标题： 契尔诺夫性质和凯莱-哈密顿霍普夫代数的判别式理想 显示英文标题

标题： Chevalley property and discriminant ideals of Cayley-Hamilton Hopf Algebras

摘要： 对于任何存在大的中心 Hopf 子代数的仿射 Hopf 代数 $H$，$H$ 都可以被赋予 De Concini-Procesi-Reshetikhin-Rosso 意义下的 Cayley-Hamilton Hopf 代数结构。 证明了任何 $H$ 的纤维代数上的有限维模范畴是 $H$ 的恒等纤维代数 $H/\mathfrak{m}_{\overline{\varepsilon}}H$ 上的有限维模张量范畴上的不可分解的精确模范畴。 对于任何仿射Cayley-Hamilton Hopf代数$(H,C,\text{tr})$，如果$H/\mathfrak{m}_{\overline{\varepsilon}}H$具有Chevalley性质，则证明了如果$(H,C,\text{tr})$的判别式理想的零点集非空，则它包含仿射代数群$\text{maxSpec}C$的单位元在左（或右）缠绕自同构群作用下的轨道。 其证明依赖于这样一个事实：$H/\mathfrak{m}_{\overline{\varepsilon}}H$ 具有 Chevalley 性质当且仅当 $(H,C)$ 的 $\overline{\varepsilon}$-Chevalley 轨迹与 $\text{maxSpec}C$ 重合。 作为应用，我们首先给出 $(H,C,\text{tr})$ 的最低判别理想的零轨迹的描述。 证明了$(H,C,\text{tr})$的最低判别理想是级别$\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))+1$，其中$\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$是有限维霍普夫代数$H/\mathfrak{m}_{\overline{\varepsilon}}H$的格罗滕迪克环，而$\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))$是$\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$的弗罗贝尼乌斯-佩龙维度。关于 Mi-Wu-Yakimov 关于最低判别理想的一些最新结果被推广了。其次，我们证明了如果$H$具有切瓦莱性质，则所有判别理想都是平凡的。 显示英文摘要

摘要： For any affine Hopf algebra $H$ which admits a large central Hopf subalgebra, $H$ can be endowed with a Cayley-Hamilton Hopf algebra structure in the sense of De Concini-Procesi-Reshetikhin-Rosso. The category of finite-dimensional modules over any fiber algebra of $H$ is proved to be an indecomposable exact module category over the tensor category of finite-dimensional modules over the identity fiber algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ of $H$. For any affine Cayley-Hamilton Hopf algebra $(H,C,\text{tr})$ such that $H/\mathfrak{m}_{\overline{\varepsilon}}H$ has the Chevalley property, it is proved that if the zero locus of a discriminant ideal of $(H,C,\text{tr})$ is non-empty then it contains the orbit of the identity element of the affine algebraic group $\text{maxSpec}C$ under the left (or right) winding automorphism group action. Its proof relies on the fact that $H/\mathfrak{m}_{\overline{\varepsilon}}H$ has the Chevalley property if and only if the $\overline{\varepsilon}$-Chevalley locus of $(H,C)$ coincides with $\text{maxSpec}C$. As applications, we first provide a description of the zero locus of the lowest discriminant ideal of $(H,C,\text{tr})$. It is proved that the lowest discriminant ideal of $(H,C,\text{tr})$ is of level $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))+1$, where $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$ is the Grothendieck ring of the finite-dimensional Hopf algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ and $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))$ is the Frobenius-Perron dimension of $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$. Some recent results of Mi-Wu-Yakimov about lowest discriminant ideals are generalized. Secondly, we prove that all the discriminant ideals are trivial if $H$ has the Chevalley property.