论文标题
霍尔谎言曲曲面的单体方案代数
Hall Lie algebras of toric monoid schemes
论文作者
论文摘要
我们将投影$ n $ n $维的复式变种$x_δ$x_δ$一对共同交互（但通常是非交通）hopf代数$ h^α_x，h^{t} _x $。这些是作为某些类别的霍尔代数出现的$ \ coh^α（x），\ coh^t（x）$ of $x_Δ$被视为单型方案的$x_δ$ - 即，通过将通勤性单体的光谱粘合在一起而不是圆环而获得的方案。 When $X_Δ$ is smooth, the category $\Coh^T(X)$ has an explicit combinatorial description as sheaves whose restriction to each $\mathbb{A}^n$ corresponding to a maximal cone $σ\in Δ$ is determined by an $n$-dimensional generalized skew shape. （非添加的）类别$ \ coh^α（x），\ coh^t（x）$是通过由Dyckerhoff-Kapranov开发的原始外科/原始 - 阿贝尔类别的形式来处理的。 大厅代数$ h^α_x，h^{t} _x $是分级和连接的，因此包裹代数$ h^α_x\ simeq u（\ n^α_x）$，$ h^{t} \ n^{t} _x $被其各自类别中的不可分解的相干滑轮跨越。 我们明确编写了几个示例，在某些情况下，我们可以将$ \ n^t_x $与已知的代数相关联。特别是，当$ x = \ mathbb {p}^1 $时，$ \ n^t_x $与$ \ mathfrak {gl} _2 _2 [t，t^{ - 1}] $中的非标准borel同构。当$ x $是$ \ mathbb {a}^2 $内部的第二个无穷小社区时，$ \ n^t_x $是$ \ mathfrak {gl} _2 _2 [t] $的sibalgebra的同构。我们还考虑了$ x = \ mathbb {p}^2 $的情况，我们通过描述$ \ coh^t（x）$中的所有不可兼容的冰淇淋来为$ \ n^t_x $提供基础。
We associate to a projective $n$-dimensional toric variety $X_Δ$ a pair of co-commutative (but generally non-commutative) Hopf algebras $H^α_X, H^{T}_X$. These arise as Hall algebras of certain categories $\Coh^α(X), \Coh^T(X)$ of coherent sheaves on $X_Δ$ viewed as a monoid scheme - i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When $X_Δ$ is smooth, the category $\Coh^T(X)$ has an explicit combinatorial description as sheaves whose restriction to each $\mathbb{A}^n$ corresponding to a maximal cone $σ\in Δ$ is determined by an $n$-dimensional generalized skew shape. The (non-additive) categories $\Coh^α(X), \Coh^T(X)$ are treated via the formalism of proto-exact/proto-abelian categories developed by Dyckerhoff-Kapranov. The Hall algebras $H^α_X, H^{T}_X$ are graded and connected, and so enveloping algebras $H^α_X \simeq U(\n^α_X)$, $H^{T}_X \simeq U(\n^{T}_X)$, where the Lie algebras $\n^α_X, \n^{T}_X$ are spanned by the indecomposable coherent sheaves in their respective categories. We explicitly work out several examples, and in some cases are able to relate $\n^T_X$ to known Lie algebras. In particular, when $X = \mathbb{P}^1$, $\n^T_X$ is isomorphic to a non-standard Borel in $\mathfrak{gl}_2 [t,t^{-1}]$. When $X$ is the second infinitesimal neighborhood of the origin inside $\mathbb{A}^2$, $\n^T_X$ is isomorphic to a subalgebra of $\mathfrak{gl}_2[t]$. We also consider the case $X=\mathbb{P}^2$, where we give a basis for $\n^T_X$ by describing all indecomposable sheaves in $\Coh^T(X)$.
