论文标题
Weyl组的群集实现和$ Q $ - 量子仿射代数的特征
Cluster realization of Weyl groups and $q$-characters of quantum affine algebras
论文作者
论文摘要
我们考虑一个无限的颤抖$ q（\ mathfrak {g}）$和一个定期Quivers $ q_m（\ Mathfrak {g}）$的家族，用于有限的尺寸简单的简单lie代数$ \ mathfrak {g} $ {g} $和$ m \ in \ in \ insbb {z} $。 Quiver $ Q（\ Mathfrak {g}）$与Hernandez和Leclerc介绍的量子仿射代数所介绍的基本相同。我们以$ q_m（\ Mathfrak {g}）$的方式构建Weyl group $ w（\ mathfrak {g}）$，其方式与作者，Ishibashi和Oya研究的方式相似，并研究了其应用于$ Q $ - $ Q $ - Q $ - Q $ - Quem-Q $ characters-qucharcter-Q $ - $ u_q（\ hat {\ mathfrak {g}}）$由Frenkel和Reshetikhin引入，以及晶格$ \ Mathfrak {G} $ - Toda Field Theory。特别是，当$ q $是团结的根源时，我们证明$ q $ - 字符在Weyl Group Action下是不变的。我们还表明，$ a $ variables for $ q（\ mathfrak {g}）$对应于lattice $ \ mathfrak {g} $ toda field方程的$τ$ - 功能。
We consider an infinite quiver $Q(\mathfrak{g})$ and a family of periodic quivers $Q_m(\mathfrak{g})$ for a finite dimensional simple Lie algebra $\mathfrak{g}$ and $m \in \mathbb{Z}_{>1}$. The quiver $Q(\mathfrak{g})$ is essentially same as what introduced by Hernandez and Leclerc for the quantum affine algebra. We construct the Weyl group $W(\mathfrak{g})$ as a subgroup of the cluster modular group for $Q_m(\mathfrak{g})$, in a similar way as what studied by the author, Ishibashi and Oya, and study its applications to the $q$-characters of quantum non-twisted affine algebras $U_q(\hat{\mathfrak{g}})$ introduced by Frenkel and Reshetikhin, and to the lattice $\mathfrak{g}$-Toda field theory. In particular, when $q$ is a root of unity, we prove that the $q$-character is invariant under the Weyl group action. We also show that the $A$-variables for $Q(\mathfrak{g})$ correspond to the $τ$-function for the lattice $\mathfrak{g}$-Toda field equation.
