论文标题
$ \ mathfrak {sl} _2 $的简单2-陈述的忠诚
Faithfulness of simple 2-representations of $\mathfrak{sl}_2$
论文作者
论文摘要
令$ \ mathcal u $为与$ \ mathfrak {sl} _2 $相关的2类。我们证明，当$ \ Mathcal U $的1个典范中，当它在每个简单的2个代理中的图像都是无效的，而仅当其图像都是无效的，则仅是无效的。在轻度界定的假设下，我们证明它实际上足以在简单的2个代理中是无环的图像。我们将此结果应用于Rickard Complex $θ$的研究，将$ \ Mathrm {SL} _2 $的简单反射的动作分类。我们证明，$θ$在$ \ u $的同型类别中是可逆的，并且有一个同型等价$θe\ simeqfθ[-1] $。
Let $\mathcal U$ be the 2-category associated with $\mathfrak{sl}_2$. We prove that a complex of 1-morphisms of $\mathcal U$ is null-homotopic if and only if its image in every simple 2-representation is null-homotopic. Under mild boundedness assumptions, we prove that it actually suffices for the image in the simple 2-representations to be acyclic. We apply this result to the study of the Rickard complex $Θ$ categorifying the action of the simple reflection of $\mathrm{SL}_2$. We prove that $Θ$ is invertible in the homotopy category of $\U$, and that there is a homotopy equivalence $ΘE \simeq FΘ[-1]$.
