Serialization of Lie algebras

[antonydellavecchia]

Implements serialization for Lie Algebras

[lgoettgens]

@antonydellavecchia I started duplicating everything that you added for the other two types of Lie algebras. All of them now have different problems I am afraid I need your help with.

1. AbstractLieAlgebra
   needs serialization of SRow
   example:

L = lie_algebra(QQ, :A, 2)

2. LinearLieAlgebra
   saving and loading works, but it yields identical (===) objects. And it seems that it does not call the specific load function at all, as loading fails after calling Oscar.reset_global_serializer_state() with some problem with deserializing matrices.
   example:

L = general_linear_lie_algebra(QQ, 2)

3. DirectSumLieAlgebra
   examples:

# empty direct sum
L = direct_sum(QQ, LieAlgebra{QQFieldElem}[])

# non-trivial example
L1 = general_linear_lie_algebra(QQ, 2)
L2 = lie_algebra(QQ, :A, 2)
L = direct_sum(L1, L1, L2) # it is important that the first two summands are still identical after saving and loading

4. Element types for all of these three
   but they are essentially the same, as they just contain a parent pointer and a coeff vector (or rather a 1xn matrix)
   examples (for all L as above):

x = basis(L, 1) # for one elem
xs = basis(L) # for a vector of elems

5. Testing
   How do I actually test if saving and loading are correct, if they give me the identical (===) object? I rather just compare some properties between original and deserialized.

[codecov]

Codecov Report

Attention: Patch coverage is 98.31461% with 3 lines in your changes missing coverage. Please review.

Project coverage is 84.64%. Comparing base (61bf7dc) to head (863cc95).
Report is 1 commits behind head on master.

Files	Patch %	Lines
experimental/LieAlgebras/test/RootSystem-test.jl	90.90%	1 Missing ⚠️
experimental/LieAlgebras/test/WeylGroup-test.jl	96.66%	1 Missing ⚠️
experimental/LieAlgebras/test/setup_tests.jl	94.44%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #3980      +/-   ##
==========================================
+ Coverage   84.58%   84.64%   +0.05%     
==========================================
  Files         597      598       +1     
  Lines       82265    82445     +180     
==========================================
+ Hits        69587    69783     +196     
+ Misses      12678    12662      -16     

Files	Coverage Δ
experimental/LieAlgebras/src/LieAlgebras.jl	100.00% <ø> (ø)
experimental/LieAlgebras/src/serialization.jl	100.00% <100.00%> (ø)
experimental/LieAlgebras/test/RootSystem-test.jl	98.52% <90.90%> (-1.48%)	⬇️
experimental/LieAlgebras/test/WeylGroup-test.jl	97.77% <96.66%> (-2.23%)	⬇️
experimental/LieAlgebras/test/setup_tests.jl	95.18% <94.44%> (-0.09%)	⬇️

... and 3 files with indirect coverage changes

[lgoettgens]

@antonydellavecchia Thanks a lot for your help with this! I added some comments to things that came up after our discussion when I played around some more.
The problem with empty vectors is worse than expected, see #3983 for details what I found while digging around.
Can you take it from here? Once you are happy with all the changes in this PR, please write your approval in a comment (since Github won't allow you to do it the usual way) and I am then happy to merge.

[lgoettgens]

@HereAround @emikelsons please wait for this TODO to be resolved (ETA end of August) before putting Lie algebras inside your FTheoryTools serialization. Or, alternatively, have a script that re-creates the serialized models so that we can just re-run it once the serialization of Lie algebras changes

[lgoettgens]

Edit: I already managed to resolve this today. But you should have a reliable way to re-generate your FTheoryTools models anyway, since there certainly will be changes to the Lie algebra serialization. (Not necessarily breaking, but adding more information about them and thus e.g. speeding up some operations on them)

[HereAround]

Sorry, this slipped my attention.

All good. I have opened #4008 to coordinate this and many other serialization changes for FTheoryTools. In other words, waiting for you and @antonydellavecchia to tell us when/how to proceed.

[lgoettgens]

I added serialization for some more Lie algebra related types. This is starting to make fun 😅
@antonydellavecchia could you have a look at the new implementations as well? They seem to work, but maybe I did something too hacky somewhere.

[lgoettgens]

#4020 looks really interesting, but will introduce conflicts. Thus I suggest we wait with this until #4020 is merged. To nevertheless get the bugfixes etc. from this PR in master in a timely manner, I extracted them to #4021

[lgoettgens]

This is now rebased on top of #4020. Once that and #4030 are merged, I can simplify this PR a bit

[lgoettgens]

This is now finished from my POV. @antonydellavecchia could you please have a look if you see any issues and if not, proceed with the merge?