AI 中文总结
研究融合范畴的分裂形式，通过标架等价性构建分裂伽罗瓦下降理论，将分裂形式存在性归约为群论分裂问题，证明单连通根数据关联的基础融合范畴在分圆域极大全实子域上有分裂形式及分裂实形式，还给出点态和有限群表示范畴的实下降判据。
AI 中文摘要
我们通过标架等价性对融合范畴构建分裂伽罗瓦下降理论，将分裂形式的存在性归约为群论分裂问题。相关的二次障碍结合了范畴相干性与单对象的稳定下降，而融合空间则施加了指标与奇偶性限制。对于单位根处的量子群，我们利用拟R矩阵使 bar 对合张量相容，并证明与单连通根数据关联的基础融合范畴$\boldsymbol{\textit{C}}(\boldsymbol{\textit{g}},\boldsymbol{\textit{k}})$在由量子参数生成的分圆域的极大全实子域上具有分裂形式，其所有分圆伽罗瓦共轭也满足该性质；特别地，每个$\boldsymbol{\textit{C}}(\boldsymbol{\textit{g}},\boldsymbol{\textit{k}})$都有一个分裂实形式。相比之下，除$\boldsymbol{\textit{C}}(\boldsymbol{\textit{E}}_8,1)$和$\boldsymbol{\textit{C}}(\boldsymbol{\textit{D}}_{4m},1)$（$m\boldsymbol{\textit{\textgreater{}}}1$）外，标准辫结构不存在分裂实形式。我们进一步证明结合 zesting 会破坏$\boldsymbol{\textit{C}}(\boldsymbol{\textit{g}},\boldsymbol{\textit{k}})$的实下降，而$\boldsymbol{\textit{C}}(\boldsymbol{\textit{g}},\boldsymbol{\textit{k}})$的每个辫 zesting 作为基础融合范畴仍具有分裂实形式。对点态和有限群表示范畴的应用给出了基于上同调与 Frobenius-Schur 指标的实下降判据。
英文摘要
We formulate split Galois descent for fusion categories through framed equivalences, reducing the existence of a split form to a group-theoretic splitting problem. The associated degree-two obstruction combines categorical coherence with stable descent of the simple objects, while fusion spaces impose index and parity restrictions. For quantum groups at roots of unity, we use the quasi-$R$-matrix to make the bar involution tensor-compatible and prove that the underlying fusion category $\mathcal C(\mathfrak g,k)$, associated with the simply connected root datum, has a split form over the maximal totally real subfield of the cyclotomic field generated by the quantum parameter. The same holds for all its cyclotomic Galois conjugates; in particular, every $\mathcal C(\mathfrak g,k)$ has a split real form. By contrast, no split real form exists for the standard braiding except on $\mathcal C(E_8,1)$ and $\mathcal C(D_{4m},1)$, $m \geq 1$. We further show that associative zesting can destroy real descent of $\mathcal C(\mathfrak g,k)$, while every braided zesting of $\mathcal C(\mathfrak g,k)$ still admits a split real form as an underlying fusion category. Applications to pointed and finite-group representation categories give criteria for real descent in terms of cohomology and Frobenius--Schur indicators.
Comments49 pages
