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Novikov-Poisson algebras. (English) Zbl 0872.17030

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

$\left(x\circ y\right)\circ z=\left(x\circ z\right)\circ y,$
$\left(x\circ y\right)\circ z-x\circ \left(y\circ z\right)=\left(y\circ x\right)\circ z-y\circ \left(x\circ z\right)·$

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

$\left(x·y\right)\circ z=x·\left(y\circ z\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(x\circ y\right)·z-x\circ \left(y·z\right)=\left(y\circ x\right)·z-y\circ \left(x·z\right)·$

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