Degree three, four, and five identities of quadratic algebras
Let \(C\) be a class of nonassociative algebras over a field \(F\). Then the polynomial identity \(f(x_1, \dots, x_n)\) is a special central identity for all algebras in \(C\) if there is an algebra in \(C\) for which \(f(x_1, \dots, x_n)\) is not an identity, but for any \(A\) in \(C\) one has \(f(a_1, \dots, a_n)\in F\) for all \(a_1, \dots, a_n\in A\). Assume that \(\text{char} (F)=0\), or is greater than the degree of the identity considered. The authors determine sets of identities which generate all degree 3, 4 and 5 polynomial identities of quadratic algebras and symmetric quadratic algebras. They also obtain the special central identities of degree 3 and 4, and some results for Jordan algebras of degree 2.
scientific article; zbMATH DE number 1214138