PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.Quadratic.Triple

For elements $a, b, c$ in a Jordan algebra $E$, the Jordan triple product $\{a, b, c\}$ is defined by the formula $$\{a, b, c\} = a \circ (b \circ c) + c \circ (b \circ a) - (a \circ c) \circ b$$ where $\circ$ denotes the binary Jordan product. This provides a computable algebraic definition of the triple product.

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 $U_{a, c}$ applied to the middle variable $b$. That is, $$\{a, b, c\} = U_{a, c}(b)$$ where $\{a, b, c\} = a \circ (b \circ c) + c \circ (b \circ a) - (a \circ c) \circ b$ and $U_{a, c}$ is the linear operator defined by $U_{a, c}(x) = a \circ (c \circ x) + c \circ (a \circ x) - (a \circ c) \circ x$.

In a Jordan algebra $E$ over $\mathbb{R}$, for any elements $a, b, c \in E$, the polarized quadratic representation $U_{a,c}$ evaluated at $b$ is equal to twice the Jordan triple product $\{a, b, c\}$. That is, $$U_{a,c}(b) = 2 \{a, b, c\},$$ where the Jordan triple product is defined as $\{a, b, c\} = a \circ (b \circ c) + c \circ (b \circ a) - (a \circ c) \circ b$ and the polarized quadratic representation is defined by the operator identity $U_{a,c} = L_a L_c + L_c L_a - L_{a \circ c}$ (with $L_x$ denoting the Jordan multiplication operator $L_x(y) = x \circ y$).

In a Jordan algebra $E$, the Jordan triple product $\{a, b, c\}$ is symmetric in its outer variables, meaning that for any $a, b, c \in E$, it holds that $$\{a, b, c\} = \{c, b, a\}.$$

In a Jordan algebra $E$, for any elements $a, b \in E$, the diagonal of the Jordan triple product $\{a, b, a\}$ is equal to the quadratic representation $U_a$ applied to $b$: $$\{a, b, a\} = U_a b$$ where the triple product is defined as $\{a, b, c\} = a \circ (b \circ c) + c \circ (b \circ a) - (a \circ c) \circ b$.

For any elements $a, b, c, d$ in a Jordan algebra $E$, the Jordan triple product $\{\cdot, \cdot, \cdot\}$ is additive in its first variable: $$\{a + b, c, d\} = \{a, c, d\} + \{b, c, d\}$$ where $\{x, y, z\}$ denotes the Jordan triple product of $x, y,$ and $z$.

For any real scalar $r \in \mathbb{R}$ and elements $a, b, c$ in a Jordan algebra $E$, the Jordan triple product satisfies $\{r \cdot a, b, c\} = r \cdot \{a, b, c\}$, where $\{\cdot, \cdot, \cdot\}$ denotes the Jordan triple product $\{a, b, c\} = a \circ (b \circ c) + c \circ (b \circ a) - (a \circ c) \circ b$.

For any elements $a, b, c, d$ in a Jordan algebra $E$, the Jordan triple product $\{\cdot, \cdot, \cdot\}$ is additive in its third argument, such that $\{a, b, c + d\} = \{a, b, c\} + \{a, b, d\}$.

For any real number $r$ and elements $a, b, c$ in a Jordan algebra $E$, the Jordan triple product $\{a, b, c\}$ satisfies $\{a, b, r \cdot c\} = r \cdot \{a, b, c\}$. This demonstrates that the Jordan triple product is real linear in its third variable.

For any elements $a, b, c, d$ in a Jordan algebra $E$, the Jordan triple product $\{a, b, c\}$ satisfies the additivity property in the middle variable: $$\{a, b + c, d\} = \{a, b, d\} + \{a, c, d\}$$

For any real number $r \in \mathbb{R}$ and elements $a, b, c$ in a Jordan algebra $E$, the Jordan triple product $\{a, b, c\}$ satisfies the property of scalar homogeneity in the middle variable: $$\{a, r \cdot b, c\} = r \cdot \{a, b, c\}$$ where $\{a, b, c\} = a \circ (b \circ c) + c \circ (b \circ a) - (a \circ c) \circ b$.