Lorentz Euclidean Jordan Algebra

A Lorentz Euclidean Jordan algebra is one of the five possible types of Euclidean Jordan algebra. In a Euclidean space Z, select some vector 1 having norm one. Define a Jordan product to make Z a Euclidean Jordan algebra in which this vector is the unit. Define X as the orthogonal complement of $$\mathbf {R}\left \{\mathbf {1}\right \}$$ , so $$\mathbf {Z}=\mathbf {X}\oplus \mathbf {R}\left \{\mathbf {1}\right \}$$ . Express a vector z as a pair $$\left ( \boldsymbol {x},y\right )$$ in the direct sum, such that $$\begin {aligned}y & =\left \langle \mathbf {1},\boldsymbol {z}\right \rangle \\ \boldsymbol {x} & =\boldsymbol {z}-y\mathbf {1}. \end {aligned}$$ The Lorentz Jordan product is $$\left ( \boldsymbol {x}^{\ast },y^{\ast }\right ) \bullet \left ( \boldsymbol {x},y\right ) =\left ( y\boldsymbol {x}^{\ast }+y^{\ast }\boldsymbol {x},\left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}\right \rangle +y^{\ast }y\right ) \!.$$ Denote the Lorentz Euclidean Jordan algebra by LX. The nonnegative cone is the $$\left ( \boldsymbol {x},y\right )$$ such that $$y\geq \left | \boldsymbol {x}\right |$$ , commonly called the “ice-cream” cone. There are an infinite number of primitive idempotents, and the nonnegative cone is not polyhedral. The linear representation $${\textit {{\sf { {L}}}}}_{\left ( \boldsymbol {x},y\right ) }\left ( \boldsymbol {x}^{\ast },y^{\ast }\right ) =\left ( \boldsymbol {x}^{\ast },y^{\ast }\right ) \bullet \left ( \boldsymbol {x},y\right )$$ is the arrow-shaped partitioned linear transformation $$\left [ \begin {array}[c]{c} \boldsymbol {x}^{\ast }\\ y^{\ast } \end {array} \right ] \mapsto \left [ \begin {array}[c]{cc} y\mathsf {I} & \boldsymbol {x}\\ \boldsymbol {x}^{\top } & y \end {array} \right ] \left [ \begin {array}[c]{c} \boldsymbol {x}^{\ast }\\ y^{\ast } \end {array} \right ] \!.$$ Here $${\textit {{\sf { {L}}}}}_{\left ( \boldsymbol {x},y\right ) }$$ is nonnegative if and only if $$\left ( \boldsymbol {x},y\right )$$ is nonnegative: all eigenvalues are nonnegative if and only if $$y\geq \left | \boldsymbol {x}\right |$$ . This book applies Lorentz Euclidean Jordan algebra to finance theory. In the vector $$\left ( \boldsymbol {x},y\right )$$ , x is a portfolio, and the optimum y is standard deviation.