*(English)*Zbl 0872.17030

A Novikov algebra $A$ is a vector space with an operation $\circ $ satisfying the identities

A Novikov-Poisson algebra is a vector space $A$ with two operations “$\xb7,\phantom{\rule{4pt}{0ex}}\circ $” such that $(A,\xb7)$ forms a commutative associative algebra (not necessary unital) and $(A,\circ )$ forms a Novikov algebra for which

Novikov algebras were introduced in *A. A. Balinskij* and *S. P. Novikov* [Sov. Math., Dokl. 32, No. 1, 228-231 (1985; Zbl 0606.58018)] in connection with the Poisson brackets of hydrodynamical type. Novikov-Poisson algebras were introduced by the author [*X. Xu*, J. Algebra 185, 905-934 (1996; Zbl 0863.17003)] in order to establish a tensor theory of Novikov algebras. All the known simple Novikov algebras have had nonzero idempotent elements.

The author constructs the first class of simple Novikov algebras without idempotent elements, using a modification of the Filippov construction of simple Novikov algebras [*V. T. Filippov*, Mat. Zametki 45, 101-105 (1989; Zbl 0659.17003)], based on a certain associative commutative algebra with a derivation. Further, he classifies finite-dimensional Novikov-Poisson algebras of characteristic $p>2$ whose Novikov algebras are simple. In the characteristic 0 case, Novikov-Poisson algebras whose Novikov algebras are simple with an idempotent element are also classified.

Finally, the author proves that a family of Novikov-Poisson algebras induces a new family of infinite-dimensional simple Lie superalgebras, which are natural analogues of the quotient algebra of the super-Virasoro algebra over its one-dimensional center.

##### MSC:

17D25 | Lie-admissible algebras |

17A30 | Nonassociative algebras satisfying other identities |

37J99 | Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems |

17B68 | Virasoro and related algebras |

17B70 | Graded Lie (super)algebras |