The structure of strongly prime quadratic Jordan algebras

Abstract

We extend the structure theory for strongly prime Jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2: we show that such J are either (1) hermitian forms H 0(A, ∗) ◀ J ⊂ H(Q(A), ∗) lying between an ample subspace of hermitian elements in a ∗-prime associative algebra A and those in its Martindale ring of quotients Q( A), (2) clifford forms with a scalar extension Ω J ̂ ⊃ J which is an algebra Ω J ̂ = J(Q, 1) of a nondegenerate quadratic from Q with basepoint 1, or (3) albert forms with a scalar extension Ω J ̂ ⊃ J which is a 27-dimensional split albert algebra Ω J ̂ = H 3(K(Ω)) of 3 × 3 hermitian matrices over an 8-dimensional split octonion algebra K(Ω). As a consequence, the simple Jordan algebras (of arbitrary dimension) are either hermitian H 0(A, ∗) for ∗-simple A, or clifford forms in J( Q, 1), or are albert algebras. In particular, our methods give an idempotent-free classification of all simple finite-dimensional algebras which are i-special. The key to this characterization is the existence of Zel'manov polynomials in the free special Jordan algebra, which are both hermitian polynomials (whose values always look like hermitian elements) and at the same time clifford polynomials (nontrivial on 3 × 3 hermitian matrices, hence on special algebras with ⩾ 3 interconnected idempotents): if a Zel'manov polynomial does not vanish on J then J is of hermitian type, and if it does vanish on J then J is of clifford type. One example of such a polynomial is q 48 = [[ p 16( x 1, y 1, z 1, w 1), p 16( x 2, y 2, z 2, w 2)], p 16( x 3, y 3, z 3, w 3)] for p 16 = [[ D 2 x, y ( z) 2, D x, y ( w)], D x, y ( w)] ( D x, y ( z) = [[ x, y], z]). Thus the structure of Jordan algebras depends very directly on polynomial identities and commutators.