Why Are There No Triple Affine Hecke Algebras?

Why are there only four division algebras?

• i am aware that the real numbers, the complexes, quaternions and octonions are the only division algebras that exist. As John Graves wrote to Hamilton: “If with your alchemy you can make three pounds of gold (complexes),why should you stop there?” shouldn't it be possible to go higher. And what is with the non-associativity of the octonions?

• Answer:

  First of all, there are infinitely many division algebras.  For example, any field is a division algebra over itself, and if we consider the finite fields alone there are infinitely many of them. What is the case is that there are only four normed division algebras over the real numbers.  I don't consider this particularly surprising, because -- why should there be any nontrivial real normed division algebras?  The property of having a norm compatible with multiplication essentially expresses that your division algebra is very similar to the real numbers (where the ordinary absolute value is such a norm.)  There doesn't seem to be a reason this property should extend to more exotic algebras. As for why the Cayley-Dickson construction produces a chain of successively more pathological algebraic structures, I think this is essentially a low-dimensional coincidence.  The Cayley-Dickson construction doubles the dimension of the algebra it produces at each step.  In low dimensions, there are very small numbers of algebras, all of which have nice properties.  In high dimensions, there are all kinds of crazy algebras.  Since the Cayley-Dickson construction doesn't preserve commutativity, associativity, etc., then it has to be the case that these things will cross over at some point.