User:Hillgentleman/computer algebra

on using the computer to study abstract algebraic structures

things that people have done


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


infinity edit

  • how do we deal with infinities in computer?

set edit

  • 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 edit

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 edit

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

commutative algebra edit

Lie algebra and Lie groups edit

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

quantum group edit

scripts edit