# Nikolaus conference 2017

Speaker: Morten Wesche (Braunschweig)

Title: Enumeration of Nilpotent Associative Algebras of Class 2 Over Arbitrary Finite Fields

Abstract:

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 $$q$$. In the talk an algorithm is described that, given a rank $$r$$, determines such polynomials for $$N_{d,r}(q)$$ for all dimensions $$d$$. Based on this the functions $$N_{d,r}(q)$$ are determined for $$r \in\{1,\ldots,5\}$$ and arbitrary $$d$$.