论文标题
联想代数的长度和深度
The length and depth of associative algebras
论文作者
论文摘要
最近，人们对研究有限群，代数组和谎言组的长度和深度引起了极大的兴趣。在本文中，我们介绍并研究代数的类似概念。让$ k $成为一个字段，让$ a $为$ k $的代数，而不一定是Unipital。 $ a $的不可限制的链条是subalgebras $ a = a_0> a_1> ...> a_t = 0 $的链条，对于某些整数$ t $，其中每个$ a_i $都是$ a_ {i-1-1} $的最大subergebra。这种不可限制的链的最大（分别为最小）长度称为$ a $的长度（分别为深度）。事实证明，有限的长度，有限的深度和有限尺寸是$ a $的等效属性。对于$ a $有限的尺寸，我们给出了$ a $的长度的公式，我们绑定了$ a $的深度，当$ a $的长度分别等于其尺寸和深度时，我们会研究。最后，我们在什么情况下调查了$ a $的尺寸在上面是其长度或深度的函数，或者长度减去其深度。
Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. In this paper we introduce and study similar notions for algebras. Let $k$ be a field and let $A$ be an associative, not necessarily unital, algebra over $k$. An unrefinable chain of $A$ is a chain of subalgebras $A=A_0>A_1>...>A_t=0$ for some integer $t$ where each $A_i$ is a maximal subalgebra of $A_{i-1}$. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of $A$. It turns out that finite length, finite depth and finite dimension are equivalent properties for $A$. For $A$ finite dimensional, we give a formula for the length of $A$, we bound the depth of $A$, and we study when the length of $A$ equals its dimension and its depth respectively. Finally, we investigate under what circumstances the dimension of $A$ is bounded above by a function of its length, or its depth, or its length minus its depth.
