TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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 gl_N evaluation formula satisfies RTT

Statement

Let $N \geq 2$ , let $W$ be a left ${g l}_{N} (C)$ -module with action $ρ$ , and let $a \in C$ . The assignment in Evaluation modules of the type-A RTT Yangian extends to a representation of $Y_{i} ({g l}_{N})$ on $W (a)$ . In particular, the dual-natural action $ρ^{\lor} (E_{i j}) f^{k} = - δ_{i k} f^{j}$ is a ${g l}_{N}$ -representation, and its unnormalized auxiliary Lax matrix equals $R_{N} (u - a)$ .

Facts & Assumptions

Given: The RTT Yangian of The type-A RTT Yangian in the rational i-normalization, a ${g l}_{N}$ -module $W$ , its action $ρ$ , and a spectral shift $a \in C$ .

[F1]

The type-A RTT Yangian in the rational i-normalization: $R_{N} (x) = x I + i P_{N}$ , and the defining relation is $R_{12} (u - v) T_{1} (u) T_{2} (v) = T_{2} (v) T_{1} (u) R_{12} (u - v)$ .

[F2]

Evaluation modules of the type-A RTT Yangian: the proposed generator action is $t_{i j} (u) = δ_{i j} I_{W} - i ρ (E_{i j}) / (u - a)$ , and the dual-natural action is $E_{i j} f^{k} = - δ_{i k} f^{j}$ .

[F3]

The type-A RTT Yangian in the rational i-normalization reindexes Matrix units $E_{i j}$ and the Kronecker delta: for $1 \leq i, j \leq N$ , $E_{i j}$ is the unit at zero-based position $(i - 1, j - 1)$ and $e_{s}$ is supported at coordinate $s - 1$ , so $E_{i j} e_{s} = δ_{j s} e_{i}$ .

[F4]

The RTT relation has an explicit component commutator form: $x [t_{i j} (u), t_{k l} (v)] = i (t_{k j} (v) t_{i l} (u) - t_{k j} (u) t_{i l} (v))$ for $x = u - v$ is equivalent to the matrix RTT relation.

Proof

1.1givenF2F3algebra

Let $e_{1}, \dots, e_{N}$ be the standard column basis. By [F3], $E_{i j} e_{s} = δ_{j s} e_{i}$ , so $E_{i j} E_{k l} e_{s} = δ_{j k} δ_{l s} e_{i} = (δ_{j k} E_{i l}) e_{s}$ for every $s$ . Hence $E_{i j} E_{k l} = δ_{j k} E_{i l}$ and $[E_{i j}, E_{k l}] = δ_{j k} E_{i l} - δ_{l i} E_{k j}$ . On a dual basis vector $f^{s}$ , the explicit commutator is $[ρ^{\lor} (E_{i j}), ρ^{\lor} (E_{k l})] f^{s} = δ_{k s} δ_{i l} f^{j} - δ_{i s} δ_{j k} f^{l}$ . The right side is also $ρ^{\lor} (δ_{j k} E_{i l} - δ_{l i} E_{k j}) f^{s}$ , by the same formula. Thus the dual-natural operators satisfy the matrix-unit bracket and form a ${g l}_{N}$ -representation.

2.1givenF2F4step 1.1algebra

Put $x = u - v$ , $U = u - a$ , and $V = v - a$ . By [F4], the required component identity is the RTT relation to check. Under [F2], its left side is $- x (δ_{j k} ρ (E_{i l}) - δ_{l i} ρ (E_{k j})) / (U V)$ by the bracket relation. In the parenthesized difference on the right, the quadratic terms cancel because both are $(- i)^{2} ρ (E_{k j}) ρ (E_{i l}) / (U V)$ . Its linear terms are $- i δ_{k j} ρ (E_{i l}) (U^{- 1} - V^{- 1}) - i δ_{i l} ρ (E_{k j}) (V^{- 1} - U^{- 1}) = i x (δ_{j k} ρ (E_{i l}) - δ_{l i} ρ (E_{k j})) / (U V)$ , since $V - U = - x$ . Multiplication by the outer factor $i$ therefore gives the same left side. Thus every component RTT relation holds.

3.1givenF1F2step 1.1step 2.1algebra∎

The component relations are exactly the coefficients of the matrix relation [F1]. Thus the images of every defining relation vanish, so the assignment on the free generators factors through the RTT quotient and defines the representation $W (a)$ . For $W = V_{N}^{\lor}$ , its auxiliary matrix is $I + i P_{N} / (u - a)$ by the explicit sum $\sum_{i, j} E_{i j} \otimes E_{j i} = P_{N}$ ; multiplying by $u - a$ gives $R_{N} (u - a)$ .

Depends on

Used by

Cited to discharge well-definedness by Evaluation modules of the type-A RTT Yangian.

Dependency tree · two levels

7 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)