Primitivity in Finitely Generated Special Quadratic Jordan Algebras

Abstract

Primitive Jordan algebras were introduced by Zelmanov in the linear case [13] and Hogben and McCrimmon in the quadratic case [4]. A Classification Theorem for quadratic primitive Jordan algebras, in the spirit of the general classification of prime nondegenerate Jordan algebras [9], is proven in [2]. In that result, besides Albert- and Clifford-type algebras, there appear special algebras containing an ideal of the form H 0( A, ∗) for a primitive associative algebra A with involution. To prove this result, a key fact is that, for a prime A, primitivity can be transferred from H 0( A, ∗) to A and back. The first part of this result stems from a more general theorem asserting that any tight associative envelope of a primitive Jordan algebra is primitive [2]. This can be extended to ∗-envelopes making use of the concept of ∗-primitivity: any ∗-tight associative envelope of a primitive Jordan algebra is ∗-primitive. The aim of this paper is to examine the reciprocal of that theorem for finitely generated Jordan algebras. We prove in Section 4 that a finitely generated special Jordan algebra having a ∗-primitive ∗-envelope is primitive. In the case of H( A, ∗) the proof makes uses of the Herstein second construction [7], for a semiprime A with involution ∗, any nonzero ideal of H( A, ∗) contains some H 0( I, ∗) with I a nonzero ∗-ideal of A. Our proof will follow a similar pattern. We prove in Section 2 that the Herstein-McCrimmon construction can be extended to prime finitely generated Jordan algebras with nonzero hermitian part. For that we make use of some basic properties of hermitian ideals as defined in [9]. In Section 3 we treat algebras an anti-hermitian type (called here simply anti-hermitian algebras) separately and we collect the previous results in Section 4 to prove our main theorem.