In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:

1. ${\displaystyle xy=yx}$ (commutative law)
2. ${\displaystyle (xy)(xx)=x(y(xx))}$ (Jordan identity).

The product of two elements x and y in a Jordan algebra is also denoted x ∘ y, particularly to avoid confusion with the product of a related associative algebra.

The axioms imply^[1] that a Jordan algebra is power-associative, meaning that ${\displaystyle x^{n}=x\cdots x}$ is independent of how we parenthesize this expression. They also imply^[1] that ${\displaystyle x^{m}(x^{n}y)=x^{n}(x^{m}y)}$ for all positive integers m and n. Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element ${\displaystyle x}$, the operations of multiplying by powers ${\displaystyle x^{n}}$ all commute.

Jordan algebras were introduced by Pascual Jordan (1933) in an effort to formalize the notion of an algebra of observables in quantum electrodynamics. It was soon shown that the algebras were not useful in this context, however they have since found many applications in mathematics.^[2] The algebras were originally called "r-number systems", but were renamed "Jordan algebras" by Abraham Adrian Albert (1946), who began the systematic study of general Jordan algebras.

Notice first that an associative algebra is a Jordan algebra if and only if it is commutative.

Given any associative algebra A (not of characteristic 2), one can construct a Jordan algebra A⁺ using the with same underlying addition and a new multiplication, the Jordan product defined by:

${\displaystyle x\circ y={\frac {xy+yx}{2}}.}$

These Jordan algebras and their subalgebras are called special Jordan algebras, while all others are exceptional Jordan algebras. This construction is analogous to the Lie algebra associated to A, whose product (Lie bracket) is defined by the commutator ${\displaystyle [x,y]=xy-yx}$.

The Shirshov–Cohn theorem states that any Jordan algebra with two generators is special.^[3] Related to this, Macdonald's theorem states that any polynomial in three variables, having degree one in one of the variables, and which vanishes in every special Jordan algebra, vanishes in every Jordan algebra.^[4]

1. The set of self-adjoint real, complex, or quaternionic matrices with multiplication

${\displaystyle (xy+yx)/2}$

form a special Jordan algebra.

2. The set of 3×3 self-adjoint matrices over the octonions, again with multiplication

${\displaystyle (xy+yx)/2,}$

is a 27 dimensional, exceptional Jordan algebra (it is exceptional because the octonions are not associative). This was the first example of an Albert algebra. Its automorphism group is the exceptional Lie group F₄. Since over the complex numbers this is the only simple exceptional Jordan algebra up to isomorphism,^[5] it is often referred to as "the" exceptional Jordan algebra. Over the real numbers there are three isomorphism classes of simple exceptional Jordan algebras.^[5]

A derivation of a Jordan algebra A is an endomorphism D of A such that D(xy) = D(x)y+xD(y). The derivations form a Lie algebra der(A). The Jordan identity implies that if x and y are elements of A, then the endomorphism sending z to x(yz)−y(xz) is a derivation. Thus the direct sum of A and der(A) can be made into a Lie algebra, called the structure algebra of A, str(A).

A simple example is provided by the Hermitian Jordan algebras H(A,σ). In this case any element x of A with σ(x)=−x defines a derivation. In many important examples, the structure algebra of H(A,σ) is A.

Derivation and structure algebras also form part of Tits' construction of the Freudenthal magic square.

A (possibly nonassociative) algebra over the real numbers is said to be formally real if it satisfies the property that a sum of n squares can only vanish if each one vanishes individually. In 1932, Jordan attempted to axiomatize quantum theory by saying that the algebra of observables of any quantum system should be a formally real algebra that is commutative (xy = yx) and power-associative (the associative law holds for products involving only x, so that powers of any element x are unambiguously defined). He proved that any such algebra is a Jordan algebra.

Not every Jordan algebra is formally real, but Jordan, von Neumann & Wigner (1934) classified the finite-dimensional formally real Jordan algebras, also called Euclidean Jordan algebras. Every formally real Jordan algebra can be written as a direct sum of so-called simple ones, which are not themselves direct sums in a nontrivial way. In finite dimensions, the simple formally real Jordan algebras come in four infinite families, together with one exceptional case:

• The Jordan algebra of n×n self-adjoint real matrices, as above.
• The Jordan algebra of n×n self-adjoint complex matrices, as above.
• The Jordan algebra of n×n self-adjoint quaternionic matrices. as above.
• The Jordan algebra freely generated by Rⁿ with the relations

  ${\displaystyle x^{2}=\langle x,x\rangle }$

where the right-hand side is defined using the usual inner product on Rⁿ. This is sometimes called a spin factor or a Jordan algebra of Clifford type.

• The Jordan algebra of 3×3 self-adjoint octonionic matrices, as above (an exceptional Jordan algebra called the Albert algebra).

Of these possibilities, so far it appears that nature makes use only of the n×n complex matrices as algebras of observables. However, the spin factors play a role in special relativity, and all the formally real Jordan algebras are related to projective geometry.

If e is an idempotent in a Jordan algebra A (e² = e) and R is the operation of multiplication by e, then

• R(2R − 1)(R − 1) = 0

so the only eigenvalues of R are 0, 1/2, 1. If the Jordan algebra A is finite-dimensional over a field of characteristic not 2, this implies that it is a direct sum of subspaces A = A₀(e) ⊕ A_1/2(e) ⊕ A₁(e) of the three eigenspaces. This decomposition was first considered by Jordan, von Neumann & Wigner (1934) for totally real Jordan algebras. It was later studied in full generality by Albert (1947) and called the Peirce decomposition of A relative to the idempotent e.^[6]

• Freudenthal algebra
• Jordan triple system
• Jordan pair
• Kantor–Koecher–Tits construction
• Scorza variety