弗罗贝尼乌斯维数的界
Bounds on Frobenius dimension
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中文总结 AI 辅助
本文研究结合代数弗罗贝尼乌斯维数，一方面改进有限维代数弗罗贝尼乌斯维数上界，指出达界代数是特定根基代数；另一方面明确计算低维代数及截断路代数的弗罗贝尼乌斯维数，依据基础箭图路径。
中文摘要 AI 辅助
在本文中，我们证明了关于结合代数弗罗贝尼乌斯维数的两类结果。首先，我们根据有限维代数作为向量空间的维数，改进了其弗罗贝尼乌斯维数的已知上界，并表明达到此界的唯一代数是与单顶点箭图相关的平方为零的根基代数。其次，我们明确计算了低维代数以及截断路代数的弗罗贝尼乌斯维数，其依据是基础箭图中无迂回的路径。
英文摘要
In this article, we prove two types of results about the Frobenius dimension of associative algebras. First, we refine the known upper bound for the Frobenius dimension of a finite dimensional algebra in terms of its dimension as a vector space, and show that the only algebras reaching this bound are radical square zero algebras associated with single-vertex quivers. Second, we compute Frobenius dimension for low-dimensional algebras explicitly, and for truncated path algebras in terms of paths with no detours in their underlying quivers.
