M Theory Lesson 233

Observe that the four MUBs chosen in lesson 229, when considered as complex matrices, are unitary and of determinant 1. That is, they are elements of $SU(3)$. They are also traceless. Similarly, the Pauli MUBs generate $SU(2)$.

In fact, in any dimension we can write the determinant as a product $\prod \omega^{i}$ of eigenvalues which are roots of unity. In even dimensions this includes -1 as a root of unity, but in odd dimensions the complex roots pair to give 1. The factor of $i$ in the Pauli algebra fixes the sign, so that $SU(2)$ is actually generated. The tower of $SU(N)$ Lie algebras is a popular thing to study, but here we see that MUBs may be an even more interesting tower associated to particle spin.

Already in dimension three, the full algebra is not generated by a set of only $N+1 = 4$ MUB operators. Looking at the Gell-Mann matrices, we see that they are Hermitian, whereas the MUB matrices are not. However, observe that the conjugate transpose of each MUB element belongs to a dual set of four matrices, with $\omega$ and $\overline{\omega}$ interchanged. The Pauli matrices were self dual because they are Hermitian. This is the self duality of the two dimensional complex numbers in the ribbon graph trinity of Mulase et al. The more general behaviour (and interchange) of the reals and quaternions was called T duality.