论文标题
扭曲的量子亲和量子的量化代数
Twisted quantum affinizations and quantization of extended affine Lie algebras
论文作者
论文摘要
在本文中，对于任意的kac-moody lie代数$ \ mathfrak g $和图表$μ$ $ $ \ $ \ mathfrak g $满足某些自然链接条件，我们介绍和研究$μ$ $ $ twist的量子量子量化代数$ \ mathcal $ \ mathcal u_ \ hbar hat G}_μ\ right）$ \ Mathfrak g $的$。 当$ \ Mathfrak G $是有限类型时，$ \ Mathcal U_ \ HBAR \ left（\ hat {\ Mathfrak g}_μ\ right）$是德林菲尔德的当前代数实现了扭曲的量子仿射代数。当$μ= \ mathrm {id} $和$ \ Mathfrak g $中的仿期类型时，$ \ Mathcal u_ \ hbar \ left（\ hat {\ hat {\ Mathfrak g}_μ\ right）$是量子toroid algebra，该代数由Ginzburg，Kapranov和kapranov和vasserot引入。作为本文的主要结果，我们首先证明了$ \ mathcal u_ \ hbar \ left（\ hat {\ mathfrak g}_μm\ right）$的三角分解。其次，我们简单地对限制的$ \ mathcal u_ \ hbar \ left（\ hat {\ mathfrak g}_μm\ right）$ - 模块在“正常订单产品”方面进行了简单的表征。第三，我们证明了限制的$ \ MATHCAL U_ \ HBAR \ left的类别（\ hat {\ Mathfrak g}_μ\右）$ - 模块是一个单个类别，因此获得了“限制限制”的拓扑结构，因此获得了$ \ hat fr的topogical Hopf代数结构。 g}_μ\右）$。最后，我们研究$ \ Mathcal U_ \ Hbar \ left的经典限制（\ hat {\ Mathfrak g}_μ\ right）$，并将其删除到扩展的仿射lie代数的量化理论中。特别是，基于Allison-Berman-Pianzola的分类结果，我们获得了所有无效的$ \ HBAR $ - $ 2 $ 2 $扩展的Aggrine Lie代数。
In this paper, for an arbitrary Kac-Moody Lie algebra $\mathfrak g$ and a diagram automorphism $μ$ of $\mathfrak g$ satisfying certain natural linking conditions, we introduce and study a $μ$-twisted quantum affinization algebra $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ of $\mathfrak g$. When $\mathfrak g$ is of finite type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ is Drinfeld's current algebra realization of the twisted quantum affine algebra. When $μ=\mathrm{id}$ and $\mathfrak g$ in affine type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$. Second, we give a simple characterization of the affine quantum Serre relations on restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$-modules in terms of "normal order products". Third, we prove that the category of restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$. Last, we study the classical limit of $\mathcal U_\hbar\left(\hat{\mathfrak g}_μ\right)$ and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the $\hbar$-deformation of all nullity $2$ extended affine Lie algebras.
