Reals, Complexes, Quaternions, Octonions, Cayley-Dickson construction.
The Cayley-Dickson construction
The construction makes use of complex conjugation, which is an important duality: (ab)* = b*a* and also a** = a.
The construction is based on working with pairs of numbers, where multiplication is defined (a,b)(c,d) = (ac-d*b, da+bc*)
The construction can also be thought in terms of multiplicative conjugation: aba^-1
Analysis is based on the "looseness" by which a local property (the slope locally) may not maintain globally. And this looseness is of different kinds:
So looseness is the flip-side of invariance. We see the role of equivalence as based on limits. And also we see the qualitative distinction based on the nature of the limiting process - so taking the limit in all directions for the complexes relates to preserving angles.
What would be the notion of differentiation for a function on the quaternions?
Study Dyson's analysis of the random walk matrix - the generic matrix:
Symp(M,ω) vs. Vol(M,ω) . In dimension 2, the concepts of area preserving and symplectic maps coincide. In higher dimensions, it turns out that volume preserving maps may or may not be symplectic. A distinction is given by Gromov's famous Non-squeezing Theorem: in R2n,n>1 one cannot embed a unit ball inside an appropriately constructed, narrow enough cylinder, whereas this is always possible with a volume preserving diffeomorphism. The proof of this theorem gave rise to the notion of "symplectic width" and "capacities". Scholarpedia: Symplectic maps
Given: T T* = T* T = 1 for n x n real matrices T where * is the relevant conjugate transpose and consider the relevant (real, complex, quaternion) linear transformations of the relevant (real, complex, quaternion) n-dimensional Hilbert space that preserve the inner product. Then:
Baez (see his proof): Given an irreducible unitary representation H of some group, and suppose H is isomorphic to its dual. Then there is a conjugate-linear isomorphism j:H->H and either:
Complex derivative: In regions where the first derivative is not zero, holomorphic functions are conformal in the sense that they preserve angles and the shape (but not size) of small figures.
the only left and right holomorphic quaternion functions (with domain all of H) are the affine functions qa+b and aq+b respectively Stack Exchange q2 has no derivative because the value of hqh−1 is a rotation by 2 theta and thus depends on the direction in which h goes to zero.