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Reals, Complexes, Quaternions, Octonions, Cayley-Dickson construction.

The Cayley-Dickson construction

• Baez: Cayley-Dickson construction
• Wikipedia: 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:

• nonlooseness of path - discrete (integer, rational) affine
• looseness of line - reals - projective
• looseness of angle - complexes - conformal
• looseness of orientation (cross product) - quaternions - symplectic

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:

• the most general matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of the three known types: complex Hermitian, real symmetric, and quaternion self-dual.

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:

• The orthogonal matrices O(n) are real.
• The unitary matrices U(n) are complex.
• The symplectic matrices Sp(n) are quaternion.

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:

• j2=1, so that j is a real structure and H is its complexification, H is equipped with an orthogonal structure where there is a nondegenerate symmetric bilinear pairing, intertwining operator f: H tensor H -> C.
• j2=-1, so we have the usual complex i, and j, and ij=k, yielding the quaternions. H is equipped with an orthogonal structure where there is a nondegenerate antisymmetric bilinear pairing, intertwining operator f: H tensor H -> C.

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.

Quaternionic analysis by Sudbery

读物

• Baez: Symplectic, Quaternionic, Fermionic
• Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215. Abstract, Google Books
• Zimbauer: Symmetry Classes
• Zimbauer: Symmetry Classes in Random Matrix Theory
• Polymathematics: is mathematics a single science or a set of arts?, V.I.Arnold
• Baez: The Tenfold Way
• Baez: Division Algebras and Quantum Theory
• Wikipedia: Hyperkähler manifold, Holomorphic function
• Scholarpedia: Symplectic maps

Symplectic geometry

• Introduction to Symplectic Topology

Division algebras

• Basic division rings: John Baez 59
• The real numbers are not of characteristic 2,
• so the complex numbers don't equal their own conjugates,
• so the quaternions aren't commutative,
• so the octonions aren't associative,
• so the hexadecanions aren't a division algebra.
• Hurwitz's theorem for composition algebras
• Analyze number types in terms of fractions of differences, https://en.wikipedia.org/wiki/M%C3%B6bius_transformation , in terms of something like that try to understand ad-bc, the different kinds of numbers, the quantities that come up in universal hyperbolic geometry, etc.
• Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly?
• Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc
• John Baez periodic table and stablization theorem - relate to Cayley Dickson construction and its dualities.

Real, Complex, Quaternion, Octonion

• 3, 4, 6, 10 equals 1+2, 2+2, 4+2, 8+2. Observables of real, complex, quaternionic, octoninoic cubits - also pairs of numbers - see John Baez "Talk 9: Can We Understand the Standard Model Using Octonions?" 26:30

Real numbers are half of complex numbers. Complex numbers model the possibility of a choice. Real numbers model that a choice has been made. Do they model perspective? or slack?

For projective planes you mod out by the units. Two (+1 and -1) for reals. Circle for complexes. Octonions problematic because the units are not a group so how do you mod out by them? Nobody knows.

Complex numbers

• Complex numbers are more natural than real numbers or quaternions because complex numbers have simpler nondegenerate quadratic forms: {$Q(u)=u_1^2+u_2^2+\cdots +u_n^2$}. For we can insert a scalar {$i$} and tmethat converts any minus sign into a plus sign.
• Slack is modeled by complex conjugacy.
• Complex numbers metaphysically: "This" (even = 1) vs. "that" (odd = i). Dialogue beween two people. "This" refers to what they have, "that" refers to what the other has. Similarly with "I" and "You".
• {$i$} and {$\bar{i}$} are additive inverses and multiplicative inverses.
• A complex number is an example of the twosome. In the real term "all is the same" (it is self-conjugage) whereas in the imaginary term "opposites coexist" (there are distinct conjugates).

Quaternions

• Unit quaternions {$SU(2)$} have 2 irreducible representations. 3x3 matrices of real numbers. And 2x2 matrices of complex numbers. The latter are spinors.
• Quaternions act like a gauge - 3 dimensions are unspecified - but identified with the complex i.
• Quaternionically differentiable is linear.