Lattice isomorphisms of Jordan Algebras

Resumen

The author considers Jordan algebras which have isomorphic lattices of subalgebras. The analogous question had been treated previously for other types of algebras [D. W. Barnes, J. Austral. Math. Soc. 4 (1964), 470--475; J. Austral. Math. Soc. 6 (1966), 106--121; E. I. Chupina, Sibirsk. Mat. Zh. 28 (1987), no. 5, 198--202; J. A. Laliena Clemente, J. Algebra 128 (1990), no. 2, 335--355]. Let M be a finite-dimensional Jordan algebra over a field F of characteristic ≠2 whose lattice of subalgebras is isomorphic to the lattice of subalgebras of the Jordan algebra H(Fn,t) of symmetric n×n matrices over F. The main result of the paper is that M is isomorphic to the Jordan matrix algebra H(Fn,Ja). In the proof the author shows that M contains n connected orthogonal idempotents and then applies Jacobson's coordinatization theorem. An important method in the existence proof of such a family of idempotents is the study of subalgebras of small (≤3) length, where the length of a Jordan algebra is the length of the longest chain of subalgebras. In fact, as preparation for the main theorem the author first proves several classification results on Jordan algebras of length ≤3.