Realizing enveloping algebras via moduli stacks
Realizing enveloping algebras via moduli stacks
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通过模栈实现包络代数
DOI：
10.4310/pamq.2015.v11.n2.a1
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Fan Xu
中科院分区：
文献类型：
--
作者：
L. Bai;Fan Xu
让$\mathop{\rm CF}\ nolimts (\mathop{\mathfrak{Obj}\kern。05em}\nolimits_\mathcal{A})$表示在给定堆栈$\mathop{\mathfrak{Obj}\kern上$\mathbb{Q}$值可构造函数的向量空间。05em}\nolimits_\mathcal{A}$的精确类别$\mathcal{A}$。通过使用Ringel—Hall代数方法，Joyce证明了$\mathop{\rm CF}\ nollimites (\mathop{\mathfrak{Obj}\kern。通过卷积乘法和子空间$\mathop{\rm CF}\nolimits^{\rm ind}\mathop{\ mathfrk {Obj}\kern, $\mathbb{Q}$-是一个结合式$\mathbb{Q}$-代数。在不可分解函数上支持的可构造函数$\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern)$的李子代数。{{{{}})$ in[10]。在本文中，我们证明了存在子代数$\mathop{\rm CF}\nolimits^{\text{KS}}(\mathop{\mathfrak{Obj}\kern。{{m}\nolimits_\mathcal{A})$ of $\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern。$\mathop{\ mathfrk {Obj}\kern .05em}\nolimits_\mathcal{A})$同构于$\mathop{\ mathfrk {Obj}\ nolimits_\mathcal{A}的泛包络代数$\mathop{\ mathfrk}\nolimits_\mathcal{A} $。此外，我们在$\mathop{\rm CF}\ nollimites ^{\text{KS}}(\mathop{\mathfrak{Obj}\kern上构造了一个乘法。{0em}\nolimits_\mathcal{A})$和格林定理的退化形式。这概括了乔伊斯的工作，以及b[3]的结果。
Let $\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ denote the vector space of $\mathbb{Q}$-valued constructible functions on a given stack $\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A}$ for an exact category $\mathcal{A}$. By using the Ringel--Hall algebra approach, Joyce proved that $\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ is an associative $\mathbb{Q}$-algebra via the convolution multiplication and the subspace $\mathop{\rm CF}\nolimits^{\rm ind}\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ of constructible functions supported on indecomposables is a Lie subalgebra of $\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ in [10]. In this paper, we show that there is a subalgebra $\mathop{\rm CF}\nolimits^{\text{KS}}(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ of $\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ isomorphic to the universal enveloping algebra of $\mathop{\rm CF}\nolimits^{\rm ind}(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$. Moreover we construct a comultiplication on $\mathop{\rm CF}\nolimits^{\text{KS}}(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ and a degenerate form of Green's theorem. This generalizes Joyce's work, as well as results of [3].