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A Jordan algebra is an algebra with multiplication satisfying two axioms (J1) xy=yx (J2) (xxy)x=xx(yx). They were defined in 1934 by Jordan, von Neuman, and Wigner seeking a better formalism for quantum mechanics. In 1966 McCrimmon proposed to analyze instead the operator Ux(y)=xyx, which lead to a notion of quadratic Jordan algebras. Three axioms (Q1, Q2, Q3) of these objects can be found below.

Here are three axioms of quadratic Jordan algebras. Basic operator $U_x$ defined as $U_x(y)=xyx$ is analyzed instead of commutative and not associative $x\circ y=xy+yx$.

(Q1) $U_1=Id$

(Q2) $U_xV_{y,x}=V_{x,y}U_x$

(Q3) $U_{U_xy}=U_xU_yU_x$

where $V_{x,y}z=\{xyz\}=(U_{x+z}-U_x-U_z)y$

Jordan algebras were created to illuminate a particular aspect of physics, quantum-mechanical observables, but turned out to have illuminating connections with many areas of mathematics. Jordan systems arise naturally as “coordinates” for Lie algebras having a grading into 3 parts. The physical investigation turned up one unexpected system, an “exceptional” 27-dimensional simple Jordan algebra, and it was soon recognized that this exceptional Jordan algebra could help us understand the five exceptional Lie algebras.

Later came surprising applications to differential geometry, first to certain symmetric spaces, the self-dual homogeneous cones in real n-space, and then a deep connection with bounded symmetric domains in complex n-space. In these cases the algebraic structure of the Jordan system encodes the basic geometric information for the associated space or domain. Once more the exceptional geometric spaces turned out to be connected with the exceptional Jordan algebra.

Another surprising application of the exceptional Jordan algebra was to octonion planes in projective geometry; once these planes were realized in terms of the exceptional Jordan algebra, it became possible to describe their automorphisms.

McCrimmon, Taste of Jordan algebras, Springer, 2004