Classification of nilpotent associative algebras and coclass theory

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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.