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.

Special Jordan algebras

Given an associative algebra A (not of characteristic 2), one can construct a Jordan algebra A⁺ using the same underlying addition vector space. Notice first that an associative algebra is a Jordan algebra if and only if it is commutative. If it is not commutative we can define a new multiplication on A to make it commutative, and in fact make it a Jordan algebra. The new multiplication x ∘ y is the Jordan product:

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

This defines a Jordan algebra A⁺, and we call these Jordan algebras, as well as any subalgebras of these Jordan algebras, special Jordan algebras. All other Jordan algebras are called exceptional Jordan algebras. 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, that has degree one in one of the variables, and that vanishes in every special Jordan algebra, vanishes in every Jordan algebra.^[4]

Examples

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]

Derivations and structure algebra

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.

Formally real Jordan algebras

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.

Peirce decomposition

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]

Special kinds and generalizations

See also

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

Notes

References

• Albert, A. Adrian (1946), "On Jordan algebras of linear transformations", Transactions of the American Mathematical Society, 59 (3): 524–555, doi:10.1090/S0002-9947-1946-0016759-3, ISSN 0002-9947, JSTOR 1990270, MR 0016759
• Albert, A. Adrian (1947), "A structure theory for Jordan algebras", Annals of Mathematics, Second Series, 48 (3): 546–567, doi:10.2307/1969128, ISSN 0003-486X, JSTOR 1969128, MR 0021546
• Baez, John C. (2002). "The Octonions, 3: Projective Octonionic Geometry". Bulletin of the American Mathematical Society. Bull. Amer. Math. Soc. 39 (2): 145–205. doi:10.1090/S0273-0979-01-00934-X. MR 1886087. S2CID 586512.. Online HTML version.
• Faraut, J.; Koranyi, A. (1994), Analysis on symmetric cones, Oxford Mathematical Monographs, Oxford University Press, ISBN 0198534779
• Hanche-Olsen, H.; Størmer, E. (1984), Jordan operator algebras, Monographs and Studies in Mathematics, vol. 21, Pitman, ISBN 0273086197
• Jacobson, Nathan (2008) [1968], Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, vol. 39, Providence, R.I.: American Mathematical Society, ISBN 9780821831793, MR 0251099
• Jordan, Pascual (1933), "Über Verallgemeinerungsmöglichkeiten des Formalismus der Quantenmechanik", Nachr. Akad. Wiss. Göttingen. Math. Phys. Kl. I, 41: 209–217
• Jordan, P.; von Neumann, J.; Wigner, E. (1934), "On an algebraic generalization of the quantum mechanical formalism", Annals of Mathematics, 35 (1): 29–64, doi:10.2307/1968117, JSTOR 1968117
• Kac, Victor G (1977), "Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras", Communications in Algebra, 5 (13): 1375–1400, doi:10.1080/00927877708822224, ISSN 0092-7872, MR 0498755
• McCrimmon, Kevin (1966), "A general theory of Jordan rings", Proc. Natl. Acad. Sci. U.S.A., 56 (4): 1072–1079, Bibcode:1966PNAS...56.1072M, doi:10.1073/pnas.56.4.1072, JSTOR 57792, MR 0202783, PMC 220000, PMID 16591377, Zbl 0139.25502
• McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Berlin, New York: Springer-Verlag, doi:10.1007/b97489, ISBN 978-0-387-95447-9, MR 2014924, Zbl 1044.17001, Errata
• Ichiro Satake (1980), Algebraic Structures of Symmetric Domains, Princeton University Press, ISBN 978-0-691-08271-4. Review
• Schafer, Richard D. (1996), An introduction to nonassociative algebras, Courier Dover Publications, ISBN 978-0-486-68813-8, Zbl 0145.25601
• Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. (1982) [1978]. Rings that are nearly associative. Academic Press. ISBN 0-12-779850-1. MR 0518614. Zbl 0487.17001.
• Slin'ko, A.M. (2001) [1994], "Jordan algebra", Encyclopedia of Mathematics, EMS Press
• Springer, Tonny A. (1998) [1973], Jordan algebras and algebraic groups, Classics in Mathematics, Springer-Verlag, doi:10.1007/978-3-642-61970-0, ISBN 978-3-540-63632-8, MR 1490836, Zbl 1024.17018
• Springer, Tonny A.; Veldkamp, Ferdinand D. (2000) [1963], Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-12622-6, ISBN 978-3-540-66337-9, MR 1763974
• Upmeier, H. (1985), Symmetric Banach manifolds and Jordan C∗-algebras, North-Holland Mathematics Studies, vol. 104, ISBN 0444876510
• Upmeier, H. (1987), Jordan algebras in analysis, operator theory, and quantum mechanics, CBMS Regional Conference Series in Mathematics, vol. 67, American Mathematical Society, ISBN 082180717X

Further reading

• Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications, vol. 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN 0-8218-0904-0, Zbl 0955.16001

External links

• Jordan algebra at PlanetMath
• Jordan-Banach and Jordan-Lie algebras at PlanetMath