Calculating Ways to Calculate

Bachelor Thesis (2023)
Author(s)

L. van der Kuil (TU Delft - Discrete Mathematics and Optimization)

Contributor(s)

N.D. Verhulst – Mentor (TU Delft - Discrete Mathematics and Optimization)

Wolter Groenevelt – Graduation committee member (TU Delft - Analysis)

Faculty
Electrical Engineering, Mathematics and Computer Science
Copyright
© 2023 Lars van der Kuil
More Info
expand_more
Publication Year
2023
Language
English
Copyright
© 2023 Lars van der Kuil
Graduation Date
06-07-2023
Awarding Institution
Delft University of Technology
Programme
['Applied Mathematics']
Faculty
Electrical Engineering, Mathematics and Computer Science
Reuse Rights

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.

Abstract

Algebras are vector spaces with a bilinear product. When we fix a finite dimension n and a finite field K with q elements, there are a finite number of non-isomorphic algebras. Seeing as vector spaces are completely determined by the dimension and scalar field, the number of non-isomorphic algebras is the number of ways we can define a bilinear product of vectors. Exploiting the bilinearity of the product we can construct a surjection from vectors of n n by n matrices over K to non-isomorphic n-dimensional algebras over K. This function can be made injective by reducing to orbits under a group action on the set of vectors of matrices, thus providing a bijection. So the number of non-isomorphic algebras is equal to the number of orbits of this group action. In 2020 Verhulst used Burnside's Lemma to count the orbits and thus the number of non-isomorphic algebras. However, the formula he derived still requires a number of complicated calculations.

The aim of this thesis is to implement Verhulst's formula in python. In Table 4.1 you can find the output of the python script. The results for n=1 merely show some trivial cases. The results for n=2 fit the formulas obtained, using completely different methods, by Petersson and Scherer. The results for n=3 and n=4 are, to my knowledge, new. Currently, the limiting factor is computation time. A more powerful computer could squeeze out a few more results, but a more efficient formula is necessary. Based on the results, it seems that the contribution of the identity matrix in Verhulst's formula provides a decently tight lower bound.

Files

License info not available