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

Сибирский математический журнал

article

V. N. Zhelyabin¹  A. S. Zakharov² 

[1] Sobolev Institute of Mathematics;Novosibirsk State University;Novosibirsk State Technical University

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;   

DOI  :  10.1134/S0037446620040072

学科分类：数学（综合）

来源: Izdatel stvo Instituta Matematiki Rossiiskoi Akademii Nauk

【 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.

【 授权许可】

CC BY