PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.StructureAlgebra

This instance allows a Jordan derivation $D$ on a Jordan algebra $E$ to be treated as a function from $E$ to $E$. It enables the notation $D(x)$ for $x \in E$, which evaluates the underlying linear map of the derivation at $x$.

There is a canonical coercion that allows a Jordan derivation $D$ on a Jordan algebra $E$ to be treated as an $\mathbb{R}$-linear map from $E$ to itself.

Let $E$ be a real vector space with a multiplication operation. If $D_1$ and $D_2$ are Jordan derivations on $E$ such that $D_1(x) = D_2(x)$ for all $x \in E$, then $D_1 = D_2$.

Let $E$ be a Jordan algebra and $D$ be a Jordan derivation on $E$. For any elements $x, y \in E$, the identity $D(x \cdot y) = D(x) \cdot y + x \cdot D(y)$ holds, where $\cdot$ denotes the Jordan product.

Let $E$ be a real vector space equipped with a multiplication operation. The set of all $\mathbb{R}$-linear endomorphisms $D: E \to E$ that satisfy the Leibniz rule $D(x \cdot y) = D(x) \cdot y + x \cdot D(y)$ for all $x, y \in E$ forms a submodule of the space of linear endomorphisms $\text{End}_{\mathbb{R}}(E)$.

The zero element in the space of Jordan derivations on a real vector space $E$ is the derivation defined by the linear map $D : E \to E$ that maps every element $x \in E$ to $0$. In the context of Jordan algebras, this represents the zero infinitesimal symmetry.

Let $E$ be a Jordan algebra. For any two Jordan derivations $D_1, D_2: E \to E$, their sum $D_1 + D_2$ is defined as the map $(D_1 + D_2)(x) = D_1(x) + D_2(x)$ for all $x \in E$. This operation defines the addition for the type of Jordan derivations, where the sum of two linear maps satisfying the Leibniz rule is again a Jordan derivation.

Let $E$ be a Jordan algebra. For any Jordan derivation $D$ on $E$, its negation $-D$ is defined as the Jordan derivation whose underlying linear map is the pointwise negation of $D$. That is, for all $x \in E$, $(-D)(x) = -(D(x))$. This negation operation ensures that the set of Jordan derivations is closed under additive inversion.

Let $E$ be a Jordan algebra. For any element $x \in E$, the zero Jordan derivation $0 : E \to E$ maps $x$ to the zero element of $E$, that is, $0(x) = 0$.

Let $E$ be a Jordan algebra. For any two Jordan derivations $D_1, D_2$ on $E$ and any element $x \in E$, the sum of the derivations $D_1 + D_2$ satisfies $(D_1 + D_2)(x) = D_1(x) + D_2(x)$.

Let $E$ be a Jordan algebra and $D$ be a Jordan derivation on $E$. For any element $x \in E$, the application of the negated derivation $-D$ to $x$ is equal to the negation of the result of applying $D$ to $x$, that is, $(-D)(x) = -D(x)$.

The scalar multiplication of a Jordan derivation $D$ on a Jordan algebra $E$ by a real number $c \in \mathbb{R}$ is defined such that for any $x \in E$, $(c \cdot D)(x) = c \cdot D(x)$. This operation preserves the property of being a Jordan derivation (an infinitesimal reversible symmetry of the algebra).

Let $E$ be a Jordan algebra. For any real number $c \in \mathbb{R}$, any Jordan derivation $D$ on $E$, and any element $x \in E$, the application of the scalar multiple $c \cdot D$ to $x$ satisfies $(c \cdot D)(x) = c \cdot D(x)$.

Let $E$ be a Jordan algebra. For any Jordan derivation $D$ on $E$, the identity element $1$ is mapped to zero, such that $D(1) = 0$.

Let $E$ be a Jordan algebra. The set of Jordan derivations on $E$ forms an additive commutative group (abelian group). The group operations are defined pointwise: for any Jordan derivations $D, D_1, D_2$ and any $x \in E$, the sum $(D_1 + D_2)(x) = D_1(x) + D_2(x)$, the zero element $0(x) = 0$, and the negation $(-D)(x) = -D(x)$. These operations satisfy the standard axioms of an additive commutative group, including associativity $(D_1 + D_2) + D_3 = D_1 + (D_2 + D_3)$, commutativity $D_1 + D_2 = D_2 + D_1$, and the existence of an additive identity and inverses.

