量子环面的辫群不变性及其应用
Braid group invariance of quantum tori and applications
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中文总结 AI 辅助
该研究证明量子仿射代数权群上的斜对称双特征在辫群作用下不变，并扩展至预基本权群，进而构造量子环面上的自同构辫群作用，导出量子Q系统公式，并给出与Hernandez-Leclerc丛代数相容的量化矩阵。
中文摘要 AI 辅助
我们证明量子仿射代数$\ell$-权群上的斜对称双特征在Chari的辫群作用下不变。进一步，我们将该双特征扩展到由预基本$\ell$-权生成的群，并证明在Frenkel-Hernandez的扩展辫群作用下该不变性仍然成立。作为直接应用，我们分别在与量子仿射代数的有限维表示范畴及其Borel子代数范畴$\mathcal{O}^{\mathfrak{b}}$相关联的量子环面上获得自同构形式的辫群作用。此外，我们在所有单连通类型中推导出量子$QQ$-系统的显式公式。最后，我们证明扩展的双特征产生一个量化矩阵，该矩阵与由$\mathcal{O}^{\mathfrak{b}}$范畴化的Hernandez-Leclerc丛代数在所有有限Dynkin类型中相容，将Bittmann在单连通情形下的构造进行了推广。
英文摘要
We establish that the skew-symmetric bicharacter on the group of $\ell$-weights of a quantum affine algebra is invariant under Chari's braid group action. Furthermore, we define an extension of this bicharacter to the group generated by prefundamental $\ell$-weights and show that this invariance continues to hold under the extended braid group action of Frenkel-Hernandez. As immediate applications, we obtain braid group actions by automorphisms on the quantum tori associated respectively with the category of finite-dimensional representations of the quantum affine algebra and with the category $\mathcal{O}^{\mathfrak{b}}$ of its Borel subalgebra. In addition, we derive explicit formulas for quantum $QQ$-systems in all simply-laced types. Finally, we show that our extended bicharacter yields a quantization matrix compatible with the Hernandez-Leclerc cluster algebra categorified by $\mathcal{O}^{\mathfrak{b}}$ for all finite Dynkin types, extending Bittmann's construction in the simply-laced setting.
