feat: Semi-direct sum of Lie algebras

[Ljon4ik4]

In this PR a construction of the semi-direct sum of two Lie algebras is added.
It is shown that the resulting object $H\rtimes G$ is a Lie algebra and that it fits into an exact sequence
$H \to H\rtimes G \to G$.

I was not sure about multiple things and would be very grateful for feedback:

• Currently the semidirect product is created via a def as a type synonym of $H\times G$ so one can define a new bracket on it, which need not coincide with the standard 'direct sum' bracket. This is analogous to e.g. PointedContMDiffMap in Mathlib.Geometry.Manifold.DerivationBundle. An alternative way would creating a structure with $H$ and $G$ as fields. This seems to be the way chosen in Mathlib.GroupTheory.SemidirectProduct.lean.

• Currently it is created as a separate file, but it could also be added to the Mathlib.Algebra.Lie.Extension file.

• I am not fully happy with some of the proofs, especially the leibniz_lie in the LieRing instance, but did not manage to get it more elegant.

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PR summary a5f6d77220

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Algebra.Lie.SemiDirect (new file)	1362

---

Declarations diff

+ SemiDirectSum
+ inclusion
+ instance : AddCommGroup (H ⋊⁅ψ⁆ G) := by
+ instance : Bracket (H ⋊⁅ψ⁆ G) (H ⋊⁅ψ⁆ G)
+ instance : LieAlgebra R (H ⋊⁅ψ⁆ G)
+ instance : LieAlgebra.IsExtension (inclusion ψ) (projection ψ)
+ instance : LieRing (H ⋊⁅ψ⁆ G)
+ instance : Module R (H ⋊⁅ψ⁆ G) := by
+ projection

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for scripts/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).