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Yangians, Quantum Affine Algebras, and Transfer Matrices: Examples and Counterexamples

1 · Prerequisites

2 · Summary

This companion contains a concrete two-site trace calculation and a counterexample to inferring completeness from a Bethe equation. Both examples refer to the exact type-A RTT and periodic-chain conventions on yangians-quantum-affine-and-transfer-matrices.

The two-site example reuses the single finite $C^{2}$ rational family The rational spin-1/2 spectral R-matrix. The counterexample uses a direct sum of counit modules to show that a scalar equation can hold while every positive-root Bethe vector vanishes.

Companion: yangians-quantum-affine-and-transfer-matrices.

Reading prerequisites

Proposed reading order: yangians-quantum-affine-and-transfer-matrices. Item dependencies state the actual logical inputs.

Existing library route

quantized-enveloping-algebras-and-quantum-serre-relations: quantized enveloping algebras Does not automatically supply spectral-parameter R-matrices.
affine-lie-algebras-and-loop-central-extensions: affine Lie algebra construction Yangian/quantum affine distinction remains.
cyclotomic-klr-algebras-and-highest-weight-categorification: cyclotomic/KLR modules Does not prove Bethe completeness.

Scope and local proof obligations

1. Worked computations, examples and failure cases for yangians, quantum affine algebras, and transfer matrices.

proposed example page.

Required supplier: yangians-quantum-affine-and-transfer-matrices.

Local obligation: For each proved example, verify the calculation from the A-page definitions and cite only already established suppliers.

Consumer: Examples clarify the A-page definitions and test parameter boundaries; they are not general theorem suppliers.

Canonical suppliers and reuse boundaries

def-shifted-yangian-and-pyramid-generators-for-the-type-a-finite-w-algebra: type-A finite W shifted-Yangian presentation. Not scattering or evaluation-module completeness.
thm-ordered-monomial-pbw-basis-for-the-shifted-yangian: ordered PBW basis. Auditor reports partial proof only; do not treat as a completed logical supplier until local proof completion is verified.

Conventions, diagrams, and sources

Apply the definition and diagram contracts and prerequisite and repair holds before treating any proposed result as a supplier. All diagrammatic operations require defined domains, codomains, orientations, coefficients, and algebraic verification. Existing carriers keep their canonical identities and placements. Physics-only mathematical carriers require the supported same-ID resolver route specified in those contracts; links here do not establish that route.

06-other-physics-algebras

Full page contract: placement inventory (entry yangians-quantum-affine-and-transfer-matrices-examples). The original audited research inventory and its historical audit receipt remain unchanged.

3 · Logical flowchart

4 · Definitions, theorems and proofs

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

An explicit two-site periodic transfer matrix and permutation Hamiltonian

Example

Take $N = 2$ , two sites, and equal inhomogeneities $a_{1} = a_{2} = i / 2$ in Finite periodic fundamental evaluation chain and transfer matrix. With $P_{12}$ the flip of the two quantum factors, direct evaluation gives

$τ (u) = (2 u^{2} + \frac{1}{2}) I - P_{12}, τ (i / 2) = - P_{12}, τ^{'} (i / 2) = 2 i I .$

Consequently the normalized logarithmic-derivative operator

$H = \frac{i}{2} τ^{'} (i / 2) τ (i / 2)^{- 1} - I$

is $P_{12} - I$ . This example is the explicit two-site specialization of the shared finite $C^{2}$ family The rational spin-1/2 spectral R-matrix; no new spin- $1 / 2$ spectral matrix is introduced.

Facts & Assumptions

Given: The two-site chain at $N = 2$ with equal inhomogeneities $i / 2$ , and the transfer-matrix definition and commutativity theorem.

[F1]

Finite periodic fundamental evaluation chain and transfer matrix: $L_{0 j} (u) = (u - a_{j}) I + i P_{0 j}$ , $M_{0} (u) = L_{01} (u) L_{02} (u)$ , and $τ (u) = {tr}_{0} M_{0} (u)$ .

[F2]

Finite periodic transfer matrices commute: the finite periodic chain satisfies $[τ (u), τ (v)] = 0$ for all complex $u, v$ .

[F3]

The rational spin-1/2 spectral R-matrix: on the ordered two-dimensional basis, $R (u) = u I + i P$ with $P (e_{a} \otimes e_{b}) = e_{b} \otimes e_{a}$ .

