Classifying Nilpotent Associative Algebras: Small Coclass and Finite Fields

Bettina Eick () and Tobias Moede ()
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Bettina Eick: TU Braunschweig, Institut Computational Mathematics
Tobias Moede: Monash University, School of Mathematical Sciences

A chapter in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, 2017, pp 213-229 from Springer

Abstract: Abstract 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 $$\mathbb F$$ .

Keywords: Coclass theory; Nilpotent associative algebras; p-groups; 16N40; 16W99; 16Z05; 20D15 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-70566-8_9

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DOI: 10.1007/978-3-319-70566-8_9

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