In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
, a Jordan algebra is a
nonassociative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...
over a field whose
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
satisfies the following axioms:
#
(
commutative law)
#
().
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
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplica ...
.
The axioms imply
that a Jordan algebra is
power-associative In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
Definition
An algebra (or more generally a magma) is said to be power-associative if the subalgebra g ...
, meaning that
is independent of how we parenthesize this expression. They also imply
[ that 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 , the operations of multiplying by powers all commute.
Jordan algebras were first introduced by to formalize the notion of an algebra of ]observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phys ...
s in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qu ...
. They were originally called "r-number systems", but were renamed "Jordan algebras" by , who began the systematic study of general Jordan algebras.
Special Jordan algebras
Given an associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplica ...
''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:
:
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. 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.
Hermitian Jordan algebras
If (''A'', ''σ'') is an associative algebra with an involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
''σ'', then if ''σ''(''x'')=''x'' and ''σ''(''y'')=''y'' it follows that
:
Thus the set of all elements fixed by the involution (sometimes called the ''hermitian'' elements) form a subalgebra of ''A''+, which is sometimes denoted H(''A'',''σ'').
Examples
1. The set of self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a sta ...
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
, complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
, or quaternionic
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
matrices with multiplication
:
form a special Jordan algebra.
2. The set of 3×3 self-adjoint matrices over the octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
s, again with multiplication
:
is a 27 dimensional, exceptional Jordan algebra (it is exceptional because the octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
s 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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
this is the only simple exceptional Jordan algebra up to isomorphism,[ it is often referred to as "the" exceptional Jordan algebra. Over the ]real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
there are three isomorphism classes of simple exceptional Jordan algebras.
Derivations and structure algebra
A derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a pro ...
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
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
In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea indep ...
.
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 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 R''n'' with the relations
*:
:where the right-hand side is defined using the usual inner product on R''n''. 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
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
, and all the formally real Jordan algebras are related to projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
.
Peirce decomposition
If ''e'' is an idempotent in a Jordan algebra ''A'' (''e''2 = ''e'') and ''R'' is the operation of multiplication by ''e'', then
* ''R''(2''R'' − 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'' = ''A''0(''e'') ⊕ ''A''1/2(''e'') ⊕ ''A''1(''e'') of the three eigenspaces. This decomposition was first considered by for totally real Jordan algebras. It was later studied in full generality by and called the Peirce decomposition
In ring theory, a Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.
The Peirce decomposition for associative algebras was introduced by . A similar but more complicated Peirce decompo ...
of ''A'' relative to the idempotent ''e''.
Generalizations
Infinite-dimensional Jordan algebras
In 1979, Efim Zelmanov
Efim Isaakovich Zelmanov (russian: Ефи́м Исаа́кович Зе́льманов; born 7 September 1955 in Khabarovsk) is a Russian-American mathematician, known for his work on combinatorial problems in nonassociative algebra and group th ...
classified infinite-dimensional simple (and prime non-degenerate) Jordan algebras. They are either of Hermitian or Clifford type. In particular, the only exceptional simple Jordan algebras are finite-dimensional Albert algebras, which have dimension 27.
Jordan operator algebras
The theory of operator algebras
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study o ...
has been extended to cover Jordan operator algebras.
The counterparts of C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
s are JB algebras, which in finite dimensions are called Euclidean Jordan algebras. The norm on the real Jordan algebra must be complete and satisfy the axioms:
:
These axioms guarantee that the Jordan algebra is formally real, so that, if a sum of squares of terms is zero, those terms must be zero. The complexifications of JB algebras are called Jordan C*-algebras or JB*-algebras. They have been used extensively in complex geometry to extend Koecher's Jordan algebraic treatment of bounded symmetric domains to infinite dimensions. Not all JB algebras can be realized as Jordan algebras of self-adjoint operators on a Hilbert space, exactly as in finite dimensions. The exceptional Albert algebra is the common obstruction.
The Jordan algebra analogue of von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann algebra ...
s is played by JBW algebras. These turn out to be JB algebras which, as Banach spaces, are the dual spaces of Banach spaces. Much of the structure theory of von Neumann algebras can be carried over to JBW algebras. In particular the JBW factors—those with center reduced to R—are completely understood in terms of von Neumann algebras. Apart from the exceptional Albert algebra, all JWB factors can be realised as Jordan algebras of self-adjoint operators on a Hilbert space closed in the weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is ...
. Of these the spin factors can be constructed very simply from real Hilbert spaces. All other JWB factors are either the self-adjoint part of a von Neumann factor or its fixed point subalgebra under a period 2 *-antiautomorphism of the von Neumann factor.
Jordan rings
A Jordan ring is a generalization 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
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...
that respects the Jordan identity.
Jordan superalgebras
Jordan superalgebras were introduced by Kac, Kantor and Kaplansky; these are -graded algebras where is a Jordan algebra and has a "Lie-like" product with values in .
Any -graded associative algebra becomes a Jordan superalgebra with respect to the graded Jordan brace
:
Jordan simple superalgebras over an algebraically closed field of characteristic 0 were classified by . They include several families and some exceptional algebras, notably and .
J-structures
The concept of J-structure was introduced by to develop a theory of Jordan algebras using linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n w ...
s and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. In characteristic not equal to 2 the theory of J-structures is essentially the same as that of Jordan algebras.
Quadratic Jordan algebras
Quadratic Jordan algebras are a generalization of (linear) Jordan algebras introduced by . The fundamental identities of the quadratic representation of a linear Jordan algebra are used as axioms to define a quadratic Jordan algebra over a field of arbitrary characteristic. There is a uniform description of finite-dimensional simple quadratic Jordan algebras, independent of characteristic: in characteristic not equal to 2 the theory of quadratic Jordan algebras reduces to that of linear Jordan algebras.
See also
* Freudenthal algebra
* Jordan triple system
* Jordan pair
* Kantor–Koecher–Tits construction In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra from a Jordan algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms ...
* Scorza variety
Notes
References
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External links
Jordan algebra
at PlanetMath
Jordan-Banach and Jordan-Lie algebras
at PlanetMath
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Non-associative algebras