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Infinite dimensional Novikov algebras of characteristic $O$. (English) Zbl 0814.17002
An algebra is called Novikov if it satisfies the identities $\left(x,y,z\right)=\left(y,x,z\right)$ and $\left(xy\right)z=\left(xz\right)y$. Let $A$ be a simple infinite-dimensional Novikov algebra over the field $F$ of characteristic 0, where each element of $F$ has an $n$-th root in $F$ for every positive integer $n$. If $A$ has an element $e$ with the properties ${e}^{2}=be$ for some $b\in F$, and $A={\sum }_{\alpha }{A}_{\alpha }^{\text{'}}$ where ${A}_{\alpha }^{\text{'}}=\left\{x\in A\mid {\left({L}_{e}-\left(\alpha +b\right)\right)}^{n}x=0$ for some $n\right\}$, then the author shows that $A$ has to belong to one of three very specific classes. He then finds the derivations of these algebras, discusses when two of these algebras can be isomorphic, and proves three results relating two of these classes to concepts developed in a forthcoming joint paper with E. I. Zelmanov [Nonassociative algebras related to Hamiltonian operators in the formal calculus of variations, J. Pure Appl. Algebra].

##### MSC:
 17A30 Nonassociative algebras satisfying other identities