无幂等元的非可解演化代数（$\mathbb{C}$ 上）
Idempotent-free non-solvable evolution algebras over $\mathbb{C}$
查看机构详情
- School of Mathematics and Statistics, Hanjiang Normal University（汉江师范大学数学与统计学院）
- School of Mathematics and Physics, Xinjiang Hetian College（新疆和田学院数学与物理学院）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文构造了最小维数（三维）复数演化代数反例，反驳了“可解等价于无非零幂等元”的猜想，并证明了稳定项为演化代数时该等价性成立。
中文摘要 AI 辅助
近期的一个猜想指出：有限维复数演化代数是可解的，当且仅当它没有非零幂等元。我们给出了一个三维反例（在 $\mathbb{C}$ 上），其同构类已出现在三维复数演化代数的分类中。由于该猜想在一维和二维情形下已知成立，因此这个反例具有可能的最小维数。该代数定义在特征不为 $2$ 的任意域上，并且是一个单参数族中两两不同构的非可解演化代数的例外成员，该族中所有代数的幂等元都被显式确定。对于例外参数，导出级数稳定在一个非零的二维子代数上，该子代数唯一的幂等元为零。与零代数的直和给出了所有维数至少为 $3$ 的复数反例。我们还证明了当导出级数的稳定项是演化代数时，所猜想的等价性成立。
英文摘要
A recent conjecture states that a finite-dimensional complex evolution algebra is solvable if and only if it has no non-zero idempotents. We exhibit a three-dimensional counterexample over $\mathbb{C}$ whose isomorphism class already appears in the classification of three-dimensional complex evolution algebras. Since the conjecture is known in dimensions one and two, this counterexample has the smallest possible dimension. The algebra is defined over every field of characteristic different from $2$ and is the exceptional member of a one-parameter family of pairwise non-isomorphic non-solvable evolution algebras whose idempotents are determined explicitly. For the exceptional parameter, the derived series stabilises at a non-zero two-dimensional subalgebra whose only idempotent is zero. Direct sums with zero algebras give complex counterexamples in every dimension at least three. We also prove the conjectured equivalence whenever the stable term of the derived series is an evolution algebra.
