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A non-semisimple Witt class

Victor Ostrik, Alexandra Utiralova

TL;DR

The work constructs and analyzes non-semisimple braided finite tensor categories $ ext{C}( rak g,e,l,q)$ arising from tilting modules for quantum groups at roots of unity, focusing on the subregular nilpotent case $e_{sr}$. It identifies block decompositions shaped by affine Dynkin diagrams and connects the principal block to McKay-type categories $ ext{Rep}( rak ext Gamma times V)$, with cohomology given by $S^ullet(V)^ rak ext Gamma$, yielding tame representation type. The paper computes explicit examples, notably $ ext{C}(G_2,G_2(a_1),7,q)$ and its de-equivariantization, producing a non-semisimple modular category with center $ ext{Rep}(S_3)$ and proving non-Witt-equivalence to semisimple categories. It also develops a framework relating exact commutative algebras, graded structures, and Witt theory, proving the existence of completely anisotropic non-semisimple examples and formulating conjectures for broader families. Overall, the results demonstrate rich non-semisimple Witt theory and modular phenomena beyond semisimple fusion categories, with concrete categorical and cohomological descriptions tied to Lie-theoretic and representation-theoretic data.

Abstract

We describe several infinite families of braided finite tensor categories. A simplest example gives a non-degenerate braided tensor category which is not Witt equivalent to a semisimple category.

A non-semisimple Witt class

TL;DR

The work constructs and analyzes non-semisimple braided finite tensor categories arising from tilting modules for quantum groups at roots of unity, focusing on the subregular nilpotent case . It identifies block decompositions shaped by affine Dynkin diagrams and connects the principal block to McKay-type categories , with cohomology given by , yielding tame representation type. The paper computes explicit examples, notably and its de-equivariantization, producing a non-semisimple modular category with center and proving non-Witt-equivalence to semisimple categories. It also develops a framework relating exact commutative algebras, graded structures, and Witt theory, proving the existence of completely anisotropic non-semisimple examples and formulating conjectures for broader families. Overall, the results demonstrate rich non-semisimple Witt theory and modular phenomena beyond semisimple fusion categories, with concrete categorical and cohomological descriptions tied to Lie-theoretic and representation-theoretic data.

Abstract

We describe several infinite families of braided finite tensor categories. A simplest example gives a non-degenerate braided tensor category which is not Witt equivalent to a semisimple category.
Paper Structure (26 sections, 44 theorems, 98 equations, 9 figures)

This paper contains 26 sections, 44 theorems, 98 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Categories $\mathcal{C}(\mathfrak{g},e,l,q)$
  4. Notations
  5. Tilting modules for quantum groups
  6. Tensor ideals
  7. The subregular cell
  8. Toy example
  9. Tilting modules for subregular cell
  10. Principal block of $\mathcal{C}(\mathfrak{g},e_{sr},l,q)$
  11. Proof of Theorem \ref{['main blocks']}
  12. Category $\mathcal{C}(G_2,G_2(a_1),7,q)$
  13. Proof of Theorem \ref{['main 7']}
  14. De-equivariantization of $\mathcal{C}(G_2, G_2(a_1), 7, q)$
  15. Proof of Theorem \ref{['main 7mod']}
  16. ...and 11 more sections

Key Result

Theorem 1.1

The category $\mathcal{C}(\mathfrak{g},e_{sr},l,q)$ decomposes into blocks which are either trivial (that is equivalent to the category of vector spaces) or of type $\Gamma$. In particular, the category $\mathcal{C}(\mathfrak{g},e_{sr},l,q)$ is of tame representation type. Also the cohomology of $\m

Figures (9)

  • Figure 1: Coxeter diagram for $\tilde{W}_l$ for $\mathfrak g$ of type $B_n$ or $C_n$
  • Figure 2: Coxeter diagram for $\tilde{W}_l$ for $\mathfrak g$ of type $G_2$
  • Figure 3: Graph $X_{sr}$ for types $B_n$ and $C_n$
  • Figure 4: Graph $X_{sr}$ for type $G_2$
  • Figure 5: Graph $X_{sr}$ for type $G_2$ (labeled)
  • ...and 4 more figures

Theorems & Definitions (88)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Corollary 2.2
  • Definition 2.3
  • Example 2.4
  • Remark 2.5
  • ...and 78 more