The set of Jordan derivations on a Jordan algebra $E$ forms a vector space over the field of real numbers $\mathbb{R}$. This structure defines how Jordan derivations are scaled by real numbers and ensures that the collection of all such derivations satisfies the axioms of a real vector space (or $\mathbb{R}$-module), including the distributivity of scalar multiplication over both derivation addition and real number addition, as well as the associativity of scalar multiplication.

For any two Jordan derivations $D_1$ and $D_2$ on a Jordan algebra $E$, their commutator $[D_1, D_2]$ is defined as the linear map $D_1 \circ D_2 - D_2 \circ D_1$. This commutator is itself a Jordan derivation.

For any two Jordan derivations $D_1$ and $D_2$ on a Jordan algebra $E$, the commutator $\text{comm}(D_1, D_2)$ applied to an element $x \in E$ is equal to $D_1(D_2(x)) - D_2(D_1(x))$.

This instance defines the canonical bracket notation $\llbracket \cdot, \cdot \rrbracket$ for the space of Jordan derivations on a Jordan algebra $E$. For any two Jordan derivations $D_1$ and $D_2$, the bracket is given by the commutator $\llbracket D_1, D_2 \rrbracket = D_1 \circ D_2 - D_2 \circ D_1$.

For any Jordan algebra $E$, any two Jordan derivations $D_1, D_2$ on $E$, and any element $x \in E$, the Lie bracket $\llbracket D_1, D_2 \rrbracket$ applied to $x$ is given by the commutator formula $\llbracket D_1, D_2 \rrbracket x = D_1(D_2(x)) - D_2(D_1(x))$.

The set of Jordan derivations on a Jordan algebra $E$ forms a Lie ring. The Lie bracket $\llbracket D_1, D_2 \rrbracket$ is defined as the commutator of derivations, given by $\llbracket D_1, D_2 \rrbracket = D_1 \circ D_2 - D_2 \circ D_1$. This structure satisfies the Lie ring axioms, including bi-additivity of the bracket, the alternating property $\llbracket D, D \rrbracket = 0$, and the Jacobi identity.

The set of Jordan derivations on a Jordan algebra $E$ forms a Lie algebra over the real numbers $\mathbb{R}$. This structure equips the real vector space of Jordan derivations with a Lie bracket $\llbracket \cdot, \cdot \rrbracket$ defined by the commutator of derivations, $\llbracket D_1, D_2 \rrbracket = D_1 \circ D_2 - D_2 \circ D_1$, satisfying the compatibility condition between scalar multiplication and the Lie bracket: $c \cdot \llbracket D_1, D_2 \rrbracket = \llbracket c \cdot D_1, D_2 \rrbracket$ for any $c \in \mathbb{R}$.

Given two elements $a$ and $b$ in a Jordan algebra $E$, the inner Jordan derivation $\text{inner}(a, b)$ is the derivation on $E$ defined by the action $x \mapsto a(bx) - b(ax)$ for all $x \in E$. This map corresponds to the commutator $[L_a, L_b]$ of the left-multiplication operators $L_a$ and $L_b$.

Let $E$ be a Jordan algebra. For any elements $a, b, x \in E$, the value of the inner derivation $\text{inner}(a, b)$ applied to $x$ is given by the formula: $$\text{inner}(a, b)(x) = a \cdot (b \cdot x) - b \cdot (a \cdot x)$$ where $\cdot$ denotes the Jordan product. This identity expresses the derivation as the commutator of the left-multiplication operators $L_a$ and $L_b$ acting on $x$.

Let $E$ be a Jordan algebra. For any elements $a, b, c, d \in E$, let $\text{inner}(a, b)$ denote the inner Jordan derivation generated by $a$ and $b$, which acts on an element $x$ as $\text{inner}(a, b)x = a(bx) - b(ax)$. The commutator of two such derivations satisfies the identity: $$[\text{inner}(a, b), \text{inner}(c, d)] = \text{inner}(\text{inner}(a, b)c, d) + \text{inner}(c, \text{inner}(a, b)d)$$ where $[\delta_1, \delta_2] = \delta_1 \circ \delta_2 - \delta_2 \circ \delta_1$ is the commutator of Jordan derivations.