An algebra is called Novikov if it satisfies the identities and . Simple finite-dimensional Novikov algebras, over a field of characteristic 0, were shown to be 1- dimensional in [E. I. Zel’manov, Dokl. Akad. Nauk SSSR 292, 1294- 1297 (1987; Zbl 0629.17002)]. In this paper simple finite-dimensional Novikov algebras with idempotent are classified as follows: Let be a simple finite-dimensional Novikov algebra, over a perfect field of characteristic , that contains an idempotent .
(a) Then for some integer and , , has a basis , where , with the products of the basis elements given by .
(b) If is contained in the maximal subalgebra of spanned by the ’s with nonnegative subscripts, and it acts as the right identity on , then there is a basis for which consists of generalized eigenvectors with respect to , and which multiplies as in (a). In this basis, .
(c) , then for some finite additive subgroup of and integer , there is a basis consisting of generalized eigenvectors for , where , , and . For these basis elements the products are given
In addition, the algebras described in (a) and (c) are simple Novikov algebras. In particular, in (c) gives the simple algebras in [V. T. Filippov, Mat. Zametki 45, 101-105 (1989; Zbl 0659.17003)].