Abstract:
We study the generalized Leibniz brackets on the coordinate algebra of the $n$-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.
Keywords:associative commutative superalgebra, Jordan superalgebra, differential algebra, Grassmann algebra, superalgebra of a bilinear form, polynomial algebra, derivation, Jordan bracket, bracket of vector type, Poisson bracket, projective module, affine space, sphere.
Fulltext:
PDF file (518 kB)
