发表机构
Tongji University; Beijing Institute of Technology（同济大学; 北京理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了仿射量子Schur-Weyl对偶：在特征零且Hecke参数非单位根时，扩展仿射Hecke代数到仿射张量空间自同态代数的自然同态是同构，并由此证明了关于仿射量子Schur代数中心和Noether性的两个猜想。
AI 中文摘要
设$\mathpzc K$为任意含可逆元素$\varepsilon$的交换环。令${\mathcal H}_{\\!\vartriangle\\!}(r)_{\mathpzc K}$为具有Hecke参数$\varepsilon$的A型扩展仿射Hecke代数，令$\Omega_{\mathpzc K}^{\otimes r}$为仿射张量空间，并令${\mathcal S}_{\\!\vartriangle\\!}(n,r)_{\mathpzc K}$为相应的仿射量子Schur代数。我们首先证明${\mathcal H}_{\\!\vartriangle\\!}(r)_{\mathpzc K}$在$\Omega_{\mathpzc K}^{\otimes r}$上的自然右作用总是忠实的。进一步假设$\mathpzc K$是特征为$0$的域且$\varepsilon$不是单位根。我们证明，对任意$n\geq 2$，自然代数同态$\xi_r:{\mathcal H}_{\\!\vartriangle\\!}(r)_{\mathpzc K}\rightarrow\operatorname{End}_{{\mathcal S}_{\\!\vartriangle\\!}(n,r)_{\mathpzc K}}(\Omega_{\mathpzc K}^{\otimes r})^{\mathrm{op}}$是同构。这证明了文献[DDF]中的猜想3.8.8。作为应用，我们证明了文献[DDF]5.2.4中提出的关于仿射量子Schur代数中心的猜想。我们还证明了当$\mathpzc K$是Noether交换环时，${\mathcal S}_{\\!\vartriangle\\!}(n,r)_{\mathpzc K}$是左、右Noether的，这验证了文献[DY]注记1.7中的一个猜想。
英文摘要
Let $\mathpzc K$ be an arbitrary commutative ring containing an invertible element $\varepsilon$. Let ${\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}$ be the extended affine Hecke algebra of type $A$ with Hecke parameter $\varepsilon$, let $Ω_{\mathpzc K}^{\otimes r}$ be the affine tensor space, and let ${\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}$ be the corresponding affine quantum Schur algebra. We first prove that the natural right action of ${\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}$ on $Ω_{\mathpzc K}^{\otimes r}$ is always faithful. Assume further that $\mathpzc K$ is a field of characteristic $0$ and that $\varepsilon$ is not a root of unity. We prove that, for any $n\geq 2$, the natural algebra homomorphism $ξ_r:{\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}\rightarrow\operatorname{End}_{{\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}}(Ω_{\mathpzc K}^{\otimes r})^{\mathrm{op}}$ is an isomorphism. This proves Conjecture~3.8.8 of \cite{DDF}. As an application, we prove the conjecture formulated in \cite[5.2.4]{DDF} concerning the center of the affine quantum Schur algebra. We also prove that ${\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}$ is left and right Noetherian whenever $\mathpzc K$ is a Noetherian commutative ring, which verify a conjecture in \cite[Rem. 1.7]{DY}.
Comments35 pages
