Deligne Categories and the Limit of Categories Rep(GL(m|n))
Deligne Categories and the Limit of Categories Rep(GL(m|n))
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DOI：
10.1093/imrn/rny144
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发表时间：
2015-11
影响因子：
1
通讯作者：
I. Entova-Aizenbud;V. Hinich;V. Serganova
中科院分区：
文献类型：
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作者：
I. Entova-Aizenbud;V. Hinich;V. Serganova
对于每个整数 $t$，构造一个张量类别 $\mathcal{V}_t$，使得精确张量函子 $\mathcal{V}_t\rightarrow \mathcal{C}$ 对 $\mathcal{C}$ 中的可对偶 $t$ 维对象进行分类，而不会被任何 Schur 函子湮没。这意味着 $\mathcal{V}_t$ 是 Deligne 范畴 $\mathcal{D}_t=\operatorname{Rep}(GL_t)$ 的“阿贝尔包络”。任何张量函子 $\operatorname{Rep}(GL_t)\longrightarrow \mathcal{C}$ 都被证明可以通过 $\mathcal{V}_t$ 或通过经典类别 $\operatorname{Rep}(GL(m|n))$ （其中 $m-n=t$）进行因式分解。 $\mathcal{V}_t$ 的普适性意味着它等价于 Deligne 建议的类别 $\operatorname{Rep}_{\mathcal{D}_{t_1}\otimes \mathcal{D}_{t_2}}(GL(X),\epsilon )$, ($t=t_1+t_2$, $t_1$ 不是整数)作为阿贝尔包络作用的候选者。
For each integer $t$ a tensor category $\mathcal{V}_t$ is constructed, such that exact tensor functors $\mathcal{V}_t\rightarrow \mathcal{C}$ classify dualizable $t$-dimensional objects in $\mathcal{C}$ not annihilated by any Schur functor. This means that $\mathcal{V}_t$ is the “abelian envelope” of the Deligne category $\mathcal{D}_t=\operatorname{Rep}(GL_t)$. Any tensor functor $\operatorname{Rep}(GL_t)\longrightarrow \mathcal{C}$ is proved to factor either through $\mathcal{V}_t$ or through one of the classical categories $\operatorname{Rep}(GL(m|n))$ with $m-n=t$. The universal property of $\mathcal{V}_t$ implies that it is equivalent to the categories $\operatorname{Rep}_{\mathcal{D}_{t_1}\otimes \mathcal{D}_{t_2}}(GL(X),\epsilon )$, ($t=t_1+t_2$, $t_1$ not an integer) suggested by Deligne as candidates for the role of abelian envelope.