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In 1974 Kharchenko proved that if a -component of an -graded associative algebra is PI then this algebra is PI. In the Novikov algebras of characteristic 0 the existence of a polynomial identity is equivalent to the solvability of the commutator ideal. We study a -graded Novikov algebra and prove that if the characteristic of the basic field is not 2 or 3 and its 0-component is associative or Lie-nilpotent of index 3 then the commutator ideal is solvable. [ABSTRACT FROM AUTHOR]

We establish the structure of the dual Lie coalgebra for a Lie algebra of the symplectic Poisson bracket (Jacobian-type Poisson bracket) on the algebra of polynomials in evenly many variables. We show that if the base field has characteristic zero then the -ary dual coalgebra for the Jacobian -Lie algebra consists of the same linear functionals as the dual coalgebra for the commutative polynomial algebra. [ABSTRACT FROM AUTHOR]

We study the generalized Leibniz brackets on the coordinate algebra of the -dimensional sphere. In the case of the one-dimensional sphere, we show that each of these is a bracket of vector type. Each Jordan bracket on the coordinate algebra of the two-dimensional sphere is a generalized Poisson bracket. We equip the coordinate algebra of a sphere of odd dimension with a Jordan bracket whose Kantor double is a simple Jordan superalgebra. Using such superalgebras, we provide some examples of the simple abelian Jordan superalgebras whose odd part is a finitely generated projective module of rank 1 in an arbitrary number of generators. An analogous result holds for the Cartesian product of the sphere of even dimension and the affine line. In particular, in the case of the 2-dimensional sphere we obtain the exceptional Jordan superalgebra. The superalgebras we constructed give new examples of simple Jordan superalgebras. [ABSTRACT FROM AUTHOR]

Let (V, Δ) be a Jordan copair over a field Φ and let V* be its dual pair. Then there exists a Lie coalgebra (Lc(V), ΔL) whose dual algebra (Lc(V))* is the Kantor–Koecher–Tits construction for the pair V*. If Φ is a field of characteristic other than 2 or 3 then the Lie coalgebra (Lc(J), ΔL) is locally finite-dimensional. As a corollary we derive that Jordan copairs over fields of characteristic other than 2 or 3 are locally finite-dimensional. [ABSTRACT FROM AUTHOR]