Tensor ideals, Deligne categories and invariant theory

Abstract

We derive some tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne’s universal categories $${\underline{\mathrm{Rep}}} O_\delta ,{\underline{\mathrm{Rep}}} GL_\delta $$ and $${\underline{\mathrm{Rep}}} P$$ . These results are then used to obtain new insight 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 $${\underline{\mathrm{Rep}}} S_t$$ and in the category of tilting modules for $${\text {SL}}_2(\Bbbk )$$ with $$\mathrm{char}(\Bbbk )>0$$ and for $$U_q(\mathfrak {sl}_2)$$ with q a root of unity. In general, for a simple Lie algebra $$\mathfrak {g}$$ of type ADE, we show that the lattice of such tensor ideals for $$U_q(\mathfrak {g})$$ corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac–Moody algebra.