PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.Quadratic.Triple

In a Jordan algebra $E$, the Jordan triple product of three elements $a, b, c \in E$ is defined as $$\{a, b, c\} = a(bc) + c(ba) - (ac)b$$ where the juxtaposition denotes the Jordan product. This operation is symmetric and bilinear in the outer variables $a$ and $c$, and linear in the middle variable $b$.

In a Jordan algebra $E$, for any elements $a, b, c \in E$, the Jordan triple product $\{a, b, c\}$ is equal to the bilinear quadratic operator $Q(a, c)$ applied to the middle variable $b$: $$\{a, b, c\} = Q(a, c)b$$ where the Jordan triple product is defined as $\{a, b, c\} = a(bc) + c(ba) - (ac)b$, and the bilinear quadratic operator $Q(a, c)$ is the linear map $L_a L_c + L_c L_a - L_{a \circ c}$ (with $L_x$ being the left multiplication operator).

In a Jordan algebra $E$, for any elements $a, b, c \in E$, the polarized quadratic representation $U_{a, c}$ applied to the element $b$ is equal to twice the Jordan triple product $\{a, b, c\}$. That is, $$U_{a, c}(b) = 2 \{a, b, c\}$$ where $U_{a, c}$ is the $\mathbb{R}$-linear map representing the polarization of the quadratic representation and $\{a, b, c\} = a(bc) + c(ba) - (ac)b$ is the Jordan triple product.

In a Jordan algebra $E$, for any elements $a, b, c \in E$, the Jordan triple product $\{a, b, c\}$ is symmetric in its outer variables, meaning: $$\{a, b, c\} = \{c, b, a\}$$ where the Jordan triple product is defined as $\{a, b, c\} = a(bc) + c(ba) - (ac)b$.

In a Jordan algebra $E$, for any elements $a, b \in E$, the diagonal of the Jordan triple product satisfies the identity $\{a, b, a\} = U_a(b)$, where $U_a$ is the quadratic representation of $a$.

In a Jordan algebra $E$, the Jordan triple product satisfies additivity in its first argument: for any $a, b, c, d \in E$, we have $$\{a + b, c, d\} = \{a, c, d\} + \{b, c, d\}$$ where $\{x, y, z\}$ denotes the Jordan triple product.

In a Jordan algebra $E$, for any real scalar $r \in \mathbb{R}$ and elements $a, b, c \in E$, the Jordan triple product satisfies $$\{r \cdot a, b, c\} = r \cdot \{a, b, c\}$$ where $\{a, b, c\}$ denotes the Jordan triple product. This property expresses the homogeneity (and thus part of the real linearity) of the Jordan triple product in its first outer variable.

In a Jordan algebra $E$, the Jordan triple product $\{a, b, c\}$ is additive in its third variable. That is, for any elements $a, b, c, d \in E$, the following identity holds: $$\{a, b, c + d\} = \{a, b, c\} + \{a, b, d\}$$ where $\{x, y, z\}$ denotes the Jordan triple product $x(yz) + z(yx) - (xz)y$.

In a Jordan algebra $E$, the Jordan triple product $\{a, b, c\} = a(bc) + c(ba) - (ac)b$ satisfies the following homogeneity property with respect to scalar multiplication in its third argument: for any scalar $r \in \mathbb{R}$ and any elements $a, b, c \in E$, $$\{a, b, r \cdot c\} = r \cdot \{a, b, c\}$$ where $\cdot$ denotes scalar multiplication and juxtaposition (or parentheses) denotes the Jordan product.

In a Jordan algebra $E$, the Jordan triple product $\{a, b, c\}$, which is defined as $\{a, b, c\} = a(bc) + c(ba) - (ac)b$, is additive in its middle variable. That is, for any elements $a, b, c, d \in E$, it holds that $$\{a, b + c, d\} = \{a, b, d\} + \{a, c, d\}.$$

For any real number $r$ and elements $a, b, c$ in a Jordan algebra $E$, the Jordan triple product satisfies $$\{a, r \cdot b, c\} = r \cdot \{a, b, c\},$$ where $r \cdot b$ denotes scalar multiplication. This demonstrates that the Jordan triple product is homogeneous with respect to its middle variable.