# Determinants in Jordan algebras of Euclidean type

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.

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:

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

• Many thanks. Is there a reference to this formula?
– asv
Apr 18, 2019 at 21:25
• @orbits : well this is just the "classical" formula for a 3*3 matrix. The remarkable fact is that it is real. I don't know who first made this observation. It's probably very old and I have no idea how to provide an acurate reference. Apr 19, 2019 at 7:32

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.