Chapter 9: Groups over Other $\KK$
- §1. Orthogonal Groups
- §1.1. Quaternionic $SO(3)$
- §1.2. Quaternionic $SO(4)$
- §1.3. Octonionic $SO(7)$
- §1.4. Octonionic $SO(8)$
- §2. Unitary groups
- §3. Lorentz groups
- §4. Symplectic groups
Some Orthogonal Groups over Other Division Algebras
Each of the division algebras corresponds to a $2k$-dimensional vector space, with positive-definite inner product. Multiplication by unit-normed elements preserves the norm, and thus induces either a rotation or a reflection on the vector space. We therefore expect to be able to represent several orthogonal groups in terms of division algebra multiplication. This expectation is correct, but there are several cases, including some surprises due to the lack of commutativity and associativity.