I am interested in the
number of isomorphism types of nilpotent associative algebras
of class two over arbitrary finite fields.
It can be shown that this number, considered as a function in q (the number
of elements of the field), is a PORC (Polynomial On Residue Classes) function,
hence there exists a natural number n and polynomials such that the function
behaves like a polynomial for all elements in the same residue class modulo n.
You can find functions giving the number of
isomorphism types of class-2 nilpotent associative algebras
over finite fields with q elements of small ranks r and all dimensions:
number of class-2 algebras.
I also wrote a small library for the computer algebra system GAP
that includes all porc functions I have computed so far.
All porc functions are considered as multivariate polynomials in GAP.
GAP library.
After the download unzip the package and copy it into the directory of your GAP packages.
Load the package by LoadPackage("ClassTwoAlgLib");
On this page you can find a short overview of this library.
To enumerate the isomorphism classes of class two nilpotent algebras over arbitrary fields
I developed together with Prof. Bettina Eick a package for the computer algebra system GAP,
which is called ClassTwoAlg.
ClassTwoAlg
After the download unzip the package and copy it into the directory of your GAP packages.
Load the package by LoadPackage("ClassTwoAlg");
The results which are available within the library classtwoalglib (see above)
were computed using this package.