PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.FreeSpecialTwo

The free associative algebra over the real numbers $\mathbb{R}$ generated by two elements. This corresponds to the algebra of polynomials in two non-commuting variables with real coefficients.

In the free associative real algebra on two generators, the natural number $2$ is an invertible element. This is established by mapping the inverse of the real number $2$ into the algebra via the canonical algebra map $\mathbb{R} \to \text{FreeAssocTwo}$.

The algebra $\text{SymAlg}(\text{FreeAssocTwo})$, which is the symmetrization of the free associative algebra over $\mathbb{R}$ on two generators, is equipped with the structure of a non-associative commutative ring. The addition is inherited from the underlying associative algebra, and the multiplication is the symmetrized Jordan product $x \circ y = \frac{1}{2}(xy + yx)$.

This definition provides a scalar multiplication of the Jordan algebra $\text{SymAlg}(\text{FreeAssocTwo})$ on itself. For any two elements $x, y \in \text{SymAlg}(\text{FreeAssocTwo})$, the scalar multiplication $x \cdot y$ is defined as their product in the algebra, which is the symmetrized Jordan product $x \circ y = \frac{1}{2}(xy + yx)$ derived from the underlying free associative algebra $\text{FreeAssocTwo}$.

Let $J = \text{SymAlg}(\text{FreeAssocTwo})$ be the Jordanization of the free associative algebra over $\mathbb{R}$ on two generators, where the Jordan product is defined as $x \circ y = \frac{1}{2}(xy + yx)$. For any scalar $r \in \mathbb{R}$ and elements $x, y \in J$, the scalar tower property $(r \cdot x) \circ y = r \cdot (x \circ y)$ holds.

In the Jordanization of the free associative real algebra on two generators, denoted by $\text{SymAlg}(\text{FreeAssocTwo})$, scalar multiplication by real numbers commutes with the Jordan product. Specifically, for any scalar $r \in \mathbb{R}$ and any elements $x, y \in \text{SymAlg}(\text{FreeAssocTwo})$, the following equality holds: $$(r \cdot x) \circ y = x \circ (r \cdot y)$$ where $\circ$ denotes the symmetrized Jordan product defined as $x \circ y = \frac{1}{2}(xy + yx)$.

The element representing the first canonical generator $X$ in the Jordanization of the free associative real algebra on two generators. The Jordanization, denoted $\text{SymAlg}(\text{FreeAssocTwo})$, is the algebra equipped with the symmetrized Jordan product $a \circ b = \frac{1}{2}(ab + ba)$, and this definition identifies the first generator (index 0) of the underlying free associative algebra as an element of this Jordan algebra.

The element in the Jordanization of the free associative real algebra on two generators that corresponds to the second canonical generator $Y$. The Jordanization $\text{SymAlg}(\text{FreeAssocTwo})$ is the algebra equipped with the symmetrized Jordan product $a \circ b = \frac{1}{2}(ab + ba)$.

The $\mathbb{R}$-submodule of the Jordanization $\text{SymAlg}(\text{FreeAssocTwo})$ generated by the two canonical elements $X$ and $Y$. Here, $\text{SymAlg}(\text{FreeAssocTwo})$ is the free associative algebra over $\mathbb{R}$ on two generators equipped with the symmetrized Jordan product $x \circ y = \frac{1}{2}(xy + yx)$. This submodule constitutes the carrier set of the free special Jordan algebra on two generators.

The type representing the free special Jordan algebra on two generators. It is defined as the unital real Jordan subalgebra of $\text{SymAlg}(\text{FreeAssocTwo})$ generated by the two canonical elements $X$ and $Y$. Here, $\text{SymAlg}(\text{FreeAssocTwo})$ denotes the Jordanization of the free associative algebra over $\mathbb{R}$ on two generators, equipped with the symmetrized Jordan product $x \circ y = \frac{1}{2}(xy + yx)$.

Let $\text{FreeAssocTwo}$ be the free associative algebra over $\mathbb{R}$ on two generators, and let $\text{SymAlg}(\text{FreeAssocTwo})$ be its Jordanization with the symmetrized product $x \circ y = \frac{1}{2}(xy + yx)$. If $X$ is the first canonical generator (denoted `freeSpecialX`) and $\text{freeSpecialJordanTwo}$ is the $\mathbb{R}$-submodule of $\text{SymAlg}(\text{FreeAssocTwo})$ generated by the two canonical generators, then $X \in \text{freeSpecialJordanTwo}$.

Let $Y$ be the second canonical generator of the free associative algebra over $\mathbb{R}$ on two generators. Let the free special Jordan algebra on two generators be the $\mathbb{R}$-submodule of the Jordanization (equipped with the symmetrized product $x \circ y = \frac{1}{2}(xy + yx)$) generated by the two canonical elements $X$ and $Y$. Then $Y$ is an element of this free special Jordan algebra.

This function maps the two indices of $\{0, 1\}$ to the two canonical generators of the free special Jordan algebra on two generators. Specifically, index $0$ is mapped to the first generator $X$ and index $1$ is mapped to the second generator $Y$, both regarded as elements of the subalgebra $\text{FreeSpecialJordanTwo} \subseteq \text{SymAlg}(\text{FreeAssocTwo})$.

The canonical unital Jordan homomorphism from the free unital real Jordan algebra on two generators, $\text{FreeJordanTwo}$, to the free special Jordan algebra on two generators, $\text{FreeSpecialJordanTwo}$. This map is defined using the universal property of the free Jordan algebra by sending its two abstract generators to the canonical generators $X$ and $Y$ of the special Jordan algebra. Shirshov's theorem states that this homomorphism is injective.

Let $\text{FreeJordanTwo}$ be the free unital real Jordan algebra on two generators and $\text{FreeSpecialJordanTwo}$ be the free special Jordan algebra on two generators. Let $\phi: \text{FreeJordanTwo} \to \text{FreeSpecialJordanTwo}$ be the canonical unital Jordan homomorphism. Let $x$ be the first abstract generator of $\text{FreeJordanTwo}$ and $X$ be the first canonical generator of $\text{FreeSpecialJordanTwo}$ (associated with index 0). Then $\phi(x) = X$.

Let $\text{FreeJordanTwo}$ be the free unital real Jordan algebra on two generators and $\text{FreeSpecialJordanTwo}$ be the free special Jordan algebra on two generators. Let $\phi: \text{FreeJordanTwo} \to \text{FreeSpecialJordanTwo}$ be the canonical unital Jordan homomorphism. If $y \in \text{FreeJordanTwo}$ is the second abstract generator and $Y \in \text{FreeSpecialJordanTwo}$ is the second canonical generator (indexed by $1$), then $\phi(y) = Y$.