1
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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:

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

$\endgroup$
2
  • $\begingroup$ Many thanks. Is there a reference to this formula? $\endgroup$ – makt Apr 18 '19 at 21:25
  • 1
    $\begingroup$ @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. $\endgroup$ – Libli Apr 19 '19 at 7:32
2
$\begingroup$

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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.