Nonassociative algebras with scalar involution (Q762595)

Let k be an arbitrary commutative associative ground ring with unity. Consider a nonassociative k-algebra A with unity which is a faithful k- module. Suppose that A admits an involution \(x^*\) such that \(n(x)=xx^*\) and \(t(x)=x+x^*\) belongs to k. The following are equivalent: (i) \(n(x+y)-n(x)-n(y)=t(xy^*)\) for any x,y\(\in A\); (ii) \(t(1)=2\), \(n(1)=1\) and \(x^ 2-t(x)x+n(x)=0\) for any \(x\in A.\) The author gives a characterization of flexible algebras A which satisfy conditions (i), (ii). Suppose A admits a nondegenerate quadratic form Q such that \(Q(xy)=Q(x)Q(y)\). Then A is alternative with conditions (i), (ii) where \(Q(x)=n(x)\). Under some restrictions any alternative k-algebra with (i), (ii) has a multiplicative norm n(x). A characterization of those alternative k-algebras A is given in which (Rad n)\({}^ 3=0.\) Finally, the author considers the Cayley-Dickson construction C(A,\(\mu)\), \(\mu\in k\setminus 0\). The paper presents a characterization of commutative, associative, alternative, flexible and simple algebras C(A,\(\mu)\). If A is a \(2^ n\)-dimensional algebra obtained from k by repeated application of the Cayley-Dickson process, then the commutative center of A coincides with the center C(A). If \(n\geq 2\), then \(C(A)=k\). If \(n\geq 3\), then the nucleus of A equals k.