The ring of twisted tilting modules of quantum $GL_n$

Quantum tilting modules over local rings

We show that tilting modules for quantum groups over local Noetherian domains exist and that the indecomposable tilting modules are parametrized by their highest weight. For this we introduce a model category ${\mathcal X}={\mathcal X}_{\mathscr A}(R)$ associated with a Noetherian ${\mathbb Z}[v,v^{-1}]$-domain ${\mathscr A}$ and a root system $R$. We show that if ${\mathscr A}$ is of quantum characteristic $0$, the model category contains all $U_{\mathscr A}$-modules that admit a Weyl filtration. If ${\mathscr A}$ is in addition local, we study torsion phenomena in the model category. This le…

Fusion rings for quantum groups

We study the fusion rings of tilting modules for a quantum group at a root of unity modulo the tensor ideal of negligible tilting modules. We identify them in type A with the combinatorial rings from [KS] and give a similar description of the sp(2n)-fusion ring in terms of noncommutative symmetric functions. Moreover we give a presentation of all fusion rings in classical types as quotients of polynomial rings extending known results in special cases. Finally we also compute the fusion rings for type G2.

通过半导出Ringel-Hall代数构造扭量子环代数

Twisted quantum loop algebras via semi-derived Ringel-Hall algebras

利用更一般加权射影直线的半导出Ringel-Hall代数，实现了与赋值星形图相关的扭量子环代数，包括Drinfeld新展示中的扭量子仿射代数。