Tensor ideals of abelian type and quantum groups
Kevin Coulembier, Pavel Etingof, Victor Ostrik
TL;DR
The paper develops a comprehensive framework for tensor ideals of abelian type in k-linear rigid monoidal categories, focusing on kernels of monoidal functors to abelian tensor categories and the ensuing notions of primeness and envelopes. It provides general principles to identify and classify such ideals by connecting them to thick tensor ideals and support theory, with pivotal results under naturality conjectures. The authors apply these ideas to tilting modules for quantum groups, establishing a prime-thick-ideal correspondence with nilpotent orbits and demonstrating abelian envelopes in key cases (and conditional classifications under naturality). Rank-two quantum group examples (A2, B2, G2) illustrate the spectrum from finite to infinite lattice structures of tensor ideals, while sections on finite groups, Duflo involutions, and the oriented Brauer category highlight the broad applicability and implications for modular representation theory and categorification. Overall, the work advances a unifying approach to tensor ideals that blends geometric, combinatorial, and categorical methods to connect representation theory with algebraic and geometric invariants.
Abstract
We initiate a study of tensor ideals in linear rigid monoidal categories that are kernels of linear monoidal functors to abelian monoidal categories. We develop general methods and apply them to the category of tilting modules over quantum groups as well as to some representation categories of finite groups. In an appendix on Duflo involutions in monoidal categories, we make a connection between Duflo involutions in the affine Weyl group and tensor ideals for quantum groups, and prove some of Lusztig's conjectures for arbitrary Coxeter groups, at equal parameters, without invoking the boundedness hypothesis.