Hence for any $A \in J_3(\mathbb{O})$, we have:

$${\det}_{\mathbb{O}}(A) = \lambda_1 \lambda_2 \lambda_3 + 2\mathfrak{Re}(a_1a_3a_2) - \lambda_2 \|a_2\|^2 - \lambda_1 \|a_3\|^2 - \lambda_3 \|a_1\|^2.$$

Answer 2

Makt wrote:

As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type.

Yes, in this paper

• Pascual Jordan, John von Neumann and Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29–64.

the authors proved:

Theorem. Every finite-dimensional Euclidean Jordan algebra is isomorphic to a direct sum of simple ones, and these are all the simple ones:

• $\mathfrak{h}_n(\mathbb{R})$: $n \times n$ self-adjoint real matrices with $a \circ b = \frac{1}{2}(ab + ba)$.

• $\mathfrak{h}_n(\mathbb{C})$: $n \times n$ self-adjoint complex matrices with $a \circ b = \frac{1}{2}(ab + ba)$.

• $\mathfrak{h}_n(\mathbb{H})$: $n \times n$ self-adjoint quaternionic matrices with $a \circ b = \frac{1}{2}(ab + ba)$.

• $\mathfrak{h}_n(\mathbb{O})$: $n \times n$ self-adjoint octonionic matrices with $a \circ b = \frac{1}{2}(ab + ba)$ where $n \le 3$.

• The spin factors $\mathbb{R}^n \oplus \mathbb{R}$, with $$ (x,t) \circ(x', t') = (t x' + t' x, x \cdot x' + tt'). $$

Every Euclidean Jordan algebra comes automatically with a cone of positive definite elements, a determinant function, a trace function, and much more. A good place to learn about these is here:

• Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, Oxford, 1994.

The determinant on $\mathfrak{h}_3(\mathbb{O})$ is given by

$$ \det \left( \begin{array}{ccc} \alpha & z & y^* \\ z^* & \beta & x \\ y & x^* & \gamma \end{array} \right) = \alpha \beta \gamma - (\alpha \|x\|^2 + \beta \|y\|^2 + \gamma \|z\|^2) + 2 \mathrm{Re}(xyz) $$

where $\alpha, \beta, \gamma \in \mathbb{R}$ and $x,y,z \in \mathbb{O}$. You can check that

$$ \mathrm{Re}((xy)z) = \mathrm{Re}(x(yz)) $$

for any octonions $x,y,z$, so this justifies us in writing either one as $\mathrm{Re}(xyz)$.

For more, including more references, try

• John Baez, The octonions, Section 3.4: $\mathbb{O}\mathrm{P}^2$ and the exceptional Jordan algebra, Bull. Amer. Math. Soc. 39 (2002), 145–205.