Complex-qutrit Jordan algebra

Definition

The complex-qutrit Jordan algebra is

\mathfrak h_3(\mathbb C) =\{X\in M_3(\mathbb C):X^*=X\}, \qquad X\circ Y=\frac12(XY+YX).

It is the nine-dimensional real Euclidean Jordan algebra of observables of a complex three-level quantum system.

Structure

The unit is $I_3$ , the rank is $3$ , and the Jordan trace and determinant are the ordinary matrix trace and determinant. The spectral theorem for Hermitian matrices is precisely the Euclidean-Jordan spectral theorem here. The algebra is simple and special. Its positive trace-one elements are the qutrit density matrices.

Symmetries

Unitary conjugation $X\mapsto UXU^*$ defines a Jordan automorphism. Scalar unitaries act trivially, so the connected automorphism group is $PU(3)$ . The full real Jordan automorphism group also has a disconnected component generated by complex conjugation; connected and full stabilizers must therefore not be conflated.

Position inside the Albert algebra

Choosing a copy $\mathbb C\subset\mathbb O$ gives a unital inclusion

\mathfrak h_3(\mathbb C)\subset\mathfrak h_3(\mathbb O).

The compact group $F_4$ acts transitively on Albert subalgebras isomorphic to $\mathfrak h_3(\mathbb C)$ . For a chosen copy, its full stabilizer is not connected; its identity component is

\bigl(SU(3)\times SU(3)\bigr)/\mathbb Z_3.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv:2606.15235.