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Novikov-Poisson algebras. (English) Zbl 0872.17030

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

(xy)z=(xz)y,
(xy)z-x(yz)=(yx)z-y(xz)·

A Novikov-Poisson algebra is a vector space A with two operations “·,” such that (A,·) forms a commutative associative algebra (not necessary unital) and (A,) forms a Novikov algebra for which

(x·y)z=x·(yz),(xy)·z-x(y·z)=(yx)·z-y(x·z)·

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:
17D25Lie-admissible algebras
17A30Nonassociative algebras satisfying other identities
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
17B68Virasoro and related algebras
17B70Graded Lie (super)algebras