Classifying Nilpotent Associative Algebras: Small Coclass and Finite Fields

We survey the state of the art in the classification of nilpotent associative 𝔽 $$\mathbb F$$ -algebras by coclass using their associated coclass graphs G 𝔽 ( r ) $$\mathcal G_{\mathbb F}(r)$$ . For arbitrary fields 𝔽 $$\mathbb F$$ , we determine up to isomorphism the nilpotent associative 𝔽 $$\mathbb F$$ -algebras of coclass 1 and their coclass graphs G 𝔽 ( 1 ) $$\mathcal G_{\mathbb F}(1)$$ . For finite fields 𝔽 $$\mathbb F$$ and arbitrary r, we propose a conjecture on the structure of the coclass graph G 𝔽 ( r ) $$\mathcal G_{\mathbb F}(r)$$ ; this conjecture is based on computational investigations. We further show how computational methods apply in an enumeration of the isomorphism types of nilpotent associative 𝔽 $$\mathbb F$$ -algebras of small dimensions over small finite fields 𝔽 $$\mathbb F$$ .