Algebra and Geometry Seminar

We develop the theory of semisimplificat​ions of tensor categories defined by Barrett and Westbury. By definition, the semisimplificat​ion of a tensor category is its quotient by the tensor ideal of negligible morphisms, i.e., morphisms $f$ such that $Tr(fg)=0$ for any morphism $g$ in the opposite direction. In particular, we compute the semisimplificat​ion of the category of representations of a finite group in characteristic $p$ in terms of representations of the normalizer of its Sylow $p$-subgroup. This allows us to compute the semisimplificat​ion of the representation category of the symmetric group $S_{n+p}$ in characteristic $p$, where $n=0,...,p-1$, and of the abelian envelope of the Deligne category, $Rep^{ab} S_t$. We also compute the semisimplificat​ion of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of $sl_2$. Finally, we study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplificat​ion construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplificat​ion). This is joint work with Victor Ostrik.