Lattice Isomorphisms of Jordan Algebras over Arbitrary Fields

Abstract

We study Jordan algebras M whose lattice of subalgebras is isomorphic to the lattice of subalgebras of a Jordan matrix algebra, J = H(D n , J A ), where D is either a quadratic extension field (if n ≥ 2), a central division quaternion algebra (if n ≥ 3) or a central division Cayley-Dickson algebra (if n = 3). We prove that M is also a Jordan matrix algebra of the same kind as J.