A Coordinatization Theorem for the Jordan algebra of symmetric 2x2 matrices

Abstract

The Jacobson Coordinatization Theorem describes the structure of unitary Jordan algebras containing the algebra $H_n(F)$ of symmetric nxn matrices over a field F with the same identity element, for $n\geq 3$. In this paper we extend the Jacobson Coordinatization Theorem for n=2. Specifically, we prove that if J is a unitary Jordan algebra containing the Jordan matrix algebra $H_2(F)$ with the same identity element, then J has a form $J=H_2(F)\otimes A_0+k\otimes A_1$, where $A=A_0+A_1$ is a $Z_2$-graded Jordan algebra with a partial odd Leibniz bracket {,} an $k=e_{12}-e_{21}\in M_2(F)$ with the multiplication given by $(a\otimes b)(c\otimes d)=ac\otimes bd + [a,c]\otimes \{b,d\},$ the commutator [a,c] is taken in $M_2(F)$.