ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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.

Root-space brackets for matrix units

Example

In ${s l}_{n} (C)$ with the diagonal Cartan subalgebra $h$ and roots $ε_{i} - ε_{j}$ of Diagonal Cartan subalgebra and roots of sl_n, the matrix units satisfy $[E_{i j}, E_{k l}] = δ_{j k} E_{i l} - δ_{l i} E_{k j},$ and the bracket of the root lines $g_{ε_{i} - ε_{j}} = C E_{i j}$ and $g_{ε_{k} - ε_{l}} = C E_{k l}$ lies in the root space of the sum of the two functionals, as required by Brackets of root spaces: $[g_{ε_{i} - ε_{j}}, g_{ε_{k} - ε_{l}}] \subseteq g_{(ε_{i} - ε_{j}) + (ε_{k} - ε_{l})} .$ For $k \neq j$ and $l \neq i$ both sides vanish; for $k = j$ and $l \neq i$ the bracket is $C E_{i l}$ , whose root is $ε_{i} - ε_{l} = (ε_{i} - ε_{j}) + (ε_{j} - ε_{l})$ .

Facts & Assumptions

Given: The algebra ${s l}_{n} (C)$ with its diagonal Cartan subalgebra and roots $ε_{i} - ε_{j}$ as in Diagonal Cartan subalgebra and roots of sl_n, the root lines $g_{ε_{i} - ε_{j}} = C E_{i j}$ , and the inclusion $[g_{α}, g_{β}] \subseteq g_{α + β}$ of Brackets of root spaces; the root-space convention is Root and root space.

Verification

technique · direct

1.1

The product formula $E_{i j} E_{k l} = δ_{j k} E_{i l}$ is immediate from matrix multiplication: the product has the single nonzero entry $1$ in position $(i, l)$ exactly when the middle indices match. Hence $[E_{i j}, E_{k l}] = δ_{j k} E_{i l} - δ_{l i} E_{k j}$ .

givenalgebra

2.1

There are four cases. If $k = j$ and $l \neq i$ , then $[E_{i j}, E_{j l}] = E_{i l}$ and the root is $(ε_{i} - ε_{j}) + (ε_{j} - ε_{l}) = ε_{i} - ε_{l}$ . If $l = i$ and $k \neq j$ , then $[E_{i j}, E_{k i}] = - E_{k j}$ and the root is $(ε_{i} - ε_{j}) + (ε_{k} - ε_{i}) = ε_{k} - ε_{j}$ . If $k = j$ and $l = i$ , then $[E_{i j}, E_{j i}] = E_{i i} - E_{j j} \in h = g_{0}$ and the functional sum is zero. Finally, if $k \neq j$ and $l \neq i$ , both Kronecker terms vanish; the functional sum has no cancellation producing a root or zero, so its root space is zero. Thus every case has the asserted bracket inclusion.

givenstep 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19–24 (standard reference, not scraped)