ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedaudited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

General and special linear Lie groups

Example

Assume ${A C}_{ω}$ and let $n \geq 1$ . The open matrix group ${GL}_{n} (R)$ has Lie algebra $M_{n} (R)$ with bracket $[X, Y] = X Y - Y X$ . Its subgroup

${SL}_{n} (R) = {A : \det A = 1}$

is an embedded Lie group with Lie algebra ${s l}_{n} (R) = {X : tr X = 0}$ .

Facts & Assumptions

Given: An integer $n \geq 1$ and the standard Euclidean structure on $M_{n} (R)$ .

[F1]

Lie groups have smooth multiplication and inversion, and the tangent bracket is defined through left-invariant fields. Lie group. Lie bracket on the tangent space of a Lie group.

[F2]

Determinant and trace use the standard finite formulas. For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix. The trace of a square matrix over a commutative ring.

[F3]

A regular level is embedded and its tangent space is the kernel of the differential. A regular level set is an embedded submanifold. The tangent space of a regular level set is the kernel.

[F4]

Countable choice is inherited by the tangent-bracket supplier. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Verification

technique · direct

1.1

The set $\det \neq 0$ is open in $M_{n} (R)$ , and matrix multiplication and inversion are smooth there, so it is a Lie group with tangent space $M_{n} (R)$ at $I$ . Its left-invariant field generated by $X$ is $A \mapsto A X$ ; differentiating two such fields gives bracket $X Y - Y X$ .

F1F2algebra

1.2

Expanding the determinant by permutations shows $\det (I + t X) = 1 + t tr X + O (t^{2})$ , hence $d (\det)_{I} (X) = tr X$ . At every $A \in \det^{- 1} (1)$ , multiplication by $A$ transports this differential to a nonzero functional, so $1$ is a regular value. By [F3], ${SL}_{n}$ is embedded and its tangent space at $I$ is the trace-zero kernel.

F2F3algebra

2.1

Determinant multiplicativity and $\det (A^{- 1}) = (\det A)^{- 1}$ make the level set a subgroup, so its induced operations are smooth and it is a Lie group. The commutator bracket preserves trace zero because $tr (X Y) = tr (Y X)$ .

F1F2step 1.1step 1.2algebra

3.1

For $n = 1$ , ${s l}_{1} = 0$ . Singular tangent matrices are allowed. No interval, endpoint, metric choice, or biconditional occurs. ${A C}_{ω}$ is used only through the current tangent-bracket interface [F1]; the displayed Euclidean coordinates are finite and add no choice.

F1F2F3F4step 1.1step 1.2step 2.1∎

Depends on

Used by

A real invertible matrix with no real logarithm Counterexample
Orthogonal and special orthogonal Lie groups Example
The real symplectic matrix group Example

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)