论文标题
Q-DATA和无链量子仿射代数的代表理论
Q-data and representation theory of untwisted quantum affine algebras
论文作者
论文摘要
对于一个复杂的有限维简单谎言代数$ \ mathfrak {g} $，我们介绍了Q-datum的概念，Q-datum的概念从Weyl group Compinatorics的角度来概括了具有高度功能的Dynkin颤动的概念。使用这个概念，我们开发了一个统一的理论，描述了扭曲的澳大利亚纪念物和[O.-Suh，J。Elgebra，2019]中介绍的扭曲的适应性类，并具有适当的广义Coxeter元素的概念。结果，我们获得了一个组合公式，该公式表达了$ \ mathfrak {g} $的量子cartan矩阵的倒数，该矩阵概括了[Hernandez-leclerc，J。Reine Angew的结果。数学，2015年]在简单的情况下。我们还发现Q-DATA组合理论的几个应用在$ \ mathfrak {g} $的无键量子仿射代数的有限维表示理论中。特别是，就Q-DATA和量子cartan矩阵的倒数而言，（i）由于[Chari-Moura，int。数学。 res。不是，2005年]和[kashiwara-kim-o.-park，arxiv：2003.03265]，（ii）我们提出了所有kirillov-reshetikhin模块和（iii）之间的归一化r-矩阵分母的统一（部分猜测）公式，我们concemant和（iii）$ unvariants $ unvariants $ sairants $ sairants $单v。 $λ^\ infty（v，w）$在[kashiwara-kim-o.-park，Compos中引入。数学，2020年]对于每对简单模块$ v $和$ w $。
For a complex finite-dimensional simple Lie algebra $\mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh, J. Algebra, 2019] with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of $\mathfrak{g}$, which generalizes the result of [Hernandez-Leclerc, J. Reine Angew. Math., 2015] in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of $\mathfrak{g}$. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to [Chari-Moura, Int. Math. Res. Not., 2005] and [Kashiwara-Kim-O.-Park, arXiv:2003.03265], (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov-Reshetikhin modules, and (iii) we compute the invariants $Λ(V,W)$ and $Λ^\infty(V, W)$ introduced in [Kashiwara-Kim-O.-Park, Compos. Math., 2020] for each pair of simple modules $V$ and $W$.
