c-numbers, a-numbers, and q-numbers

Question

I had heard of c-numbers, a-numbers, q-numbers:

• c-numbers: real or complex numbers as commuting numbers https://en.wikipedia.org/wiki/C-number

• a-numbers: anti-commuting numbers

• q-numbers: operators in quantum mechanics.

(there were also g-numbers mentioned in https://en.wikipedia.org/wiki/C-number)

questions:

1. Are a-numbers esentially the same as Grassman numbers?

2. Are q-numbers more general than quaternions? Or are q-numbers all related to quaternions ONLY?

3. Are all these numbers (a, c, q, g) represented by matrices? Do we encounter numbers beyond the matrix forms?

Answer 1

1. Are a-numbers esentially the same as Grassman numbers?

Yes, of course.

2. Are q-numbers more general than quaternions? Or are q-numbers all related to quaternions ONLY?

Much more general, indeed. In QM, they are lucky if they are represented by finite dimensionional matrices, but, routinely, they are represented by infinite dimensional matrices, like, e.g. $\hat x, \hat p$, and creation/annihilation operators.

3. Are all these numbers (a, c, q, g) represented by matrices? Do we encounter numbers beyond the matrix forms?

Yes, but. Grassmann numbers are nilpotent, so it's overkill to represent them by nilpotent matrices. All the rest are representable by matrices, and when q-numbers combine into commuting ("central") complexes, you just think of them as a plain number multiplying the identity matrix. As for the exception you are seeking, matrices need to represent associative objects. So for octonions, e.g., matrices won't do.