AI 中文总结
该研究引入两类伪欧几里得近结合代数，建立其与李、若当结构的对应，通过双扩张程序实现递归构造，为两类代数的描述与分类提供统一框架。
AI 中文摘要
我们引入反左不变伪欧几里得近结合代数与左不变伪欧几里得近结合代数的概念，二者分别源自与伪欧几里得李代数和若当代数相关的近结合Levi-Civita乘积。我们建立了这些代数类与其关联的李和若当结构的Levi-Civita乘积之间的对应关系。对于反左不变伪欧几里得近结合代数，我们证明其幂零指数至多为5，并将其刻画为雅可比-若当容许近结合代数；进一步证明其关联的伪欧几里得雅可比-若当代数是循环的。受Medina和Revoy经典双扩张的启发，我们为这类代数引入双扩张程序，并证明每个反左不变伪欧几里得近结合代数均可通过有限次此类扩张从平凡伪欧几里得代数得到。对于左不变伪欧几里得近结合代数，我们证明其关联的伪欧几里得李代数是两步可解且循环的；随后开发块、平面和线性双扩张，并证明每个左不变伪欧几里得近结合代数均可通过块双扩张从二次交换结合代数递归构造，且在实数域上，每个此类代数均可通过平面双扩张从二次交换结合代数递归构造。这些递归构造为在反左不变和左不变两种情形下描述和分类伪欧几里得近结合代数提供了统一框架。
英文摘要
We introduce the notions of anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras, which arise as nearly associative Levi-Civita products associated with pseudo-Euclidean Lie and Jordan algebras, respectively. We establish a correspondence between these classes of algebras and the Levi-Civita products of their associated Lie and Jordan structures. For anti-left-invariant pseudo-Euclidean nearly associative algebras, we prove that they are nilpotent of index at most five and characterize them as Jacobi--Jordan-admissible nearly associative algebras. We further show that the associated pseudo-Euclidean Jacobi--Jordan algebras are cyclic. Motivated by the classical double extension of Medina and Revoy, we introduce a double extension procedure for this class of algebras and prove that every anti-left-invariant pseudo-Euclidean nearly associative algebra can be obtained from a trivial pseudo-Euclidean algebra by a finite sequence of such extensions. For left-invariant pseudo-Euclidean nearly associative algebras, we prove that the associated pseudo-Euclidean Lie algebras are two-step solvable and cyclic. We then develop block, planar, and linear double extensions and show that every left-invariant pseudo-Euclidean nearly associative algebra can be recursively constructed from a quadratic commutative associative algebra by means of block double extensions. Moreover, we prove that over the field of real numbers, every such algebra can be recursively constructed from a quadratic commutative associative algebra using planar double extensions. These recursive constructions provide a unified framework for describing and classifying pseudo-Euclidean nearly associative algebras in both the anti-left-invariant and left-invariant settings.
Comments43 pages