Proof

1.1givenF1F3algebra

Set $q = u - i / 2$ . By [F1] and [F3], $M_{0} (u) = (q I + i P_{01}) (q I + i P_{02}) = q^{2} I + i q (P_{01} + P_{02}) - P_{01} P_{02}$ . The explicit partial traces are ${tr}_{0} I = 2 I$ , ${tr}_{0} P_{01} = {tr}_{0} P_{02} = I$ , and ${tr}_{0} (P_{01} P_{02}) = P_{12}$ . For the last identity, expand $P_{0 j} = \sum_{r, s} E_{r s}^{(0)} E_{s r}^{(j)}$ , multiply using $E_{r s} E_{t u} = δ_{s t} E_{r u}$ , and trace the auxiliary factor to obtain $\sum_{r, s} E_{s r}^{(1)} E_{r s}^{(2)} = P_{12}$ . Therefore $τ (u) = 2 (q^{2} + i q) I - P_{12} = (2 u^{2} + \frac{1}{2}) I - P_{12}$ .

2.1givenF1step 1.1algebra

Substitution $u = i / 2$ gives $τ (i / 2) = - P_{12}$ , and differentiating the polynomial gives $τ^{'} (i / 2) = 4 (i / 2) I = 2 i I$ . Since $P_{12}^{2} = I$ , the inverse of $- P_{12}$ is $- P_{12}$ . Thus $(i / 2) τ^{'} (i / 2) τ (i / 2)^{- 1} - I = (i / 2) (2 i I) (- P_{12}) - I = P_{12} - I$ .

3.1givenF2step 2.1algebra∎

The whole polynomial family commutes by [F2], including this specialization; the calculation above identifies one normalized logarithmic derivative and uses no limiting inverse at a singular spectral difference.

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

A one-root Bethe equation need not produce a complete set of vectors

Facts & Assumptions

Given: The Hopf counit representation, its direct sum on a two-dimensional space, and the one-root Bethe-vector notation from The one-root gl2 Bethe equation cancels the unwanted transfer term.

[F1]

The RTT Yangian has its standard Hopf structure: $ϵ (t_{i j} (u)) = δ_{i j}$ defines the counit representation.

[F2]

The one-root gl2 Bethe equation cancels the unwanted transfer term: the cancellation statement uses a common regular spectral domain realizing the RTT identities; its one-root condition is $a (v) = d (v)$ , and it assumes the associated Bethe vector is nonzero to conclude it is an eigenvector.

Counterexample

1.1givenF1F2algebra

Let $H = C e_{0} \oplus C e_{1}$ be the direct sum of two counit representations and set $Ω = e_{0}$ . Applying the counit on each summand gives $t_{i j} (u) = δ_{i j} I_{H}$ , hence $A = D = I$ and $B = C = 0$ . These constant operator families are regular on the common spectral domain $D = C$ and satisfy the pointwise RTT identities. Thus the pseudovacuum conditions hold there with $a (u) = d (u) = 1$ , and the one-root equation in [F2] holds for every $v \in D$ .

2.1givenF1step 1.1algebra∎

For every $k \geq 1$ and parameters $v_{1}, \dots, v_{k}$ , the rightmost $B (v_{k})$ sends $Ω$ to zero, so $B (v_{1}) \dots B (v_{k}) Ω = 0$ . The empty product is $e_{0}$ . Hence all Bethe vectors from this pseudovacuum span $C e_{0}$ , a proper subspace because $e_{1}$ is linearly independent from $e_{0}$ . Thus the one-root equation by itself does not guarantee completeness.

5 · Examples, counterexamples and false statements

None yet.

Yangians, Quantum Affine Algebras, and Transfer Matrices: Examples and Counterexamples

1 · Prerequisites

2 · Summary

Reading prerequisites

Existing library route

Scope and local proof obligations

1. Worked computations, examples and failure cases for yangians, quantum affine algebras, and transfer matrices.

Canonical suppliers and reuse boundaries

Conventions, diagrams, and sources

3 · Logical flowchart

4 · Definitions, theorems and proofs

An explicit two-site periodic transfer matrix and permutation Hamiltonian

Example

Facts & Assumptions

Proof

A one-root Bethe equation need not produce a complete set of vectors

Facts & Assumptions

Counterexample

5 · Examples, counterexamples and false statements

Sources