论文标题
量子组的簇性质
Cluster Nature of Quantum Groups
论文作者
论文摘要
我们提出了一个刚性群集模型，以实现量子组$ {\ bf u} _q（\ mathfrak {g}）$ for $ \ mathfrak {g} $ a ade的类型。也就是说，我们证明，量子组$ {\ bf u} _q（\ mathfrak {g}）$的天然HOPF代数同构。 $ \ Mathcal {O} _Q（\ Mathscr {p} _ {{\ rm g}，\ odot}）$。通过应用群集代数的量子二元性，我们表明$ {\ bf u} _q（\ mathfrak {g}）$允许自然基础$ \ bar {\bfθ} $，其结构系数在$ \ mathbb {N} [q^n} [q^Q^{q^{Q^{Q^{1}}中q^{ - \ frac {1} {2}}} $。基础$ \ bar {\bfθ} $在Lusztig的编织组动作，Dynkin Kutomormormisms和Star Anti Antivolution下满足不变性属性。
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.
