[Paper Review] Finite symmetric integral tensor categories with the Chevalley property

Pavel Etingof, Shlomo Gelaki|arXiv (Cornell University)|Jan 2, 2019

Algebraic structures and combinatorial models26 references4 citations

TL;DR

This paper proves that every finite symmetric integral tensor category with the Chevalley property over an algebraically closed field of characteristic p > 2 admits a symmetric fiber functor to supervector spaces, establishing a complete classification via finite supergroup schemes and grouplike elements of order ≤2. The key result confirms Ostrik's conjecture in this setting and extends to classifications of triangular quasi-Hopf algebras and Sweedler cohomology of restricted enveloping algebras.

ABSTRACT

We prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber functor to $ ext{sVec}$. This proves Ostrik's conjecture \cite[Conjecture 1.3]{o} in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $\mathcal{G}$ over $k$ and a grouplike element $ε\in k[\mathcal{G}]$ of order $\le 2$, whose action by conjugation on $\mathcal{G}$ coincides with the parity automorphism of $\mathcal{G}$, such that $\mathcal{C}$ is symmetric tensor equivalent to $\Rep(\mathcal{G},ε)$. In particular, when $\mathcal{C}$ is unipotent, the functor lands in $\Vect$, so $\mathcal{C}$ is symmetric tensor equivalent to $\Rep(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of \cite{g} to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p>0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p>0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper \cite{Co}, and, more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $ e 2$ is always a Serre subcategory.

Motivation & Objective

• To classify finite symmetric integral tensor categories with the Chevalley property over algebraically closed fields of characteristic p > 2.
• To prove Ostrik's conjecture that such categories admit a symmetric fiber functor to supervector spaces.
• To classify finite-dimensional triangular quasi-Hopf algebras with the Chevalley property using group scheme-theoretic methods.
• To compute Sweedler cohomology of restricted enveloping algebras and classify associators for their duals.
• To study finite-dimensional local quasi-Hopf algebras and unipotent tensor categories in positive characteristic.

Proposed method

• Uses the theory of finite (super)group schemes and symmetric fiber functors to realize tensor categories as representation categories.
• Applies results from [O, Theorem 1.1] and [EOV, Theorem 8.1] to reduce the classification to supergroup schemes with specific automorphisms.
• Employs the notion of pseudotwist equivalence to classify triangular quasi-Hopf algebras up to equivalence.
• Utilizes cohomological techniques, including Sweedler cohomology, to classify associators for duals of restricted enveloping algebras.
• Applies Tannakian and super-Tannakian duality, showing that maximal Tannakian and super-Tannakian subcategories are Serre subcategories.
• Leverages functorial cohomology computations for coordinate algebras of finite connected group schemes, generalizing results from [FN].

Experimental results

Research questions

• RQ1 Does every finite symmetric integral tensor category with the Chevalley property over a field of characteristic p > 2 admit a symmetric fiber functor to supervector spaces?
• RQ2 Can such categories be classified via finite supergroup schemes and grouplike elements of order ≤2?
• RQ3 What is the structure of finite-dimensional triangular quasi-Hopf algebras with the Chevalley property in positive characteristic?
• RQ4 How can Sweedler cohomology be computed for restricted enveloping algebras in characteristic p > 0?
• RQ5 Are the maximal Tannakian and super-Tannakian subcategories of symmetric tensor categories always Serre subcategories in characteristic ≠ 2?

Key findings

• Every finite symmetric integral tensor category with the Chevalley property over an algebraically closed field of characteristic p > 2 admits a symmetric fiber functor to the category of supervector spaces.
• Such categories are symmetric tensor equivalent to Rep(G, ǫ) for a unique finite supergroup scheme G and grouplike element ǫ ∈ kG of order ≤2 whose conjugation action coincides with the parity automorphism of G.
• When the category is unipotent, the fiber functor lands in Vec, so the category is symmetric tensor equivalent to Rep(U) for a unique finite unipotent group scheme U.
• The maximal Tannakian and super-Tannakian subcategories of a symmetric tensor category over a field of characteristic ≠ 2 are always Serre subcategories.
• The Sweedler cohomology of restricted enveloping algebras over algebraically closed fields of characteristic p > 0 is computed, and associators for their duals are fully classified.
• Finite-dimensional local quasi-Hopf algebras and unipotent tensor categories in characteristic p > 0 are fully classified via group scheme-theoretic means.