Zelmanov's prime theorem for quadratic Jordan algebras

Abstract

E. I. Zelmanov has recently shown that any prime linear Jordan algebra without nil ideals (but no finiteness conditions imposed) is either a homomorphic image of a special algebra or is a form of the 27-dimensional exceptional algebra. In particular, all division algebras and all simple unital algebras are either derived from associative algebras or are 27.dimensional over their centers. This effectively ends the search for infinite-dimensional exceptional algebras. In the present paper, Zelmanov’s result is extended to quadratic Jordan algebras. Jordan algebras were invented in the 1930s in the search for an exceptional algebraic setting for quantum mechnics: an algebraic system which behaved like the usual algebra of quantum mechanical observables (hermitian operators on Hilbert space), but was exceptional in the sense that its structure was not determined behind the scenes by some unobservable associative algebra. In their pioneering paper of 1933, [ 91 Jordan, von Neumann, and Wigner classified the finite-dimensional formally real linear Jordan algebras, and A. A. Albert showed [ 11 that the only simple algebra in the list which was exceptional was a certain 27.dimensional algebra of 3 X 3 hermitian matrices with entries from an gdimensional Cayley algebra. All subsequent investigations of exceptional Jordan algebras led back to this same Albert algebra. The next breakthrough in the structure of Jordan algebras was N. Jacobson’s 1965 introduction [6 J of inner ideals and the resultant quadratilication of the theory (in terms of the quadratic product xyx instead of the linear product xy + .vx). The final breakthrough was the astounding 1979 result of a young Russian mathematician, E. I. Zelmanov, that the only simple exceptional linear Jordan are Albert algebras. Thus the search for an infinite-dimensional setting for exceptional quantum mechanics is doomed to failure: the only simple exceptional structure allotted to mortals is the 27-dimensional Albert algebra. This came as a complete surprise to Western researchers (who hoped the free Jordan algebra might lead to an 291 002 I-8693/82/060297-30$02.00/0