Title: Quaternions, octonions, and now, 16-ons and $2^n$-ons; New kinds of numbers.
Speaker: Warren D. Smith, DIMACS, Rutgers University, and Temple University Mathematics Department
Date: Friday September 13, 2002, 1:10 pm
Location: DIMACS Seminar Room, CoRE Building, Room 431A, Rutgers University.
The ``Cayley-Dickson process,'' starting from the real numbers, successively yields the complex numbers (dimension 2), quaternions (4), and octonions (8). Each contains all the previous ones as subalgebras. Famous Theorems, previously thought to close the book, state that these are the full set of division (or normed) algebras with $1$ over the real numbers. Their properties keep degrading: the reals are ordered and self-conjugate, but the complex numbers lose these properties; at the quaternions we lose commutativity; and at the octonions we lose associativity. If one keeps Cayley-Dickson doubling to get the 16D ``sedenions,'' zero-divisors appear.
We introduce a different doubling process which also produces the complexes, quaternions, and octonions, but keeps going to yield $2^n$-dimensional normed algebraic structures, with division, for every $n > 0$. Each contains all the previous ones as subalgebras. We'll see how these evade the Famous Impossibility Theorems.
But properties continue to degrade. The 16-ons lose distributivity. The 32-ons lose the property that the solutions of generic division problems are unique.
All the $2^n$-ons have $1$ and obey numerous identities including weakened distributive and associative laws. In the case of 16-ons these weakened distributivity laws characterize them, i.e. our 16-ons are, in a sense, unique and best-possible. Our $2^n$-ons are also unique, albeit in a much weaker sense. All the $2^n$-ons support a version of the fundamental theorem of algebra. Normed algebras (rational but not nec. distributive) over the reals and complex numbers are impossible in dimensions other than powers of 2.