LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

The RTT relation has an explicit component commutator form

Statement

For the presentation in The type-A RTT Yangian in the rational i-normalization, set $x = u - v$ . For every $1 \leq i, j, k, l \leq N$ , its matrix RTT relation is equivalent coefficientwise to

$x [t_{i j} (u), t_{k l} (v)] = i (t_{k j} (v) t_{i l} (u) - t_{k j} (u) t_{i l} (v)) .$

Facts & Assumptions

Given: The two-auxiliary-leg RTT matrix relation with $R_{N} (x) = x I + i P_{N}$ .

[F1]

The type-A RTT Yangian in the rational i-normalization: $R_{N} (x) = x I + i P_{N}$ with the auxiliary flip $P_{N} (e_{a} \otimes e_{b}) = e_{b} \otimes e_{a}$ , and $R_{12} (u - v) T_{1} (u) T_{2} (v) = T_{2} (v) T_{1} (u) R_{12} (u - v)$ .

Proof

1.1givenF1algebra

In the coefficient of the auxiliary row $(i, k)$ and column $(j, l)$ , the $x I$ part of the left side is $x t_{i j} (u) t_{k l} (v)$ and the $i P$ part is $i t_{k j} (u) t_{i l} (v)$ , because the flip sends the row $(i, k)$ to $(k, i)$ . On the right side the corresponding terms are $x t_{k l} (v) t_{i j} (u)$ and $i t_{k j} (v) t_{i l} (u)$ , since the flip acts on the column. Thus equality of those matrix entries gives $x t_{i j} (u) t_{k l} (v) + i t_{k j} (u) t_{i l} (v) = x t_{k l} (v) t_{i j} (u) + i t_{k j} (v) t_{i l} (u)$ .

2.1givenF1step 1.1algebra∎

Subtract the two $x$ -terms and move the two $i$ -terms to the other side. The result is exactly the displayed commutator relation. Conversely, if that relation holds for every four indices, the coefficient computation in step 1.1 makes every auxiliary matrix entry of the RTT difference zero, hence the full matrix relation holds.

Depends on

The type-A RTT Yangian in the rational i-normalization

Used by

Dependency tree · two levels

4 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

A. I. Molev, M. L. Nazarov, and G. I. Olshanski, Yangians and classical Lie algebras (standard reference, not scraped)
L. D. Faddeev, How Algebraic Bethe Ansatz Works for Integrable Model (standard reference, not scraped)