feat(Algebra/Lie/Cochain): 2-coboundaries and second cohomology of a Lie algebra

[KevorkianPhilippe]

Adds 2-coboundaries and the second cohomology of a Lie algebra with coefficients in a module, completing the coboundaries, cohomology item of the TODO list of Mathlib/Algebra/Lie/Cochain.lean.

• twoCoboundary: the image of d₁₂ inside the 2-cochains, with membership criteria in general and for a trivial module.
• twoCoboundary_le_twoCocycle: coboundaries are cocycles, from the existing d₂₃_comp_d₁₂.
• twoCoboundaryIn: the coboundaries seen inside the cocycles, and secondCohomology (H²(L, M)) as the quotient.
• secondCohomology_mk_eq_zero_iff and its trivial-module form: a cocycle vanishes in H² exactly when it is a coboundary, stated on the explicit formulas, which is what one uses when computing H² of a concrete Lie algebra.
• subsingleton_secondCohomology_of_le: H² vanishes when every cocycle is a coboundary.

The remaining TODO items (comparison with the Chevalley-Eilenberg complex, construction and classification of central extensions) are untouched.

Note: the ## Main definitions block of this file prefixes the names with LieAlgebra. although the namespace is LieModule.Cohomology. I followed the existing convention rather than change it here; happy to fix it in this PR if preferred.

Use of AI: this contribution was written with AI assistance (Claude), iterated against the Lean compiler; the module builds, lint-style and lake exe runLinter Mathlib.Algebra.Lie.Cochain are clean, and #print axioms on the new declarations gives only propext, Classical.choice, Quot.sound. The mathematical content is the standard low-degree Chevalley-Eilenberg picture, and I am happy to justify any of the design choices.

[KevorkianPhilippe]

LLM-generated

[github-actions]

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PR summary dbc1075191

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ mem_twoCoboundaryIn_iff
+ mem_twoCoboundary_iff
+ mem_twoCoboundary_iff_of_trivial
+ secondCohomology
+ secondCohomology_mk_eq_zero_iff
+ secondCohomology_mk_eq_zero_iff_of_trivial
+ secondCohomology_mk_surjective
+ subsingleton_secondCohomology_of_le
+ twoCoboundary
+ twoCoboundaryIn
+ twoCoboundary_le_twoCocycle

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit dbc1075).

• +12 new declarations
• −0 removed declarations

+LieModule.Cohomology.mem_twoCoboundaryIn_iff
+LieModule.Cohomology.mem_twoCoboundary_iff
+LieModule.Cohomology.mem_twoCoboundary_iff_of_trivial
+LieModule.Cohomology.secondCohomology
+LieModule.Cohomology.secondCohomology_mk_eq_zero_iff
+LieModule.Cohomology.secondCohomology_mk_eq_zero_iff_of_trivial
+LieModule.Cohomology.secondCohomology_mk_surjective
+LieModule.Cohomology.subsingleton_secondCohomology_of_le
+LieModule.Cohomology.twoCoboundary
+LieModule.Cohomology.twoCoboundary.congr_simp
+LieModule.Cohomology.twoCoboundaryIn
+LieModule.Cohomology.twoCoboundary_le_twoCocycle

---

No changes to strong technical debt.
No changes to weak technical debt.

Current commit dbc1075191
Reference commit b1007d8abf

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.py 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).