Find commutation relations for differential chnages that generate continuous transforms in a group algebra over a finite non-abelian group In the paper "Symmetries of preon interactions modeled as a finite group" (J.N. Bellinger J.Math.Phys.38: 3414-3426,1997), I showed that non-abelian finite groups can sustain a special type of continuous transformations in a group algebra space, one in which each transform maps the group elements in such a way that the result new "elements" maintain the original group structure. This is in addition to the obvious permutations possible among elements of the finite group. These are obviously a subset of the "units" of the group.
The same transformations appear when working with the antiautomorphism ( https://en.wikipedia.org/wiki/Antihomomorphism )
The question I want to answer is "Is there a simple way to predict from aspects of the finite group what those symmetries will be?"
I find that examples make analysis easier, so I am providing some tools for generating the symmetries. It expects a finite group presented as a text Cayley table with elements numbered from 0 to N-1.
This python program generates a set of commutation relations among the differential transformations, but does not attempt to resolve which continuous groups these might be. I don't know a systematic procedure for comparing sets of commutation relations to find if the sets are equivalent--and would be very interested in hearing about such things.
Consider the set of elements in a group algebra K(G) corresponding to the elements g of the finite group G
{g} => {(1)g}
A transformation V in the group algebra will map these into other elements {g'}. Sometimes the natural operation on this new set will be isomorphic to the original finite group operations. That's what I'm interested in here. Because I'm requiring that the transformation be linear, since it satisfies V{g_i} x V{g_j} = V{g_ij}, if a x b = c, it will also satisfy V(a) x V(b) = V(c).
Sometimes these transformations are simple permutations, but in some circumstances one finds continuous transformations as well. (My work assumes that the field K is the complex numbers.) One requirement is that the finite group G is non-abelian. So far I haven't found any other requirements.
If I represent the group elements as i,j,k and their products as ij, jk^{-1}, etc, then the equation governing the relationships of the elements of small deviations from I (V == I + \delta V)
\delta V_{i,tj^{-1}} + \delta V_{j,i^{-1}t} = \delta V_{ij,t}
where the sum over any row or column of \delta V is zero.
Test a few things.
- pip install coverage
- coverage run -m unittest tests/test_*.py
- coverage html
- python3 Builder/run.py tests/6group
Several non-abelian finite groups are provided as examples in the tests/ directory, such as the a4group referenced above. For example, the file tests/6group is just
0 1 2 3 4 5
1 2 0 4 5 3
2 0 1 5 3 4
3 5 4 0 2 1
4 3 5 1 0 2
5 4 3 2 1 0
Its symmetries are generated by 3 differential transformations, the commutators of whose matrices are
[C0, C1] = (-2)C2
[C0, C2] = (-2)C1
[C1, C2] = (2)C0
This is isomorphic to the transformations generated by the generators c_i satisfying [c_i, c_j]= e_ijk c_k
The tests/a4group example has 8 generators, and 28 commutation relations, which can be manually simplified if you please.
This is not lightning-fast. You will probably not have the patience to wait for a set of commutators even when the group size is well less than the implied size limit of 100. The a4group example takes a minute and a half on my machine. The 16_9group example takes a little over 9 minutes. The s4group example (output is s4.output) took 2 1/2 hours.
The equation-solving is simple, and I do not warrant that it will be adequate for all situations arising. If it isn't adequate, one can try using lapack instead of exact fractions, and rounding off the numbers when done. I probably should have taken that approach from the beginning, but was worried that there might be stability issues which, combined with the rounding, would give me bogus answers.
The approach I used working on the original paper used a Fortran program to create and pre-simplify the equations, but the final solution and creation of commutation relations was manual.
The program seems to fail with the groups of size 18. There is a bug, and the differential matrices do not all satisfy the fundamental equation. I am investigating.
The Builder directory contains the python source files.
The docs directory contains a couple of LaTeX files: the original results collection (not the same as the shorter JMP paper, and not cross-checked. E.g. example 16_9 is different from the result I get with this program) and an as-yet unpublished paper on transformations of dihedral groups. (Executive summary: a dihedral group of order p has at most 3(p-1)/2 generators if p is odd, and 3(p-2)/2 generators if p is even.)
The tests directory contains python testing files (it does not have 100% coverage) for use with the coverage tool, and a few example group Cayley tables. Please do not laugh too hard at the code.