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Root Jones–Wenzl recursion and its negligible ideal

Statement

Fix even $p \geq 4$ , a primitive $2 p$ -th root $A$ , and $r = p / 2$ . In the Temperley–Lieb diagram category over $C$ with loop value $δ = - A^{2} - A^{- 2}$ , call $f : m \to n$ negligible when ${Tr}_{m} (g \circ f) = 0$ for every reverse morphism $g : n \to m$ , where $Tr$ is the unnormalized diagram-closure trace. A tensor ideal is a family of linear subspaces of the morphism spaces closed under composition on either side and tensor product on either side by arbitrary morphisms. The negligible morphisms form a tensor ideal. This ideal is generated, as a tensor ideal, by the Jones–Wenzl projector $p_{r - 1}$ , defined by the standard recursion through its first zero quantum integer; in particular the quotient has precisely the surviving projector labels $0, \dots, r - 2$ .

Statement references

Root and loop conventions

Existence and zero trace of the boundary projector

Temperley–Lieb diagram category

Chen, The Temperley-Lieb Categories and Skein Modules, §§5.1–5.4

Kauffman and Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 3–7

Facts & Assumptions

Given: The specialized Temperley–Lieb category, its unnormalized closure trace, and the root-order projectors.

[F1]

Jones–Wenzl projectors exist up to the first zero quantum integer: $p_{r - 1}$ is a nonzero idempotent with zero closure trace.

[F2]

Temperley–Lieb planar diagram data and endomorphism modules: each morphism is a finite $C$ -linear combination of noncrossing matching diagrams; cups and caps bend strands, and composition is stacking with each closed component weighted by $δ$ .

[F3]

Negligible morphisms in the Temperley–Lieb category: $f : m \to n$ is negligible exactly when ${Tr}_{m} (g \circ f) = 0$ for every $g : n \to m$ .

[F4]

Negligible Temperley–Lieb morphisms form a tensor ideal: the negligible subspaces form a two-sided tensor ideal in the planar Temperley–Lieb category for every complex loop value.

[F5]

Diagram closure trace is cyclic and satisfies the tower rules: the unnormalized closure trace is cyclic and its identities do not divide by $δ$ .

[F6]

Jones–Wenzl projectors: annihilation, uniqueness, partial trace, and closure: the corner of each defined $p_{n}$ is $C p_{n}$ .

[F7]

The rank-one tilting quotient matches the Temperley-Lieb ribbon convention: at the canonical root with order parameter $r \geq 3$ , the negative-loop quotient has kernel equal to the embedded ideal generated by $p_{r - 1}$ and its ribbon quotient is equivalent to the specified rank-one tilting quotient.

[F8]

The root negligible quotient is semisimple with truncated fusion: for the positive-loop root $q = e^{π i / r}$ , the boundary projector ideal is the exact negligible ideal and the quotient has simple labels $0, \dots, r - 2$ with the truncated fusion rule.

Proof

Given: The data in the Statement.

1.1F1F2F3F5F6givenalgebra

By the boundary-projector result, $p_{r - 1}$ is an idempotent with trace zero. The diagram category [F2] supplies the endomorphism algebra ${T L}_{r - 1}$ and its closure trace. For any $g \in {T L}_{r - 1}$ , [F6] gives $p_{r - 1} g p_{r - 1} = λ p_{r - 1}$ for some $λ \in C$ . Cyclicity [F5] gives ${Tr}_{r - 1} (g \circ p_{r - 1}) = {Tr}_{r - 1} (p_{r - 1} g p_{r - 1}) = λ {Tr}_{r - 1} (p_{r - 1}) = 0$ . Thus $p_{r - 1}$ itself is negligible by [F3].

2.1F3F4step 1.1algebra

By [F4], the negligible morphisms form a tensor ideal. Since [F3] and step 1.1 place $p_{r - 1}$ in that ideal, the tensor ideal it generates is contained in the negligible ideal.

2.2F1F7F8step 1.1algebra

Put $A_{0} = e^{π i / (2 r)}$ . For $r \geq 3$ , $A_{0}$ is a primitive $4 r$ -th root, $q_{0} = A_{0}^{2} = e^{π i / r}$ , and $- A_{0}^{2} - A_{0}^{- 2} = - (q_{0} + q_{0}^{- 1})$ . Thus the negative-loop category in [F7] is exactly the BHMV Temperley–Lieb category at $A_{0}$ ; its quotient kernel is the ideal generated by $p_{r - 1}$ , and [F8] gives the complete label list.

2.3F1F3F4step 1.1algebracases

If $r = 2$ , the loop value is zero and $p_{1} = 1_{1}$ has zero trace by step 1.1. The tensor ideal it generates contains every identity $1_{n}$ for $n \geq 1$ , while the object $0$ has endomorphism algebra $C$ with trace $1$ . Hence the quotient is equivalent to finite-dimensional vector spaces with the single simple label $0$ ; its negligible ideal is exactly the ideal generated by $p_{1}$ .

3.1F1F2step 2.2step 2.3algebra∎

For arbitrary primitive $A$ , write $A = A_{0}^{k}$ with $\gcd (k, 4 r) = 1$ . The cyclotomic automorphism $σ_{k}$ of $Q (A_{0})$ sends $A_{0}$ to $A$ , sends $δ (A_{0})$ to $δ (A)$ , and sends every coefficient of each recursively defined projector and every closure-trace pairing matrix to its specialized coefficient at $A$ . For each fixed pair of object labels, the matching set is finite by [F2], so the Hom spaces and trace-pairing matrices are finite-dimensional. The field automorphism preserves the ranks and kernels of all these matrices, the exact boundary ideal, and the quotient's simple labels. This completes the canonical-to-arbitrary-root transport and the proof.

Remarks

The canonical-root quotient and its boundary ideal use the full-rank path decomposition and the exact diagram-to-tilting comparison. Both supplier statements and their complete local proofs were inspected. The proof is transported to other primitive roots by the finite cyclotomic-field argument in step 3.1; no source-only acceptance is asserted.

Remarks references

Full-rank path decomposition and root quotient

Exact diagram-to-tilting comparison

Depends on

Used by

Root fusion rules and completeness of the simple objects Lemma

Dependency tree · two levels

34 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

Joshua Chen, The Temperley-Lieb Categories and Skein Modules, §§5.1–5.4 (standard reference, not scraped)
Louis H. Kauffman and Sostenes L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 3–7 (standard reference, not scraped)