代数的增长函数及其在Leavitt路径代数中的应用
Growth functions of algebras and an application to Leavitt path algebras
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中文总结 AI 辅助
本文研究代数的三种增长函数（Gelfand-Kirillov维数、超维数和熵），精确刻画其适用场景，并应用于Leavitt路径代数，给出其为PI的新判据。
中文摘要 AI 辅助
本文考虑了代数的三个增长函数：Gelfand-Kirillov维数、超维数和熵，以及后者的一种变体，该变体已成功用于Leavitt路径代数的研究。我们证明了若干结果，精确化了以下启发式事实：Gelfand-Kirillov维数适用于研究多项式增长的代数，超维数适用于次指数增长的代数，而熵适用于指数增长的代数。我们受分次代数的熵的启发，引入了分次模的熵的定义。作为我们研究的一个应用，我们给出了Leavitt路径代数为PI的一个新判据。
英文摘要
In this paper, we consider three growth functions of algebras: the Gelfand-Kirillov dimension, superdimension, and entropy -- as well as a variation of the latter successfully used in the study of Leavitt path algebras. We prove results that make precise the heuristic fact that the Gelfand-Kirillov dimension is suitable for the study of algebras with polynomial growth, the superdimension for algebras with subexponential growth, and the entropy for algebras with exponential growth. We introduce a definition of entropy for graded modules motivated by the entropy for graded algebras. As an application of our study, we give a new criterion for Leavitt path algebras to be PI.
