bibkey: florespeltola2018generators
authors: Steven M. Flores and Eveliina Peltola
year: 2018
title: "Generators, projectors, and the Jones-Wenzl algebra"
doi: 10.48550/arXiv.1811.12364
url: "https://arxiv.org/abs/1811.12364v2"
claim: "Theorem 1.1 proves that the unit P_ς and the elements U_i = P_ς U_{s_1+...+s_i} P_ς form a minimal generating set of the Jones–Wenzl algebra JW_ς(ν) = P_ς TL_{n_ς}(ν) P_ς when n_ς < p̄(q); Conjecture 1.2 asks whether item 1 holds whenever max ς < p̄(q)."
strata_touched:
  - D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation
license: citation-only
triage: anchor

Flores and Peltola, Generators, projectors, and the Jones-Wenzl algebra

S. M. Flores and E. Peltola, arXiv:1811.12364v2 (11 December 2018), math-ph. Quotations are from the arXiv v2 TeX source WJalgebra_gen-sec1-intro.tex; macros are expanded.

The Temperley–Lieb algebra ${\mathrm{TL}}_n(\nu )$ is presented by generators $U_1,\dots ,U_{n-1}$ with $U_i{U}_{i\pm 1}{U}_i=U_i$, $U_i^2=\nu U_i$ and $U_i{U}_j=U_j{U}_i$ for $|i-j|>1$, with $\nu =-q-q^{-1}$. For $q\notin \{\pm 1\}$, $\bar{\mathfrak{p}}(q)$ is $p$ when $q=e^{\pi i{p}^{\prime }/p}$ with $p,p^{\prime }$ coprime and $\infty$ when $q$ is not a root of unity; $\bar{\mathfrak{p}}(\pm 1)=\infty$. The Jones–Wenzl projectors satisfy $P_1=1$ and $P_{s+1}=P_s+\frac{[s]}{[s+1]}{P}_s{U}_s{P}_s$. For a multiindex $\varsigma$ with $\max \varsigma <\bar{\mathfrak{p}}(q)$, ${\mathrm{JW}}_{\varsigma }(\nu )=P_{\varsigma }{\mathrm{TL}}_{n_{\varsigma }}(\nu )P_{\varsigma }$, where $P_{\varsigma }$ is the composite projector.

Theorem 1.1, item 1, assuming $n_{\varsigma }<\bar{\mathfrak{p}}(q)$:

The unit $1_{{\mathrm{JW}}_{\varsigma }}=P_{\varsigma }$ together with the $\varsigma$-Jones-Wenzl link diagrams $U_i:=P_{\varsigma }{U}_{s_1+s_2+\cdots +s_i}^{\mathrm{TL}}{P}_{\varsigma }$ with $i\in \{1,2,\dots ,d_{\varsigma }-1\}$, forms a minimal generating set for the Jones-Wenzl algebra ${\mathrm{JW}}_{\varsigma }(\nu )$.

Conjecture 1.2. Item 1 in theorem 1.1 holds whenever $\max \varsigma <\bar{\mathfrak{p}}(q)$.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.1811.12364
• Source: https://arxiv.org/abs/1811.12364v2