Associative bilinear forms in some baric algebras

Abstract

Let ( A, ω) be a K-algebra, e ϵ A a nontrivial idempotent, K an infinite field whose characteristic is different from 2, and A = K e ⊕ U ⊕ V, where U = { x ϵ ker ω| e x = 1 2 x}. If B : A × A → K is an associative bilinear form, we prove the two following main facts: (1) If A is a kth order Bernstein algebra and B is a nondegenerate form, then A is a kth order quasiconstant algebra and the idempotent is unique. (2) If x 3 - (1 + γ)ω( x) x 2 + γω( x) 2 x = 0 is the rank equation of A and if B is nondegenerate, then A is a Jordan algebra. If 0 ≠ γ ≠ 1, then B , the symmetry of B only depends on the symmetry of B | v , and B ( e 1, e 1) = B ( e, e) for any idempotent elements e, e 1 in A.