Twisted quantum affinizations and quantization of extended affine Lie algebras
Twisted quantum affinizations and quantization of extended affine Lie algebras
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DOI：
10.1090/tran/8706
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发表时间：
2020-06
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影响因子：
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通讯作者：
Fulin Chen;N. Jing;Fei Kong;Shaobin Tan
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文献类型：
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作者：
Fulin Chen;N. Jing;Fei Kong;Shaobin Tan
本文对任意Kac-Moody李代数$\mathfrak{g}和满足两个连结条件的图自同构$\m u=mathfrak{g}$，引入并研究了$\mathfrak{g}$的$\m扭曲量子仿射代数$\m o l{U}_{\hbar}(\ch{\mathfrak{g}}_u)$.当$\mathfrak{g}$是有限类型时，$\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\u)$是Drinfeld当前的扭曲量子仿射代数的代数实现。当$\Mu=\mathm{ID}$时，$\Mathcal{U}_{\hbar}(\HAT{\mathfrak{g}}_\Mu)$是Ginzburg-Kapranov-Vasserot引入的量子仿射代数。作为本文的主要结果，我们首先证明了$\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$.的一个三角分解其次，我们用“正规序积”给出了受限$\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$-modules上仿射量子Serre关系的一个简单刻画。第三，我们证明了受限$\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$-modules范畴是一个么半群范畴，从而得到了关于$\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$.的“受限完备”的拓扑Hopf代数结构第四，我们研究了数学式{U}_{\hbar}({\mathfrak{g}}_u)}的经典极限，并将其简化为扩展仿射李代数的量子化理论。特别地，基于Allison-Berman-Pianzola的分类结果，我们得到了零度$2$扩展仿射李代数的$-形变。
In this paper, for an arbitrary Kac-Moody Lie algebra $\mathfrak{g}$ and a diagram automorphism $\mu$ of $\mathfrak{g}$ satisfying two linking conditions, we introduce and study a $\mu$-twisted quantum affinization algebra $\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$ of $\mathfrak{g}$. When $\mathfrak{g}$ is of finite type, $\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$ is Drinfeld's current algebra realization of the twisted quantum affine algebra. And, when $\mu=\mathrm{Id}$, $\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$ is the quantum affinization algebra introduced by Ginzburg-Kapranov-Vasserot. As the main results of this paper, we first prove a triangular decomposition of $\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$. Second, we give a simple characterization of the affine quantum Serre relations on restricted $\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$-modules in terms of "normal order products". Third, we prove that the category of restricted $\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of $\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$. Fourth, we study the classical limit of $\mathcal{U}_{\hbar}(\hat{\mathfrak{g}}_\mu)$ 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 nullity $2$ extended affine Lie algebras.