量子群的簇性质
Cluster Nature of Quantum Groups
摘要：
我们提出了一个刚性团簇模型来实现ADE型$\mathfrak{g}$的量子群${\bf U}_q(\mathfrak{g})$.也就是说，我们证明了从量子群${\bf U}_q(\mathfrak{g})$到Fock-Goncharov量子团簇代数$\mathcal{O}_q(\mathscr{P}_{{\rm G},\odot})$的Weyl群不变量的商代数存在一个自然的Hopf代数同构.利用簇代数的量子对偶性，证明了${\bf U}_q(\mathfrak{g})$存在一个自然基$\bar{\bf Θ}$，其结构系数在$\mathbb{N}[q^{\frac{1}{2}}, q^{-\frac{1}{2}}]$中.基$\bar{\bf Θ}$在Lusztig的辫群作用、Dynkin自同构和星反对合下满足不变性。
英文摘要：
We present a rigid cluster model to realize the quantum group ${\bf U}_q(\mathfrak{g})$ for $\mathfrak{g}$ of type ADE. That is, we prove that there is a natural Hopf algebra isomorphism from the quantum group ${\bf U}_q(\mathfrak{g})$ to a quotient algebra of the Weyl group invariants of the Fock-Goncharov quantum cluster algebra $\mathcal{O}_q(\mathscr{P}_{{\rm G},\odot})$. By applying the quantum duality of cluster algebras, we show that ${\bf U}_q(\mathfrak{g})$ admits a natural basis $\bar{\bf Θ}$ whose structural coefficients are in $\mathbb{N}[q^{\frac{1}{2}}, q^{-\frac{1}{2}}]$. The basis $\bar{\bf Θ}$ satisfies an invariance property under Lusztig's braid group action, the Dynkin automorphisms, and the star anti-involution.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
