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Selecta Mathematica

, Volume 24, Issue 5, pp 4659–4710 | Cite as

Tensor ideals, Deligne categories and invariant theory

  • Kevin Coulembier
Article
  • 16 Downloads

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 ABCDP. 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.

Keywords

Monoidal (super)category Tensor ideal Thick tensor ideal Deligne category Algebraic (super)group Second fundamental theorem of invariant theory Tilting modules Quantum groups 

Mathematics Subject Classification

18D10 17B45 17B10 15A72 

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Notes

Acknowledgements

The research was supported by ARC grant DE170100623. The author thanks Jon Brundan, Michael Ehrig, Inna Entova, Jim Humphreys, Gus Lehrer, Andrew Mathas, Geordie Williamson and Yang Zhang for interesting discussions and the referee for very useful remarks.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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