论文标题
连贯的滑轮的大厅代数内部的宇宙在复曲面的分辨率上
The universe inside Hall algebras of coherent sheaves on toric resolutions
论文作者
论文摘要
令$ \ mathfrak {g} \ neq \ mathfrak {so} _8 $成为$ a，d，e $的简单谎言代数，带有$ \ wideHat {\ mathfrak {\ mathfrak {g}} $ \ wideHat {\ mathfrak {g}} $ a nilpotent subalgebra。给定的$ \ mathfrak {n} _- $如上所述，我们提供了以下属性的$ sl_3（\ mathbb {c}）$的无限有限的Abelian子组的集合。令$ g $为集合中的任何组，$ y = g \ operatorName { - } \ mbox {hilb}（\ m马理{c}^3）$和$ψ：d^b_g（coh（\ sathbb {c}^3））\ rightArrow d^b（y）我们提出了$ coh_g（\ Mathbb {c}^3）$，s.t。中的对象的（明确描述的）子集。其图像在$ψ$下生成的霍尔代数是$ u（\ mathfrak {n} _-）$的同构。 In case the field $\Bbbk$ (in place of $\mathbb{C}$) is finite and $\mbox{char}(\Bbbk)$ is coprime with the order of $G$, we conjecture the isomorphisms of the corresponding 'counting' Ringel-Hall algebras and the specializations of quantized universal enveloping algebras $ u_v（\ Mathfrak {n} _-）$ v = \ sqrt {| \ bbbk |} $。
Let $\mathfrak{g}\neq \mathfrak{so}_8$ be a simple Lie algebra of type $A,D,E$ with $\widehat{\mathfrak{g}}$ the corresponding affine Kac-Moody algebra and $\mathfrak{n}_-\subset \widehat{\mathfrak{g}}$ a nilpotent subalgebra. Given $\mathfrak{n}_-$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $SL_3(\mathbb{C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname{-}\mbox{Hilb}(\mathbb{C}^3)$ and $Ψ: D^b_G(Coh(\mathbb{C}^3))\rightarrow D^b(Coh(Y))$ the derived equivalence of Bridgeland, King and Reid. We present an (explicitly described) subset of objects in $Coh_G(\mathbb{C}^3)$, s.t. the Hall algebra generated by their images under $Ψ$ is isomorphic to $U(\mathfrak{n}_-)$. In case the field $\Bbbk$ (in place of $\mathbb{C}$) is finite and $\mbox{char}(\Bbbk)$ is coprime with the order of $G$, we conjecture the isomorphisms of the corresponding 'counting' Ringel-Hall algebras and the specializations of quantized universal enveloping algebras $U_v(\mathfrak{n}_-)$ at $v=\sqrt{|\Bbbk|}$.
