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 link-state quotient is a right Temperley–Lieb module

Statement

For $R$ , $δ$ , $n$ , $t$ and $W_{n, t} (R, δ)$ as in Temperley–Lieb link-state standard modules by defect number, the rule $x \cdot a = [x \circ a]$ defines a unital right ${T L}_{n} (R, δ)$ -module structure on $W_{n, t} (R, δ)$ .

Facts & Assumptions

Given: A commutative ring $R$ with unit, $δ \in R$ , $n \geq 0$ , an allowed defect number $t$ , and the quotient $M_{R} (n, t) / M_{R}^{< t} (n, t)$ .

[F1]

The modules $M_{R} (m, n)$ are free on noncrossing matching diagrams; composition stacks the left factor above the right factor, deletes each closed cycle with a factor $δ$ , and extends bilinearly (Temperley–Lieb planar diagram data and endomorphism modules).

[F2]

Stacking is associative and the vertical matching $1_{n}$ is a two-sided identity; $End (n) = {T L}_{n} (R, δ)$ is a unital algebra (Stacking is associative and horizontal juxtaposition is strict monoidal).

[F3]

$M_{R}^{< t} (n, t)$ is spanned by diagrams with fewer than $t$ propagating components, and $W_{n, t}$ is the quotient by this span (Temperley–Lieb link-state standard modules by defect number).

Proof

technique · direct

1.1F1F2algebra

For basis diagrams $x : n \to t$ and $a, b : n \to n$ , both $(x \cdot a) \cdot b$ and $x \cdot (a b)$ are represented by the same three-layer glued graph, with the same exterior paths and the same number of closed components. By [F1] their coefficients are the same power of $δ$ , and [F2] gives $(x \cdot a) \cdot b = x \cdot (a b)$ . The identity diagram leaves $x$ unchanged. Bilinearity extends these equalities to $M_{R} (n, t)$ , making it a right ${T L}_{n} (R, δ)$ -module before quotienting.

2.1F1F3step 1.1

If $prop (x) < t$ , every propagating component of $x \circ a$ must pass through a distinct propagating component of $x$ at the glued $n$ -boundary; stacking cannot create a new path from the top $t$ -boundary of $x$ to the exterior bottom boundary of $a$ when no such path existed in $x$ . Thus every basis term of $x \cdot a$ has fewer than $t$ propagating components. By [F3], $M_{R}^{< t} (n, t)$ is a right submodule.

3.1F1F3step 1.1step 2.1∎

The action therefore descends to the quotient $W_{n, t} (R, δ)$ , and the module identities from step 1.1 descend with it. Since the diagram basis of $M_{R} (n, t)$ is partitioned into diagrams with exactly $t$ propagating components and those with fewer, the surviving cosets form the stated link-state basis.

Depends on

Used by

Cited to discharge well-definedness by Temperley–Lieb link-state standard modules by defect number.

Dependency tree · two levels

12 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

David Ridout and Yvan Saint-Aubin, Standard Modules, Induction and the Structure of the Temperley-Lieb Algebra (standard reference, not scraped)
John Enyang and Frederick M. Goodman, Cellular bases for algebras with a Jones basic construction (standard reference, not scraped)