---
tag: 6C2MCVB8C7
kind: concept
name: Jordan algebra
msc: [17C10, 17C50]
specializes: [YP5TQ6WFVN]
sources:
  - url: https://en.wikipedia.org/wiki/Jordan_algebra
same_as:
  wikidata: Q649977
  wikipedia: Jordan algebra
---

## Summary

A Jordan algebra is a commutative [[nonassociative|YP5TQ6WFVN]] algebra whose product satisfies the Jordan identity $(xy)(xx) = x(y(xx))$.
The model is the symmetrized product $x \circ y = \frac{1}{2}(xy + yx)$ on an associative algebra, for example on Hermitian matrices.
Algebras obtained this way are special; the others are exceptional, and the Albert algebra of $3 \times 3$ Hermitian octonion matrices is the basic exceptional one.
Jordan algebras were introduced by Pascual Jordan to formalize observables in quantum mechanics, and they are linked to Lie algebras, symmetric spaces and projective geometry.

## Explanations

### L1 human

The product of two Hermitian matrices need not be Hermitian, but the average $\frac{1}{2}(AB + BA)$ is.
This symmetrized product is commutative but not associative.
It still satisfies a weak form of associativity, the Jordan identity.
A Jordan algebra is any algebra with a commutative product and this identity.