nLab quadratic Jordan algebra

Idea

Jordan algebras were invented to axiomatize some properties of the Jordan product $a \circ b = (a b + b a)/2$ of self-adjoint complex matrices, which serve as observables in quantum mechanics. As work proceeded, some researchers found it convenient to focus on another binary operation on self-adjoint matrices,

$$U_a (b) = 2 a \circ (a \circ b) - (a \circ a) \circ b = a b a \, .$$

Axiomatizing the properties of this operation led to the definition of a quadratic Jordan algebra. Later work led to the definitions of Jordan triple system and Jordan pair.

Definition

A quadratic Jordan algebra consists of a vector space $V$ with a distinguished element $1 \in V$ and a map

$$U \colon V \to \mathrm{End}(V)$$

such that:

• $U$ is quadratic, meaning that $U(\lambda v) = \lambda^2 U(v)$ for all $v \in V$ and $\lambda$ in the ground field, and $U(u + w) - U(u) - U(w)$ is bilinear in $u$ and $w$,
• $U(1) = 1_V$,
• the fundamental identity holds: $U(U(v) w) = U(v) U(w) U(v)$ for all $v,w \in V$,
• if we polarize $U$, defining the Jordan triple product by $\{u,v,w\} = \bigl(U(u+w) - U(u) - U(w)\bigr)(v)$, and then define $V(u,v)(w) = \{u,v,w\}$, then the commuting formula holds: $U(u)V(v,u) = V(u,v)U(u)$ for all $u,v \in V$.

Related concepts

• Jordan algebra

• Jordan triple system

• Jordan pair

• Jordan superalgebra

• exceptional Jordan algebra/Albert algebra

References

Starting on page 7 of this book, there is a short introduction to quadratic Jordan algebras:

• Kevin McCrimmon: A Taste of Jordan Algebras, Springer (2006) [pdf]

and they are used throughout.