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At the first root boundary the Jones trace degenerates before semisimplicity fails

Statement

Let $q^{2}$ have finite order $ℓ \geq 3$ , put $δ = q + q^{- 1}$ , and let $n = ℓ - 1$ . Then ${T L}_{n} (C, δ)$ is semisimple but the bilinear trace form $(x, y) \mapsto τ_{n} (x y)$ of Normalized Temperley–Lieb trace from diagram closure is degenerate.

Facts & Assumptions

Given: The complex order criterion, the normalized closure trace, and the Jones–Wenzl recursion.

[F1]

Since $ℓ \geq 3$ , the complex semisimplicity theorem gives semisimplicity at rank $n = ℓ - 1$ (Complex Temperley–Lieb semisimplicity by the order of q squared).

[F2]

The Jones–Wenzl idempotent $p_{n}$ is nonzero and primitive, and $τ_{n} (p_{n}) = [n + 1] / δ^{n}$ when the required denominators are nonzero (The Jones–Wenzl recursion gives annihilating idempotents and their trace).

[F3]

The closure trace is cyclic (The normalized closure trace is cyclic and satisfies the Markov rules).

[F4]

If $q^{2}$ has order $ℓ \geq 3$ , then $[r]_{q} \neq 0$ for $1 \leq r < ℓ$ and $[ℓ]_{q} = 0$ (Temperley–Lieb quantum integers and quantum characteristic).

Proof

1.1F2F4given

Since $δ \neq 0$ and $[r]_{q} \neq 0$ for $1 \leq r \leq n = ℓ - 1$ , [F2] supplies the nonzero primitive idempotent $p_{n}$ and gives $τ_{n} (p_{n}) = [ℓ]_{q} / δ^{n} = 0$ by [F4].

2.1F2F3step 1.1algebra

For any $x \in {T L}_{n}$ , cyclicity [F3] and $p_{n}^{2} = p_{n}$ imply $τ_{n} (p_{n} x (1 - p_{n})) = τ_{n} ((1 - p_{n}) p_{n} x) = 0$ , so $τ_{n} (p_{n} x) = τ_{n} (p_{n} x p_{n})$ . The one-dimensional corner in [F2] gives $p_{n} x p_{n} = λ_{x} p_{n}$ for some $λ_{x} \in C$ . Hence $τ_{n} (p_{n} x) = λ_{x} τ_{n} (p_{n}) = 0$ for every $x$ , while $p_{n} \neq 0$ . Thus $p_{n}$ is a nonzero vector in the radical of the trace form, so the form is degenerate.

3.1F1step 2.1∎

Semisimplicity is the independent assertion in [F1]; combining it with step 2.1 proves that trace degeneracy occurs at rank $ℓ - 1$ while the algebra is still semisimple.

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Sources

K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras (standard reference, not scraped)
Joshua Chen, The Temperley-Lieb Categories and Skein Modules (standard reference, not scraped)