稳定$q$-埃尔米特坐标演算：Capelli--PBW传递、混合极化代数与局域化障碍
Stable $q$-Hermitian coordinate calculus: Capelli--PBW transport, mixed polarization algebra, and localization obstructions
- University of Ostrava（俄斯特拉发大学）
- Charles University（查理大学）
- VSB–Technical University of Ostrava（俄斯特拉发科技大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究将有限支撑的$q$-埃尔米特径向演算扩展到克利福德值坐标多项式，引入混合极化代数，确定局域化所需的最小Ore条件，排除了未变形坐标代数的固定有限阶微分与位移实现。
AI中文摘要：
我们将有限支撑的$q$-埃尔米特径向演算扩展到克利福德值坐标多项式。对于$N$个玻色埃尔米特向量标记和有限费米支撑，稳定范围$m\ge2N$通过Howe分离得到典范的Gram-调和范式。标记-Capelli逆元对克利福德收缩进行归一化，而有限Wick-Chevalley共轭将分幂标量规范提升到全克利福德PBW模。将归一化收缩与可缩费米种子复形耦合，得到两个保持多项式的共轭$q$-埃尔米特坐标族，它们是平方零的，在每个极化内反交换，精确限制到径向PBW演算，具有标量边界迹$[2m-2n]_q$，且当$q\to1$时恢复经典埃尔米特超狄拉克对。对于混合极化，我们引入相对三角传递。归一化Capelli收缩的格拉斯曼关系在全坐标模上成立，产生逐标记因式分解，因此所有高阶过滤混合项是显式局域缺陷的有限子集乘积，且混合极化代数对任意有限标记集和有限费米支撑封闭。经典复结构对两个极化是共同的，而当$0<q<1$时，两个完全传递的克利福德-外尔乘积是不同的。我们还证明费米嘉当分母在自然多项式一阶Berezin-Weyl类中是必需的，并确定共轭全电荷同伦所需的最小逐因子Ore局域化。当$m\ge\max\{2N,n\}$时，$q$-依赖的嘉当轨道是非共振的。最后，排除了未变形坐标代数上的固定有限阶微分和有限非零位移实现。
英文摘要：
We extend a finite-support $q$-Hermitian radial calculus to Clifford-valued coordinate polynomials. For $N$ bosonic Hermitian vector labels and finite fermionic support, the stable range $m\ge 2N$ admits a canonical Gram--harmonic normal form by Howe separation. A label--Capelli inverse normalizes the Clifford contractions, while a finite Wick--Chevalley conjugation lifts the divided-power scalar gauge to the full Clifford PBW module. Coupling the normalized contractions to contractible fermionic seed complexes gives two conjugate polynomial-preserving $q$-Hermitian coordinate families. They are square-zero, anticommute within each polarization, restrict exactly to the radial PBW calculus, have scalar boundary trace $[2m-2n]_q$, and recover the classical Hermitian super Dirac pair as $q\to1$. For the mixed polarizations we introduce a relative triangular transport. The Grassmann relations for the normalized Capelli contractions hold on the full coordinate module and yield a labelwise factorization. Hence all higher-filtration mixed terms are finite subset products of explicit local defects, and the mixed polarization algebra closes for arbitrary finite label sets and finite fermionic support. The classical complex structure is common to both polarizations, whereas the two full transported Clifford--Weyl products are distinct for $0<q<1$. We also show that the fermionic Cartan denominator is forced in the natural polynomial first-order Berezin--Weyl class and determine the minimal factorwise Ore localization needed for the conjugate all-charge homotopies. The $q$-dependent Cartan orbit is nonresonant for $m\ge\max\{2N,n\}$. Finally, fixed finite-order differential and finite nonzero-shift realizations on the undeformed coordinate algebra are ruled out.
