Classification of nilpotent associative algebras and coclass theory

结合代数自然地出现在数学的各个领域。例如，它们在表示论和上同调中发挥作用;它们在李理论中作为泛包络代数出现，或者在群论中作为群代数出现。我们的中心目标是发展有效的算法的分类，建设和计数的有限维幂零结合代数。首先，我们使用的维数作为主要的不变量，并制定有效的算法来构造或枚举同构类型的幂零结合代数的一个给定的维在一个单一的有限域。然后，我们扩展我们的结果，以涵盖所有的有限域，我们认为无限域的情况下。Coclass理论在有限幂零群的分类中是一个非常成功的工具。我们将coclass理论的中心思想转化到有限维幂零结合代数上，然后利用coclass给出了有限维幂零结合代数的分类和研究算法。这一研究结果将对幂零结合代数的结构产生重要的新见解。

Associative algebras arise naturally in various areas of mathematics. For example, they play a role in representation theory and in cohomology; they arise as universal enveloping algebras in Lie theory, or as group algebras in group theory. Our central aim is to develop effective algorithms for the classification, construction and enumeration of finite dimensional nilpotent associative algebras. We first use the dimension as primary invariant and develop effective algorithms to construct or enumerate the isomorphism types of nilpotent associative algebras of a given dimension over a single finite field. We then extend our results to cover all finite fields and we consider the case of infinite fields. Coclass theory has been a highly successful tool in the classification of finite nilpotent groups. We translate the central ideas of coclass theory to finite dimensional nilpotent associative algebras and then develop algorithms to classify and investigate finite dimensional nilpotent associative algebras by coclass. The results of this research will yield significant new insights into the structure of nilpotent associative algebras.