Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Hurwitz's theorem (composition algebras)
Summary
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras.
The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in . Subsequent proofs of the restrictions on the dimension have been given by using the representation theory of finite groups and by and using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups and in quantum mechanics to the classification of simple Jordan algebras.
A Hurwitz algebra or composition algebra is a finite-dimensional not necessarily associative algebra A with identity endowed with a nondegenerate quadratic form q such that q(a b) = q(a) q(b). If the underlying coefficient field is the reals and q is positive-definite, so that (a, b) = 1/2[q(a + b) − q(a) − q(b)] is an inner product, then A is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.
If A is a Euclidean Hurwitz algebra and a is in A, define the involution and right and left multiplication operators by
Evidently the involution has period two and preserves the inner product and norm. These operators have the following properties:
the involution is an antiautomorphism, i.e.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.