AI 中文总结
研究有限维非退化演化代数的格罗滕迪克环，通过分析与自然基相关的有向图平衡，证明其与基无关，且平衡控制张量积消去，还分析了由循环演化代数生成的子环并证明零因子准则。
AI 中文摘要
我们研究了域$\mathbb{K}$上有限维非退化演化代数的格罗滕迪克环，其加法由直和诱导，乘法由张量积诱导。尽管其基础阿贝尔群由不可分解同构类自由生成，但环结构要小得多：张量积会产生系统的非消去现象和许多零因子。关键不变量是与自然基相关的有向图的平衡。我们证明，对于非退化演化代数，这种平衡与所选自然基无关。然后我们表明平衡控制张量积后的消去：平衡为1的不可分解因子会消去，而平衡较大的因子在温和假设下会产生零因子关系。最后，我们分析了由循环演化代数生成的子环，并证明了循环代数所有有限直和的预测零因子准则。
英文摘要
We study the Grothendieck ring of finite-dimensional non-degenerate evolution algebras over a field $\mathbb{K}$, with addition induced by direct sum and multiplication induced by tensor product. Although its underlying abelian group is freely generated by indecomposable isomorphism classes, the ring structure is much smaller: tensor products create systematic non-cancellation phenomena and many zero-divisors. The key invariant is the balance of the directed graph associated with a natural basis. We prove that, for non-degenerate evolution algebras, this balance is independent of the chosen natural basis. We then show that balance controls cancellation after tensoring: indecomposable factors of balance $1$ cancel, whereas factors of larger balance produce zero-divisor relations under mild hypotheses. Finally, we analyze the subring generated by cyclic evolution algebras and prove the predicted zero-divisor criterion for all finite direct sums of cyclic algebras.
Comments32 pages, no figures
