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Contragredient Lie algebras in symmetric categories

Iván Angiono, Julia Plavnik, Guillermo Sanmarco

Abstract

We define contragredient Lie algebras in symmetric categories, generalizing the construction of Lie algebras of the form $\mathfrak{g}(A)$ for a Cartan matrix $A$ from the category of vector spaces to an arbitrary symmetric tensor category. The main complication resides in the fact that, in contrast to the classical case, a general symmetric tensor category can admit tori (playing the role of Cartan subalgebras) which are non-abelian and have a sophisticated representation theory. Using this construction, we obtain and describe new examples of Lie algebras in the universal Verlinde category in characteristic $p\geq5$. We also show that some previously known examples can be obtained with our construction.

Contragredient Lie algebras in symmetric categories

Abstract

We define contragredient Lie algebras in symmetric categories, generalizing the construction of Lie algebras of the form for a Cartan matrix from the category of vector spaces to an arbitrary symmetric tensor category. The main complication resides in the fact that, in contrast to the classical case, a general symmetric tensor category can admit tori (playing the role of Cartan subalgebras) which are non-abelian and have a sophisticated representation theory. Using this construction, we obtain and describe new examples of Lie algebras in the universal Verlinde category in characteristic . We also show that some previously known examples can be obtained with our construction.
Paper Structure (35 sections, 31 theorems, 99 equations)

This paper contains 35 sections, 31 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Analogs of Deligne's theorem in positive characteristic
  3. Affine group schemes in pre-Tannakian categories
  4. Contragredient Lie superalgebras
  5. A summary of our construction and main results
  6. Preliminaries
  7. Symmetric tensor categories and Deligne's Theorem
  8. Semisimplification of symmetric tensor categories
  9. Ind-completions of symmetric tensor categories
  10. Vanishing of alternating powers
  11. Moderate growth
  12. The Verlinde category and Ostrik's Theorem
  13. Frobenius exact categories and Coulembier-Etingof-Ostrik's Theorem
  14. Operadic, PBW, and genuine Lie algebras
  15. Free operadic Lie algebras
  16. ...and 20 more sections

Key Result

Theorem 2.1

Del90Del02 Assume $\operatorname{char} \Bbbk=0$ and let $\mathcal{C}$ be a pre-Tannakian category over $\Bbbk$. Then $\mathcal{C}$ admits a symmetric tensor functor $\mathcal{C}\to\mathsf{Vec}_{\Bbbk}$ if and only if for each $X$ in $\mathcal{C}$ there is some $n\ge 1$ such that $\operatorname{A}^n(

Theorems & Definitions (105)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • ...and 95 more