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Marked-point modules and central nilpotents at roots of unity

A mix of human-written and AI-generated textHuman understanding: some partsmath.RT — Representation Theorymath.QA — Quantum Algebramath-ph — Mathematical Physics

Contributed by Zhenyu Sun ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We study extensions in the two-marked-point representation of the periodic Temperley--Lieb algebra at q=eiπ/pq=e^{i\pi/p}, p∈Z≥3p\in\mathbb Z_{\geq3}. For every positive even size NN, we construct a submodule ZNZ_N extending TN=WN;2,εT_N=W_{N;2,\varepsilon} by WN=WN;0,εqW_N=W_{N;0,\varepsilon q}, where ε=±1\varepsilon=\pm1. The dual trees of noncrossing matchings give an explicit injection ΘN:TN→WN\Theta_N:T_N\to W_N. Both central braid transfer elements have nilpotent part gΘNπNg\Theta_N\pi_N on ZNZ_N, where πN\pi_N is the quotient map and g=x+x−1+ε(y+y−1)g=x+x^{-1}+\varepsilon(y+y^{-1}) for the marked-point parameters x,y∈C×x,y\in\mathbb{C}^\times. The extension splits exactly when g=0g=0, and the formula determines the central Jordan forms at every even size. In the window 2≤N<2p−22\leq N<2p-2, ZNZ_N is a joint central summand and, when g≠0g\ne0, has simple layers TN/SN/TNT_N/S_N/T_N. An explicit intertwiner and annular localization identify its two-layer quotient with a periodic XXZ sector. The associated basic image algebras have dimensions five and four; the quotient preserves first extensions between simples and kills a nonzero Yoneda product in degree two. An annular operator detects the extension with coefficient 2εg2\varepsilon g. Beyond the isolated window, we determine a five-simple image algebra and compute a 4343-dimensional joint central summand at p=3,N=6p=3,N=6, on which the two central nilpotents generate a four-dimensional algebra. Exact matrix calculations and verification code accompany these finite computations.

Provenance statement

The text is a mix of human-written and AI-generated material. Some parts, but not all, have been checked and understood by a human. The submitted manuscript was reviewed and edited with OpenAI Codex. Exact finite-dimensional computations were checked using Python, SymPy, NumPy, and python-flint; these computational checks do not constitute human verification of the entire paper. Supporting scripts and exact data are included in anc/.

Tools used

OpenAI
Codex

References

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Version history

  1. v1 · 28 pagesSubmitted by Zhenyu SunInitial depositCurrentNov 11, 2026