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Enumeration of nilpotent associative algebras of class 2 over arbitrary finite fields. (English) Zbl 1432.16018

Summary: Higman’s PORC theory implies that the number \(N_{d, r}(q)\) of isomorphism types of nilpotent associative algebras of dimension \(d\), rank \(r\) and class 2 over a finite field with \(q\) elements, considered as a function in \(q\), can be described by a polynomial on residue classes in \(q\). We describe an algorithm that, given a rank \(r\), determines such polynomials for \(N_{d, r}(q)\) for all dimensions \(d\). Using this, we determine \(N_{d, r}(q)\) for \(r \in \{1, \dots, 5 \}\) and arbitrary \(d\).

MSC:

16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16Z05 Computational aspects of associative rings (general theory)

Software:

GAP; AutPGrp; ClassTwoAlg
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Full Text: DOI

References:

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