User:Hillgentleman/computer algebra

on using the computer to study abstract algebraic structures

things that people have done

• http://www.gap-system.org/
• magnus
• a good list -[1]
• AXIOM-a platform
• symmetrica
  • character tables and other interesting stuff, run online- [2]

starting points

• google: google:algebraic+structure+data+type, google:object+oriented+algebraic+structure
• Modelling Algebraic Structures. in a Symbolic Computation Environment by Stephan A. Missura. - [3]
• A case study in w:Eiffel: [4]
• psu: [5]
• ieee: [6]
• wikipedia: [7]
• Uniform representation of basic algebraic structures in computer algebra [8], springer lecture notes in computer science, ISBN 978-3-540-61697-9
• A Computer Algebra Solution to a Problem in Finite Groups [9]
• python abstract algebra objects: [10]

• Not quite what I want, but still: wikipedia:computer algebra systems ,e.g. Xcas

• computer algebra in abstract algebra, by Kulich, (abstract only)[11]

• [12]
• [13]

infinity

• how do we deal with infinities in computer?

set

• Python has the finite set and list types already.
  • Need: given a map between two sets, show that it is a bijection

• What about infinite sets?
  • Exercise: describe the category of finite-dimensional complex vector spaces in python

group

OR more precisely, we want to have a way to work with the category of groups (or a subcategory thereof) in the computer environment

• What is a group? A group is a multiplet: (set, multiplication, inverse, identity).
• how do we describe a group? simplest - by generators and relations; what else?
• How are two groups related? Homomorphisms
  • given a set map (which is a set of ordered pairs), we should construct a way to verify that it is a homomorphism - we can check (heuristically at least) that a map is a homomorphism by brute force computations on the generators
• When do we know two groups are isomorphic?
  • when we are given a bijective homomorphism
    • What if it is not given???

• What are the operations on groups? direct product, quotient by a subgroup, semi-direct product, ... and a lot more

problem

• is it possible to write a script to classify finite groups of small order ?

Lie algebra and Lie groups

The theory of Lie groups, with its vast machinary of representation theory and with Dynkin diagrams as their DNAs, should be highly automatic!

scripts

• user:hillgentleman/FiniteGroup.py - still working on it