Primitive noncommutative Jordan algebras with nonzero socle

Abstract

Let A A be a nondegenerate noncommutative Jordan algebra over a field K K of characteristic ≠ 2 \ne 2 . Defining the socle S ( A ) S(A) of A A to be the socle of the plus algebra A + {A^ + } , we prove that S ( A ) S(A) is an ideal of A A ; then we prove that if A A has nonzero socle, A A is prime if and only if it is primitive, extending a result of Osborn and Racine [6] for the associative case. We also describe the prime noncommutative Jordan algebras with nonzero socle and in particular the simple noncommutative Jordan algebras containing a completely primitive idempotent. In fact we prove that a nondegenerate prime noncommutative Jordan algebra with nonzero socle is either (i) a noncommutative Jordan division algebra, (ii) a simple flexible quadratic algebra over an extension of the base field, (iii) a nondegenerate prime (commutative) Jordan algebra with nonzero socle, or (iv) a K K -subalgebra of L W ( V ) ( λ ) {L_W}{(V)^{(\lambda )}} containing F W ( V ) {F_W}(V) or of H ( L V ( V ) , ∗ ) ( λ ) H{({L_V}(V), * )^{(\lambda )}} containing H ( F V ( V ) , ∗ ) H({F_V}(V), * ) where in the first case ( V , W ) (V,W) is a pair of dual vector spaces over an associative division K K -algebra D D and λ ≠ 1 / 2 \lambda \ne 1/2 is a central element of D D , and where in the second case V V is self-dual with respect to an hermitian inner product ( | ) , D (|),D has an involution α → α ¯ \alpha \to \bar \alpha and λ ≠ 1 / 2 \lambda \ne 1/2 is a central element of D D with λ + λ ¯ = 1 \lambda + \bar \lambda = 1 .