Semi-isotopies and the lattice of inner ideals of certain quadratic Jordan algebras

Abstract:

The concept of isotopy plays an extremely important role in the structure theory of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. We take Koecher’s characterization of isotopy and use it as the basis of a definition of semi-isotopy. It is clear that semi-isotopies induce, in a natural way, automorphisms of the lattice of inner ideals. We concern ourselves with the converse problem; namely, if $\eta$ is a semilinear bijection of a quadratic Jordan algebra such that $\eta$ induces an automorphism of the lattice of inner ideals, is $\eta$ necessarily a semi-isotopy? We answer the above question in the affirmative for a large class of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals (said class includes all such algebras of capacity at least three over fields of characteristic unequal to two). Moreover, we prove that the only such maps which induce the identity automorphism on the lattice are the scalar multiplications.