On commutative algebras of degree two

Abstract

Let 9 be a simple, commutative, power-associative algebra of degree 2 over an algebraically closed field a of characteristic not equal to 2, 3 or 5. The degree of 9 is defined to be the number of elements in the maximal set of pairwise orthogonal idempotents in W. This algebra has a unit element 1 [1, Theorem 3]. The algebras Wt of characteristic zero were considered by Kokoris [8] and found to be Jordan algebras. Kokoris also gave examples of algebras 9X that were not Jordan [6]. This left the problem of determining those algebras WI that are not Jordan algebras. Since 1 = e +f where e and f are primitive orthogonal idempotents, we have a decomposition 2 = Wfe(1) + 1e(1/2) + We(O) where x E We(A) if and only if ex = Ax. We have %,(A) = If(l A); ze(A)9e(1/2) C ,(1 A) + 9c(1/2) for A = 1,0; and 9We(1) = ea + 91., 9We(?) =f + So where 91 and 90 are nilideals of We(l) and 91e(0) respectively. If 2e(A)MIe(1/2) c SlXe(1/2) for A = 1,0 we say that e is a stable idempotent. If 9fe(A)9e(1/2) c 9I0(1/2) + 91At for A = 1,0 we say that e is a nilstable idempotent. The results of Albert extend the characteristic zero case to include algebras of characteristic p $ 2,3,5 for which every idempotent is stable [2]. He also characterized those algebras of characteristic p : 2,3,5 that have at least one stable idempotent [3; 4]. Recently Kokoris announced [9] that every simple, flexible, power-associative algebra over an algebraically closed field of characteristic : 2, 3 that is of degree two and in which every idempotent is nilstable is a J-simple algebra. It is the purpose of this paper to fill in the remaining gap by giving a characterization of those algebras ER that have an idempotent that is not nilstable. An example is also given of an algebra 9R that does not have a stable idempotent. 1. Let 9 be an algebra that is simple, commutative, power-associative, of degree two and whose base field a is an algebraically closed field of characteristic p $ 2,3,5. Let e be a primitive idempotent of E( that is not nilstable. Since 9 is power-associative we have X2X2 = X4 for all x E 9 and the linearization of this identity