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Complex Temperley–Lieb semisimplicity by the order of q squared

Statement

Let $q \in C^{\times}$ , $δ = q + q^{- 1}$ , and let $ℓ$ be the order of $q^{2}$ when $q^{2}$ is a root of unity, with $ℓ = \infty$ otherwise. For every finite $n$ :

if $ℓ = 1$ (equivalently $q^{2} = 1$ and $δ = \pm 2$ ), then ${T L}_{n} (C, δ)$ is semisimple;
if $ℓ = 2$ (equivalently $q^{2} = - 1$ and $δ = 0$ ), then it is semisimple exactly when $n = 0$ or $n$ is odd;
if $3 \leq ℓ < \infty$ , then it is semisimple exactly when $n < ℓ$ ;
if $ℓ = \infty$ , then it is semisimple.

In particular, the phrase “ $q$ is a root of unity” alone is not a fixed-rank nonsemisimplicity criterion.

Facts & Assumptions

Given: The complex parameter convention $δ = q + q^{- 1}$ , the determinant formula for all link-state cell forms, and the criterion that the algebra is semisimple exactly when every such form is nondegenerate.

[F1]

The determinant formula and its polynomial-continuation convention (The Temperley–Lieb link-state Gram determinant).

[F2]

The link-state forms detect semisimplicity over $C$ (Temperley–Lieb cell forms detect semisimplicity over the complex numbers).

[F3]

For $q \neq \pm 1$ , $[r]_{q} = 0$ exactly when $q^{2 r} = 1$ ; at $q = \pm 1$ , $[r]_{q} = r q^{r - 1}$ (Temperley–Lieb quantum integers and quantum characteristic).

Proof

technique · Split into the four possible orders of $q^2$ and the rank cases at order two and above

1.1F1F2F3givenassume-case

Case $ℓ = \infty$ . No positive quantum integer vanishes by [F3], so every factor in [F1] is nonzero and [F2] gives semisimplicity at every rank.

1.2F1F2F3assume-case

Case $ℓ = 1$ . Then $q = \pm 1$ and $[r]_{q} = r q^{r - 1} \neq 0$ for every positive $r$ by [F3]. Every Gram determinant in [F1] is nonzero, so [F2] gives semisimplicity at every rank.

1.3F1F2F3assume-casealgebra

Case $3 \leq ℓ < \infty$ and $n < ℓ$ . For every allowed $p$ , each denominator index $j$ and each numerator index $n - 2 p + 1 + j$ in [F1] is positive and less than $ℓ$ . Neither vanishes by [F3], so all forms are nondegenerate and [F2] gives semisimplicity.

1.4F1F2F3assume-casechoosealgebra

Case $3 \leq ℓ < \infty$ and $n \geq ℓ$ . Write $n = a ℓ + s$ with $a \geq 1$ and $0 \leq s < ℓ$ . If $s = 0$ , take $p = j = 1$ ; if $1 \leq s \leq ℓ - 2$ , take $p = j = s + 1$ ; if $s = ℓ - 1$ , take $p = ℓ - 1, j = ℓ - 2$ . In every case $1 \leq j \leq p < ℓ$ and $p \leq ⌊ n / 2 ⌋$ : for $s = 0$ , $p = 1$ and $n \geq 3$ ; for $1 \leq s \leq ℓ - 2$ , $2 p = 2 s + 2 \leq ℓ + s \leq n$ ; and for $s = ℓ - 1$ , $2 p = 2 ℓ - 2 \leq 2 ℓ - 1 \leq n$ . Direct substitution gives $n - 2 p + 1 + j = a ℓ$ . The exponent $d_{n, p - j}$ is positive because $p - j$ is $0$ or $1$ , and $[j]_{q} \neq 0$ because $j < ℓ$ . Thus the determinant in [F1] is zero, one form is degenerate, and [F2] gives nonsemisimplicity.

1.5F1F2F3assume-casealgebra

Case $ℓ = 2$ . At $n = 0$ , ${T L}_{0} = C$ is semisimple. For even $n \geq 2$ , $p = j = 1$ gives $\det G_{n, 1} = [n]_{q} / [1]_{q} = 0$ , so [F2] gives nonsemisimplicity. If $n$ is odd, then $n - 2 p + 1 = 2 m > 0$ for every allowed $p$ . For odd $j$ , both quantum integers in each factor of [F1] are nonzero; for $j = 2 k$ , numerator and denominator have simple zeros at $q^{2} = - 1$ , and their ratio tends to $(- 1)^{m} (m + k) / k \neq 0$ . Hence all continued determinants are nonzero and [F2] gives semisimplicity for odd $n$ .

2.1step 1.1step 1.2step 1.3step 1.4step 1.5cases-exhaustive∎

The cases $ℓ = \infty$ , $ℓ = 1$ , $3 \leq ℓ < \infty$ with $n < ℓ$ or $n \geq ℓ$ , and $ℓ = 2$ exhaust all possibilities. Steps 1.1–1.5 prove exactly the branches in the Statement, including the rank-zero order-two endpoint.

Depends on

Used by

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15 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)
K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras (standard reference, not scraped)