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

[摘要] 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.

[学科分类] 数学（综合）

[关键词] 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