关联单项式代数的增长及其在Manturov群中的应用
Growth of associated monomial algebras with application to Manturov groups
AI总结：
本文研究了关联单项式代数与原代数的GK维度关系，并应用该结果分析了Manturov (k,n)-群的增长性，发现其在特定条件下具有指数增长。
摘要：
众所周知，一个结合代数与其按照度数词法顺序的关联单项式代数具有相同的增长性和Gelfand-Kirillov维度（GK维度）。本文主要研究一个代数与其按照单项式顺序的关联单项式代数的GK维度之间的关系。我们获得了关于单项式顺序的充分条件，使得这两个代数具有相同的GK维度。我们的结果推广了已知的结果，并有多种应用。特别是，作为应用，我们研究了正整数n>k时的Manturov (k,n)-群的增长性。证明了Manturov (1,n)-群对于所有n>1具有增长率为0；Manturov (2,3)-群具有增长率为2；而对于所有n>k≥3，Manturov (k,n)-群包含一个秩为2的自由子群，因此具有指数增长。
英文摘要：
It is well-known that an associative algebra shares the same growth and Gelfand-Kirillov dimension (GK-dimension) as its associated monomial algebra with respect to a degree-lexicographic order. This article mainly investigates the relationship between the GK-dimension of an algebra and that of its associated monomial algebra with respect to a monomial order. We obtain sufficient conditions on a monomial order such that these two algebras have the same GK-dimension. Our result generalizes the well-known result and has several applications. In particular, as an application, we study the growth of Manturov $(k,n)$-groups for positive integers $n>k$. It is shown that the Manturov $(1,n)$-group has growth equal to $0$ for all $n>1$; the Manturov $(2,3)$-group has growth equal to $2$; and, for all $n>k\geq3$, the Manturov $(k,n)$-group contains a free subgroup of rank $2$ and thus has exponential growth.
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