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A Uniform Calculation of Weight-Preserving Infinitesimal Braid Deformations in Type D

Primarily AI-generated textHuman understanding: some partsmath.QA — Quantum Algebramath.RT — Representation Theory

Contributed by Hangang Wang ↗

Version 1 / Oct 27, 2026 / CC BY 4.0

Abstract

Over Q(q)\mathbb Q(q), for every n≥4n\ge4, we calculate the full space of classical-weight-preserving solutions of the linearized braid equation at the standard type-DnD_n vector braiding. Its dimension is (n+1)(n+2)/2(n+1)(n+2)/2. A coordinate slice for scalar and parameter variations, diagonal vector-basis changes, and alternating twists reduces the upper bound to a homogeneous vanishing problem. Outer-pair coefficients eliminate the zero-weight block through a recurrence closed by the two central weights. We also give a functorial formal local parametrization over local Artin algebras. The explicit integrable family belongs to earlier multiparameter constructions; an exact parameter dictionary is provided. No bibliographic priority claim is made for the full tangent-space exhaustion. The proof is uniform in nn; finite-rank computations are diagnostics.

Provenance statement

This work originated from a computational investigation undertaken by Hangang Wang in January 2026, following preliminary research-direction discussions with Rushu Zhuang. The historical MATLAB implementations and computational experiments were carried out by Wang with AI-assisted programming, based on mathematical constructions documented in the cited literature. The general-rank deformation theorem and its proof were not part of the January 2026 work. The subsequent investigation of weight-preserving infinitesimal braid deformations, including the proposed uniform type-D theorem, its proof, exact symbolic computations, literature comparisons, and manuscript preparation, was developed in October 2026 with extensive assistance from OpenAI ChatGPT and Codex. Substantial portions of the present manuscript were AI-generated. The human contributor understands the historical computational work and has reviewed some aspects of the subsequent research, but does not claim full understanding or independent human verification of the complete general-rank proof. Multiple AI-assisted mathematical audits and independently implemented computational checks have found no counterexample to the main theorem. These checks do not constitute independent verification by a human specialist or formal proof-assistant verification. The explicit multiparameter R-matrix constructions are acknowledged as prior work. The proposed mathematical contribution concerns the exhaustion of the full classical-weight-preserving infinitesimal deformation space and its formal local parametrization. Bibliographic priority for this exhaustion result remains unresolved. The manuscript and its accompanying verification materials are made available to facilitate independent mathematical scrutiny, reproducibility, and further research. The historical computational work and the subsequent mathematical developments are documented separately.

Tools used

OpenAI
ChatGPT
OpenAI
CodexVersion GPT-6.1 Sol
Python Software Foundation
PythonVersion 3.10+
SymPy
SymPy
MathWorks
MATLAB

References

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Version history

  1. v1Submitted by Hangang WangInitial depositCurrentOct 27, 2026