On a class of power-associative algebras
This paper is about finite-dimensional, commutative, power-associative algebras \(A\) over a field \(K\) of characteristic \(\neq 2\) which have an idempotent \(e\) such that the associated Peirce decomposition \(A=A_0 + A_{1/2} + A_1\) satisfies \(\dim (A_1)=1\) or \(A_0 = \{0\}\). It follows from an earlier result of the author that such algebras are train algebras. The algebras in the first class are shown to be Bernstein algebras of some order \(n\) provided they are Jordan. Conversely, any power-associative \(n\)-th order Bernstein algebra is contained in this class. The Jordan algebras in the second class are shown to admit a train equation \((L(x)-\omega (x) \text{id})^m(x) = 0\) for some \(m\).
scientific article; zbMATH DE number 871110
scientific article