The Superalgebras of Jordan Brackets Defined by the n-Dimensional Sphere

We study the generalized Leibniz brackets on the coordinatealgebra of the $ n $-dimensional sphere. In the case of the one-dimensionalsphere, we show that each of these is a bracket of vector type.Each Jordan bracket on the coordinatealgebra of the two-dimensional sphereis a generalized Poisson bracket. We equip the coordinate algebraof a sphere of odd dimension with a Jordan bracketwhose Kantor double is a simple Jordan superalgebra.Using such superalgebras, we provide some examplesof the simple abelian Jordan superalgebras whose odd part isa finitely generated projective module of rank 1in an arbitrary number of generators.An analogous result holds for theCartesian product of the sphere of even dimension and the affine line.In particular, in the case of the 2-dimensional spherewe obtain the exceptional Jordan superalgebra. Thesuperalgebras we constructed give new examples of simple Jordan superalgebras.