顶点李代数与分圆共不变量
Vertex Lie algebras and cyclotomic coinvariants
摘要：
给定一个顶点李代数$\mathscr L$，它具有一个由循环群$Γ$的自同构所产生的作用，我们定义了黎曼球面上的分圆共不变空间。这些是张量积的光滑模在“本地”李代数$\mathsf L(\mathscr L)_{z_i}$分配给标记点$z_i$，由一个“全球”李代数${\mathsf L}^Γ_{\{z_i \}}(\mathscr L)$的$Γ$ -等变函数的行动。另一方面，$\mathscr L$的泛包络顶点代数$\mathbb V (\mathscr L)$本身是一个顶点李代数，其诱导作用是$Γ$。这给出了上述李代数的“大”类似物。从这些，我们构建的空间“大”分圆的共不变，即关于${\mathsf L}^Γ_{\{z_i \}}(\mathbb V(\mathscr L))$的共不变。我们证明，这两个定义的分圆coinvariants实际上是一致的，提供的起源是包括作为一个标记点。作为一个推论，我们证明了一个结果的分圆共不变量，我们需要分圆高丁模型的解决方案arXiv：1409.6937的函。在由$Γ$确定的原点处，必须在$\mathsf L(\mathscr L)$的稳定子代数$\mathsf L(\mathscr L)^Γ$上分配一个模。这个模成为Li意义下的$\mathbb V(\mathscr L)$ -拟模。作为一个双积，我们得到了这样的拟模的一个双积公式。
英文摘要：
Given a vertex Lie algebra $\mathscr L$ equipped with an action by automorphisms of a cyclic group $Γ$, we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over `local' Lie algebras $\mathsf L(\mathscr L)_{z_i}$ assigned to marked points $z_i$, by the action of a `global' Lie algebra ${\mathsf L}^Γ_{\{z_i \}}(\mathscr L)$ of $Γ$-equivariant functions. On the other hand, the universal enveloping vertex algebra $\mathbb V (\mathscr L)$ of $\mathscr L$ is itself a vertex Lie algebra with an induced action of $Γ$. This gives `big' analogs of the Lie algebras above. From these we construct the space of `big' cyclotomic coinvariants, i.e. coinvariants with respect to ${\mathsf L}^Γ_{\{z_i \}}(\mathbb V(\mathscr L))$. We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in arXiv:1409.6937. At the origin, which is fixed by $Γ$, one must assign a module over the stable subalgebra $\mathsf L(\mathscr L)^Γ$ of $\mathsf L(\mathscr L)$. This module becomes a $\mathbb V(\mathscr L)$-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.
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