TheoremStatement: 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.

Finite periodic transfer matrices commute

Statement

For every finite chain in Finite periodic fundamental evaluation chain and transfer matrix, its ordinary un-twisted transfer matrices satisfy $[τ (u), τ (v)] = 0$ for every $u, v \in C$ . No genericity condition on the inhomogeneities is needed. The proof uses invertibility only for $u - v \notin {i, - i}$ and then extends the resulting polynomial identity to all spectral pairs.

Facts & Assumptions

Given: The finite-dimensional periodic chain, its polynomial monodromy and trace, and the monodromy RTT relation.

[F1]

Finite periodic fundamental evaluation chain and transfer matrix: $H$ is finite dimensional, $M_{0} (u)$ is polynomial in $u$ , and $τ (u) = {tr}_{0} M_{0} (u)$ is the ordinary auxiliary trace.

[F2]

The periodic fundamental monodromy satisfies RTT: $R_{00^{'}} (u - v) M_{0} (u) M_{0^{'}} (v) = M_{0^{'}} (v) M_{0} (u) R_{00^{'}} (u - v)$ for every $u, v$ .

[F3]

The type-A rational R-matrix satisfies Yang–Baxter and has exact singular parameters: $R_{N} (d)$ has a two-sided inverse if $d \notin {i, - i}$ .

Proof

1.1givenF1F2F3algebra

Suppose $u - v \notin {i, - i}$ . Multiplying the RTT identity [F2] on the right by $R_{00^{'}} (u - v)^{- 1}$ , which exists by [F3], gives $R_{00^{'}} M_{0} (u) M_{0^{'}} (v) R_{00^{'}}^{- 1} = M_{0^{'}} (v) M_{0} (u)$ . Take the ordinary trace over the two auxiliary spaces. For a finite-dimensional auxiliary matrix $X$ with coefficients in $End (H)$ and an invertible scalar auxiliary matrix $R$ , $tr (R X R^{- 1}) = tr (X)$ : writing entries, $\sum_{i, j, k} R_{i j} X_{j k} (R^{- 1})_{k i} = \sum_{j, k} X_{j k} \sum_{i} (R^{- 1})_{k i} R_{i j} = \sum_{j} X_{j j}$ , since scalar entries commute with the coefficients and $R^{- 1} R = I$ . Thus the trace of the left side is ${tr}_{00^{'}} (M_{0} (u) M_{0^{'}} (v))$ .

2.1givenF1step 1.1algebra

The tensor-space trace of $M_{0} (u) M_{0^{'}} (v)$ equals $τ (u) τ (v)$ , since the two auxiliary factors are independent and the coefficient order is $u$ then $v$ . The trace of $M_{0^{'}} (v) M_{0} (u)$ equals $τ (v) τ (u)$ by the same index expansion with the coefficient order reversed. Step 1.1 therefore proves $[τ (u), τ (v)] = 0$ whenever $u - v \notin {i, - i}$ .

3.1givenF1step 2.1algebra∎

Each matrix entry of $[τ (u), τ (v)]$ is a polynomial in $u, v$ by [F1]. For every fixed $v$ , it vanishes for all but at most two values of $u$ by step 2.1, hence is the zero polynomial in $u$ ; consequently it vanishes for all $u, v$ . This proves the assertion also at $u - v = \pm i$ , where the auxiliary $R$ is singular.

Depends on

Used by

An explicit two-site periodic transfer matrix and permutation Hamiltonian Example

Cited to discharge well-definedness by The periodic fundamental monodromy satisfies RTT.

Dependency tree · two levels

8 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

L. D. Faddeev, How Algebraic Bethe Ansatz Works for Integrable Model (standard reference, not scraped)
F. Loebbert, Lectures on Yangian Symmetry (standard reference, not scraped)
A. P. Isaev, Lectures on Quantum Groups and Yang-Baxter Equations (standard reference, not scraped)