Tensor ideals, Deligne categories and invariant theory

We derive some tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals inDeligne's universal categories RepO(delta), RepGL(delta) and RepP. These results are then used to obtain newinsight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A, B, C, D, P. We also find new short proofs for the classification of tensor ideals in RepSt and in the category of tilting modules for SL2(k) with char(k) > 0 and for U-q (sl2) with q a root of unity. In general, for a simple Lie algebra g of type ADE, we show that the lattice of such tensor ideals for Uq (g) corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac-Moody algebra.