AI 中文总结
研究配备弗罗贝尼乌斯形式的交换非结合实代数中诺顿不等式所强制的融合规则，回答相关问题，给出非退化和退化情况下的子代数包含关系及反例。
AI 中文摘要
我们研究配备弗罗贝尼乌斯形式的交换非结合实代数中由诺顿不等式强制的融合规则。我们回答了T. M. Mudziiri Shumba和S. Shpectorov关于与任意幂等元\(e\in A\)相关的特征子空间\(A_0(e)\)是否一定是\(A\)的子代数的问题。如果弗罗贝尼乌斯形式是非退化的，如在马约拉纳代数中，那么对于每个幂等元\(e\in A\)，\(A_0(e)\)和\(A_1(e)\)都是\(A\)的子代数，且\(A_0(e)A_1(e)\subseteq A_{1/2}(e)\)。在退化情况下，相应的包含关系在弗罗贝尼乌斯形式的根模下成立。我们还给出了一个具有退化弗罗贝尼乌斯形式且满足诺顿不等式的显式轴向代数，对于其一个轴，\(A_0(e)\)不是子代数。
英文摘要
The Norton inequality is one of the fundamental axioms in the theory of Majorana and axial algebras, yet its precise structural consequences have remained only partially understood. In this paper, we show that the Norton inequality alone forces the $0$- and $1$-eigenspace fusion rules for arbitrary idempotents in a commutative real algebra $A$ equipped with a Frobenius form. More precisely, if the Frobenius form is nondegenerate (as in Majorana algebras), we prove that the eigenspaces $A_0(e)$ and $A_1(e)$ of an arbitrary idempotent $e \in A$ are subalgebras and annihilate one another: \[ A_0(e)A_1(e)=\{0\}, \] while in the degenerate case the corresponding inclusions hold modulo the radical of the Frobenius form. This answers a question of T. M. Mudziiri Shumba and S. Shpectorov concerning the closure of the $0$-eigenspace $A_0(e)$.
Comments8 pages. Theorem 3.3 was improved in v2
