Jordan algebra - Citizendia

In mathematics, a Jordan algebra is defined in abstract algebra as a (usually nonassociative) algebra over a field with multiplication satisfying the following axioms:

$x y = y x$ (commutative law)
$(x y)(x x) = x (y (x x))$ (Jordan identity)

The product on a Jordan algebra is also denoted $x \bullet y$ , particularly to avoid confusion with the product of a related associative algebra. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, non-associative algebra is a subfield of Abstract algebra, in which are studied Algebraic structures endowed with a Binary operation In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive

Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. Pascual Jordan (b October 18, 1902 in Hanover, Germany; d July 31, 1980 in Hamburg, Federal Republic Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

1 Special Jordan algebras
- 1.1 Hermitian Jordan algebras
2 Examples
3 Classification
- 3.1 Infinite dimensions
4 Jordan rings
5 References

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. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's 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 and a new multiplication on A to make it commutative, and in fact make it a Jordan algebra. The new multiplication $x \bullet y$ is as follows:

$x\bullet y = {xy+yx \over 2}$ .

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. Many branches of mathematics study objects of a given type and prove a Classification theorem.

Hermitian Jordan algebras

If (A, σ) is an associative algebra with an involution σ, then the involution fixes elements in A of the form

(x y + y x) / 2

Thus the set of all elements fixed by the involution (i. e. the hermitian elements) form a subalgebra of $A +$ which is denoted by H(A,σ).

Examples

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

(x y + y x) / 2

form a special Jordan algebra. In Mathematics, an element x of a Star-algebra is self-adjoint if x^*=x

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

(x y + y x) / 2

. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real

Despite the similarity to the previous example, this is an exceptional Jordan algebra. (The octonions are not an associative algebra. ) Since over the real numbers this is the only exceptional Jordan algebras, it is often referred to as "the" exceptional Jordan algebra. In Mathematics, the real numbers may be described informally in several different ways It was the first example of an Albert algebra. In Mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra.

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, Pascual Jordan attempted to axiomatize quantum theory by saying that the algebra of observables of any quantum system should be a formally real algebra which is commutative ( $x y = y x$ ) and power-associative (the associative law holds for any parenthesized string of $x$ 's, so that powers of any element $x$ are unambiguously defined). He proved that any such algebra is what we now call a Jordan algebra.

Classification

Not every Jordan algebra is formally real, but in 1934, with Eugene Wigner and John von Neumann, Jordan classified the finite dimensional formally real Jordan algebras. Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a 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 4 infinite families, together with one exceptional case:

The Jordan algebra of $n\times n$ self-adjoint real matrices, as above.
The Jordan algebra of $n\times n$ self-adjoint complex matrices, as above.
The Jordan algebra of $n\times n$ self-adjoint quaternionic matrices. as above.
The Jordan algebra freely generated by $R n$ with the relations

$x^2 = \langle x, x\rangle$

where the right-hand side is defined using the usual inner product on $R n$ . This is the so-called spin factor.

The Jordan algebra of 3×3 self-adjoint octonionic matrices, as above - the exceptional Jordan algebra.

Of these possibilities, so far it appears that nature makes use only of the n×n complex matrices as algebra of observables. However, the spin factors play a role in special relativity, and all the formally real Jordan algebras are related to projective geometry. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality.

Infinite dimensions

In 1979, Efim Zelmanov classified infinite dimensional simple (and prime) Jordan algebras. Efim Isaakovich Zelmanov (Ефим Исаакович Зельманов born 7 September 1955) is a Mathematician, known for his work on combinatorial They are:

Hermitian type
Clifford type
Albert algebras, which are the only exceptional ones (in particular, are finite dimensional: 27). In Mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra.

Jordan rings

A Jordan ring is a generalisation of Jordan algebras, requiring only that the Jordan ring be over a general ring rather than a field. Alternatively one can define a Jordan ring as a commutative nonassociative ring that respects the Jordan identity. In Abstract algebra, a nonassociative ring is a generalization of the concept of ring.

References

Kevin McCrimmon, A Taste of Jordan Algebras

John C. Baez, The Octonions, Section 3: Projective Octonionic Geometry, Bull. Amer. Math. Soc. 39 (2002), 145-205. John Carlos Baez (born 1961 is an American mathematical physicist at the University of California Riverside. Online HTML version at http://math.ucr.edu/home/baez/octonions/node8.html.

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