Implement the BGG resolution of a simple finite dimensional Lie algebra representation

[tscrim]

The goal of this PR is to implement the BGG resolution of a finite dimensional simple Lie algebra representation $L_{\lambda}$. In order to implement this, we need to implement a number of features:

• Simple modules (this does not assume finite dimensional).
• Dual modules in BGG category $\mathcal{O}$.
• Fix embeddings of Verma modules Not all Verma module morphisms are found #36793.
• Improve construction of elements of free abelian monoids when strings are indices.
• Move to_root_vector() to a method of fundamental weights and cache the inverse Cartan matrix.
• Add is_dominant_weight() method to weight lattice realization elements.
• Add is_verma_dominant() to define a dominant weight in the sense of the Humphreys 2008 reference, which means the Verma module is projective (that is, maximal in its dot action orbit).
• Add the category of FiniteDimensionalKacMoodyAlgebras.
• @cache_in_parent_method marked weak_le for Coxeter groups.
• Subclass of PBW bases for semisimple Lie algebras.

We also add methods to compute the contravariant form of a Verma module and add tester methods for them being simple or projective.

It is possible to split this up into smaller parts, but it is natural for them to go together given the final goal.

One major TODO is to speed up the computation of the bases of the simple modules. The main place to optimize this would be to improve the multiplication of the PBW basis code, which for finite dimensional Lie algebras (not necessarily semisimple), this could be improved by using the fixed basis order and using the monomials as vectors to avoid lots of calls for getting the sort order and comparing. In particular, the triangular sorting order for the Chevalley basis implementation has error messages being raised (and caught) for comparing the roots and coroots. This will be done at some point on a separate PR.

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[fchapoton]

some problems with imports, apparently

[tscrim]

That should take care of all of the failures.

[mkoeppe]

Needs rebase

[github-actions]

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[tscrim]

I have made all suggested changes.

[mkoeppe]

Lint complains

[tscrim]

Fixed, although I don't know why the linter didn't catch that before.

[mkoeppe]

LGTM.

[tscrim]

Thank you!