Abstract In this paper, we study derived versions of the fusion category associated to Lusztig's quantum group U q . The categories that arise in this way are not semisimple but recovers the usual fusion ring when passing to Grothendieck rings. On the derived level it turns out that it is possible to define fusion for U q without using the notion of tilting modules. Hence, we arrive at a definition of the fusion ring that makes sense in any spherical category. We apply this new definition to the small quantum group and we relate it with some rings appearing in [22] .

中文翻译：

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量子群融合范畴的衍生对应物

摘要 在本文中，我们研究了与 Lusztig 的量子群 U q 相关的融合范畴的衍生版本。以这种方式出现的类别不是半简单的，而是在传递到格洛腾迪克环时恢复了通常的融合环。在导出的层次上，可以在不使用倾斜模块的概念的情况下定义 U q 的融合。因此，我们得出了在任何球形类别中都有意义的融合环的定义。我们将这个新定义应用于小量子群，并将其与 [22] 中出现的一些环联系起来。