Isomorphisms of the lattice of inner ideals of certain quadratic Jordan algebras

Abstract:

The inner ideals play a role in the theory of quadratic Jordan algebras analogous to that played by the one-sided ideals in the associative theory. In particular, the simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals play a role analogous to that of the simple artinian algebras in the associative theory. In this paper, we investigate the automorphism group of the lattice of inner ideals of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. For the case $\mathfrak {H}(\mathfrak {A}{,^ \ast })$ where $(\mathfrak {A}{,^ \ast })$ is a simple artinian algebra with hermitian involution, we show that the automorphism group of the lattice of inner ideals is isomorphic to the group of semilinear automorphisms of $\mathfrak {A}$. For the case $\mathfrak {H}({\mathfrak {Q}_n}{,^ \ast })$ where $\mathfrak {Q}$ is a split quaternion algebra, we obtain only a partial result. For the cases $J = \mathfrak {H}({\mathfrak {O}_3})$ and $J = {\text {Jord}}(Q,1)$ for $\mathfrak {O}$ an octonion algebra, $(Q,1)$ a nondegenerate quadratic form with base point of Witt index at least three and J finite dimensional, it is shown that every automorphism of the lattice of inner ideals is induced by a norm semisimilarity. Finally, we determine conditions under which two algebras of the type under consideration can have isomorphic lattices of inner ideals.