发表机构
The University of Hong Kong（香港大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过引入加厚实现，将修正量子群的典范基与更大量子群负部分的典范基相联系，从而证明典范基在余乘法、双线性形式及特定乘法下的正性，并验证了Lusztig猜想在单连通有限型情形成立。
AI 中文摘要
设 $\mathbf{U}$ 为与对称 Cartan 数据相关联的量子群，$\dot{\mathbf{U}}$ 为其修正形式，$\dot{\mathbf{B}}$ 为 $\dot{\mathbf{U}}$ 的典范基。Lusztig 猜想 $\dot{\mathbf{U}}$ 中关于 $\dot{\mathbf{B}}$ 的乘法、余乘法和双线性形式的结构常数属于 $\mathbb{N}[v,v^{-1}]$。我们引入\emph{加厚实现}，它将 $\dot{\mathbf{B}}$ 与更大量子群 $\tilde{\mathbf{U}}^-$ 的负部分的典范基联系起来。更精确地说，它将 $\dot{\mathbf{U}}$ 中的相关结构常数等同于 $\tilde{\mathbf{U}}^-$ 中的结构常数。作为推论，我们证明对于任意对称 Cartan 数据，$\dot{\mathbf{B}}$ 具有余乘法和双线性形式的正性性质，并且当其中一个因子为球面抛物型时，乘法也具有正性。特别地，Lusztig 猜想对单连通有限型成立。我们还证明了广泛一类可积模张量积的典范基具有转移矩阵和 $\dot{\mathbf{B}}$ 作用的正性性质。
英文摘要
Let $\mathbf{U}$ be a quantum group associated with a symmetric Cartan datum, let $\dot{\mathbf{U}}$ be its modified form, and let $\dot{\mathbf{B}}$ be the canonical basis of $\dot{\mathbf{U}}$. Lusztig conjectured that the structure constants of the multiplication, comultiplication, and bilinear form in $\dot{\mathbf{U}}$ with respect to $\dot{\mathbf{B}}$ belong to $\mathbb{N}[v,v^{-1}]$. We introduce the \emph{thickening realization}, which relates $\dot{\mathbf{B}}$ to the canonical basis of the negative part of a larger quantum group $\tilde{\mathbf{U}}^-$. More precisely, it identifies the relevant structure constants in $\dot{\mathbf{U}}$ with the structure constants in $\tilde{\mathbf{U}}^-$. As consequences, we prove, for arbitrary symmetric Cartan datum, $\dot{\mathbf{B}}$ has the positivity properties for the comultiplication and bilinear form, as well as the positivity for the multiplication whenever one factor is spherical parabolic. In particular, Lusztig's conjecture holds for simply-laced finite type. We also prove the canonical bases of a broad class of tensor products of integrable modules have the positivity properties for the transition matrices and actions by $\dot{\mathbf{B}}$.
Comments46 pages. This paper supersedes arXiv:2510.12154 and establishes substantially stronger results
