量子Magnusian的Hopf代数理论
A Hopf Algebraic Theory of the Quantum Magnusian
- Rutgers University（罗格斯大学）
- California Institute of Technology（加州理工学院）
- CERN（欧洲核子研究组织）
- Seoul National University（首尔大学）
- Korea Institute for Advanced Study（韩国高等科学研究院）
- University of Seville（塞维利亚大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文为量子Magnus展开中的图系数建立了Hopf代数理论，引入混合箭图的收缩Hopf代数，给出连通函数的普适闭式公式，并推导出量子Murua公式及与Tutte和色多项式相关的恒等式。
AI中文摘要：
我们发展了量子Magnus展开中出现的图系数的Hopf代数理论。在经典层面，图展开由有向树控制，而其量子对应物涉及圈图、重边和不同类型的边数据。我们引入了一个混合箭图的收缩Hopf代数，该代数包含这些物理图并在收缩下封闭。在由箭图张成的Hopf子代数上，我们定义了一个特征$e$，它来自归一化的线性扩展数据和蝌蚪处方，并令$\omega$为其卷积逆。我们的主要结果是对每个有限箭图上的连通函数$\omega_c$的普适闭式公式。该公式是对顶点集的有序集合划分的有限和，且在无环性或简单性假设下均有效。我们还给出了一个局部刻画：一个两顶点收缩恒等式，连同蝌蚪分解、在非连通图上消失以及边界数据，唯一确定了$\omega_c$。对于无环箭图，一般公式简化为一个置换公式，其系数仅依赖于下降数。我们独立地从量子Magnus展开通过算子乘积和Wick收缩推导出同一公式，表明其物理图系数由收缩Hopf代数的卷积结构控制。作为进一步的结果，我们获得了精细的量子Murua公式以及与Tutte和色多项式相关的方向求和恒等式。
英文摘要：
We develop a Hopf-algebraic theory of the graph coefficients arising in the quantum Magnus expansion. At the classical level, the graph expansion is governed by directed trees, whereas its quantum counterpart involves loop graphs, multiple edges, and different types of edge data. We introduce a contraction Hopf algebra of mixed quivers containing these physical diagrams and closed under contraction. On the Hopf subalgebra spanned by quivers, we define a character $e$ from normalized linear-extension data and a tadpole prescription, and let $ω$ be its convolution inverse. Our main result is a universal closed formula for the connected function $ω_c$ on every finite quiver. The formula is a finite sum over ordered set partitions of the vertex set and is valid without acyclicity or simplicity assumptions. We also give a local characterization: a two-vertex contraction identity, together with tadpole factorization, vanishing on disconnected graphs, and boundary data, determines $ω_c$ uniquely. For acyclic quivers, the general formula reduces to a permutation formula whose coefficients depend only on descent numbers. We derive the same formula independently from the quantum Magnus expansion via operator products and Wick contractions, showing that its physical graph coefficients are governed by the convolution structure of the contraction Hopf algebra. As further consequences, we obtain a refined quantum Murua formula and orientation-sum identities related to Tutte and chromatic polynomials.
