Local Jordan algebras

Abstract:

A local Jordan algebra $\mathfrak {J}$ is a unital quadratic Jordan algebra in which $\operatorname {Rad} \mathfrak {J}$ is a maximal ideal, $\mathfrak {J}/\operatorname {Rad} \mathfrak {J}$ satisfies the DCC, and ${ \cap _k}\operatorname {Rad} {\mathfrak {J}^{(k)}} = 0$ where ${K^{(n + 1)}} = {U_K}(n){K^{(n)}}$. We show that the completion of a local Jordan algebra is also local Jordan, and if $\mathfrak {J}$ is a complete local Jordan algebra over a field of characteristic not 2, then either (1) $\mathfrak {J}$ is a complete completely primary Jordan algebra, (2) $\mathfrak {J} \cong {\mathfrak {J}_1} \oplus {\mathfrak {J}_2} \oplus S$ where each ${\mathfrak {J}_i}$ is a completely primary local Jordan algebra, or (3) $\mathfrak {J} \cong \mathfrak {H}({D_n},{J_a})$ where $(D,j)$ is either a not associative alternative algebra with involution or a complete semilocal associative algebra with involution.