论文标题
随机产生联想代数
Random generation of associative algebras
论文作者
论文摘要
在最近几十年中，人们对有限和涂鸦群体的随机产生，特别是有限的简单群体引起了人们的兴趣。在本文中，我们研究了有限和涂鸦联想代数的类似概念。令$ k = f_q $为有限字段。让$ a $为有限的维度，关联，Unital代数超过$ k $。令$ p（a）$是$ a $ a $（均匀和独立）的两个元素随机产生$ a $作为Unital $ k $ -Algebra的可能性。众所周知，如果$ a $很简单，则$ p（a）\至1 $ as $ | a | \ to \ infty $。我们将此结果扩展到一大批有限的联想代数。对于$ a $ simple，我们找到了$ p（a）$的最佳下限，我们估计任何适当的$ a $ a $ a $的适当子代理的指数$ m（a）$的增长率（a）$。我们还通过两个具有给定特征多项式的元素研究了简单代数$ a $的随机生成（分别为A给定等级）。此外，我们在一般有限代数的发电机数量最少的上方和下方结合。最后，我们让$ a $成为超过$ k $的代数。我们表明，$ a $在且仅当$ a $具有多项式的最大亚位bra增长时会产生积极的有限生成。还建立了相关的定量结果。
There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, and finite simple groups in particular. In this paper we study similar notions for finite and profinite associative algebras. Let $k=F_q$ be a finite field. Let $A$ be a finite dimensional, associative, unital algebra over $k$. Let $P(A)$ be the probability that two elements of $A$ chosen (uniformly and independently) at random will generate $A$ as a unital $k$-algebra. It is known that, if $A$ is simple, then $P(A) \to 1$ as $|A| \to \infty$. We extend this result to a large class of finite associative algebras. For $A$ simple, we find the optimal lower bound for $P(A)$ and we estimate the growth rate of $P(A)$ in terms of the minimal index $m(A)$ of any proper subalgebra of $A$. We also study the random generation of simple algebras $A$ by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let $A$ be a profinite algebra over $k$. We show that $A$ is positively finitely generated if and only if $A$ has polynomial maximal subalgebra growth. Related quantitative results are also established.
