论文标题
$ v（ρ）\ otimes v（ρ）$的组成部分和主要的重量Polyhedra for affine kac-moody lie代数
Components of $V(ρ) \otimes V(ρ)$ and dominant weight polyhedra for affine Kac-Moody Lie algebras
论文作者
论文摘要
Kostant问以下问题：让$ \ Mathfrak {g} $在复数上是一个简单的谎言代数。令$λ$为主要的整体权重。然后，$ v（λ）$是$ v（ρ）\ otimes v（ρ）$的组成部分，仅当$λ\ leq2ρ$下的$λ\ leq2ρ$下的平常bruhat-chevalley订单下。在与R. chirivi和A. Maffei的早期工作中，第二作者对这个问题给出了肯定的答案，直到饱和因素。当前工作的目的是将此结果扩展到与任何与任何简单的Lie lie algebra $ \ Mathring {\ Mathfrak {\ Mathfrak {g}} $相关的无关的affine affine kac-moody lie lie代数。实际上，我们证明了Aggine $ sl_n $而无需任何饱和因子的结果。我们的证明需要一些其他技术，包括戈达德 - 肯特 - 奥利维尔结构以及对非紧凑型多面体的特征性锥体的研究。
Kostant asked the following question: Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers. Let $λ$ be a dominant integral weight. Then, $V(λ)$ is a component of $V(ρ)\otimes V(ρ)$ if and only if $λ\leq 2 ρ$ under the usual Bruhat-Chevalley order on the set of weights. In an earlier work with R. Chirivi and A. Maffei the second author gave an affirmative answer to this question up to a saturation factor. The aim of the current work is to extend this result to untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ associated to any simple Lie algebra $\mathring{\mathfrak{g}}$ (up to a saturation factor). In fact, we prove the result for affine $sl_n$ without any saturation factor. Our proof requires some additional techniques including the Goddard-Kent-Olive construction and study of the characteristic cone of non-compact polyhedra.
