arXiv 1801.044092018-01math.RT

关于张量范畴的半化简

On semisimplification of tensor categories

Pavel Etingof, Victor Ostrik

备注：: 32 pages, latex; in v2 minor changes have been made, and the end of Section 7 as well as Section 8 are new; in v3 the nonsymmetric version of Andre-Kahn results on p.11-12 has been added

摘要：

我们发展了Barrett和Westbury定义的张量范畴的半单化理论。特别是，我们计算半简化的表示的有限群的类别的特征$p$在其Sylow $p$ -子群的正规化表示。这使我们能够计算半简化的表示类别的对称群$S_{n+p}$在特征$p$，其中$0\le n\le p-1$，和德利涅类别$\underline{\rm Rep}^{\rm ab}S_t$，其中$t\in \Bbb N$。我们还计算了$\mathfrak{sl}_2$的Borel子代数的Kac-De Concini量子群的表示范畴的半简化。我们还研究了半单代数群的Verlinde范畴与有限群的模表示范畴中的有限型对象之间的张量函子（即，在半简化中生成融合类别的对象）。最后，我们确定了特征$2$中$GL(n)$，$SL(n)$和$PGL(n)$的倾斜范畴的半单化.在附录中，我们对$SO(3)$的表示的Grothendieck环及其截断进行了分类。

英文摘要：

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic $p$ in terms of representations of the normnalizer of its Sylow $p$-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group $S_{n+p}$ in characteristic $p$, where $0\le n\le p-1$, and of the Deligne category $\underline{\rm Rep}^{\rm ab}S_t$, where $t\in \Bbb N$. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of $\mathfrak{sl}_2$. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of $GL(n)$, $SL(n)$ and $PGL(n)$ in characteristic $2$. In the appendix, we classify categorifications of the Grothendieck ring of representations of $SO(3)$ and its truncations.

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摘要：

英文摘要：

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