Tensor ideals, Deligne categories and invariant theory

TL;DR

This paper develops tools for classifying tensor ideals in monoidal categories and applies them to various categories, providing new insights into invariant theory and simplifying existing classifications.

Contribution

It introduces methods for classifying tensor ideals in monoidal categories and applies these to Deligne categories, supergroups, and quantum groups, offering new proofs and theoretical insights.

Findings

Classified tensor ideals in Deligne's categories RepO, RepGL, RepP

Provided new understanding of the second fundamental theorem of invariant theory for algebraic supergroups

Simplified classification proofs for tensor ideals in RepS and tilting modules for SL2 and quantum groups

Abstract

We derive several tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne's universal categories RepO, RepGL and RepP. 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 short proofs for the classification of tensor ideals in RepS and in the category of tilting modules for SL2(k) with char(k)>0 and for Uq(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.