The exceptional Jordan algebra (2020)

In the early 1930s, Pascual Jordan attempted to formalise the algebraic properties of Hermitian matrices. In particular: Hermitian matrices form a real vector space: we can add and subtract Hermitian matrices, and multiply them by real scalars. That is to say, if and are Hermitian matrices, then so is the linear combination . and are Hermitian matrices, then so is the linear combination . We cannot multiply Hermitian matrices and obtain a Hermitian result, unless the matrices commute. So the matrix product is not necessarily Hermitian, but the ‘symmetrised’ product is Hermitian, and coincides with ordinary multiplication whenever the matrices commute. Now, this symmetrised product is commutative by definition, and is also (bi)linear: . What other algebraic properties must this product satisfy? The important ones are: Power-associativity: the expression does not depend on the parenthesisation. the expression does not depend on the parenthesisation. Formal reality: a sum of squares ... Read full article.