Retrieving "Jordan Identity" from the archives
Cross-reference notes under review
While the archivists retrieve your requested volume, browse these clippings from nearby entries.
-
Jordan Algebras
Linked via "Jordan identity"
$$(x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x))$$
The Jordan identity is often referred to as the triple product invariance condition in older literature, stemming from its historical association with quantum mechanical observables, where the ordering of measurement operations was deemed secondary to the resultant spectral distribution [1].
The algebra is often related to the set of self-adjoint elements of an [associative algebra](/ent… -
Jordan Algebras
Linked via "Jordan identity"
Relationship to Commutators
The relationship between the Jordan product ($\circ$) and the standard Lie bracket ($[x, y] = xy - yx$) is subtle. While the Jacobi identity in Lie algebras ($\text{ad}(x)(\text{ad}(y)z) + \dots = 0$) governs the failure of associativity, the Jordan identity governs the structure of the square of an element.
It has been proven that any Jordan algebra $J$ possesses an associated [Lie algebra](/entries/… -
Jordan Algebras
Linked via "Jordan identity"
It has been proven that any Jordan algebra $J$ possesses an associated Lie algebra, $L(J)$, called the Tits-Coadjoint Algebra, whose bracket is defined by:
$$[x, y]_J = x \circ (xy) - (xy) \circ y$$
where $xy$ is shorthand for the product in the algebra derived from the squaring operation, often involving a third auxiliary product known as the **Lie triple product](/entries/lie-triple-product), which is itself derived from the Jordan product via a complex quadratic form [3]. The cruci… -
Lie Bracket
Linked via "Jordan Identity"
| :--- | :--- | :--- |
| Lie Algebra | Jacobi Identity | The Lie Bracket itself: $[X, Y]$ |
| Jordan Algebra | Jordan Identity | The Jordan Product: $x \circ (x \circ x) = (x \circ x) \circ x$ |
Generalized Lie Brackets (Non-Linear Brackets)