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Simple Novikov algebras with an idempotent. (English) Zbl 0772.17001

An algebra is called Novikov if it satisfies the identities $\left(x,y,z\right)=\left(y,x,z\right)$ and $\left(xy\right)z=\left(xz\right)y$. 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 $A$ be a simple finite-dimensional Novikov algebra, over a perfect field $F$ of characteristic $p>2$, that contains an idempotent $e$.

(a) Then for some integer $n$ and $a$, $c\in F$, $A$ has a basis $\left\{{x}_{i}\right\}$, where $-1\le i\le s={p}^{n}-2$, with the products of the basis elements given by ${x}_{i}{x}_{j}=\left(\genfrac{}{}{0pt}{}{i+j+1}{j}\right){x}_{i+j}+{\delta }_{i,-1}{\delta }_{j,-1}a{x}_{s}+{\delta }_{i,-1}{\delta }_{j,0}c{x}_{s}$.

(b) If $e$ is contained in the maximal subalgebra $S$ of $A$ spanned by the ${x}_{i}$’s with nonnegative subscripts, and it acts as the right identity on $S$, then there is a basis for $A$ which consists of generalized eigenvectors with respect to ${L}_{e}$, and which multiplies as in (a). In this basis, ${x}_{0}=e$.

(c) $e\notin S$, then for some finite additive subgroup ${\Delta }$ of $F$ and integer $r$, there is a basis $\left\{{x}_{\alpha k}\right\}$ consisting of generalized eigenvectors for $A$, where $\alpha \in {\Delta }$, $0\le k\le {p}^{r}-1$, and ${x}_{00}=e$. For these basis elements the products are given

${x}_{\alpha k}{x}_{\beta \ell }=\left(\beta +1\right)\left(\genfrac{}{}{0pt}{}{k+\ell }{k}\right){x}_{\alpha +\beta ,k+\ell }+\left(\genfrac{}{}{0pt}{}{k+\ell -1}{k}\right){x}_{\alpha +\beta ,k+\ell -1}·$

In addition, the algebras described in (a) and (c) are simple Novikov algebras. In particular, $r=0$ in (c) gives the simple algebras in [V. T. Filippov, Mat. Zametki 45, 101-105 (1989; Zbl 0659.17003)].

##### MSC:
 17A30 Nonassociative algebras satisfying other identities 17D99 Other nonassociative rings and algebras