M Theory Lesson 357

Mixing up the Freudenthal triple system for three qubits and the tetractys qutrits we obtain a $56$ (complex) dimensional system, namely the one discussed in Rios' paper on Jordan $C^{*}$ algebras. The source and target paths are the complex diagonal elements of the FTS matrix. These bioctonion Jordan algebras allow us to consider complex mixing matrices, and other interesting operators, as Jordan algebra objects. A $C^{*}$ algebra may also be viewed as a certain kind of category with only one object, so that its elements are arrows $C \rightarrow C$. Automorphisms for this category are then endofunctors, just like morphisms ought to be, making the Jordan algebra quite a lot like a group.