[Paper Review] Tensor categories (after P. Deligne)

Victor Ostrik|ArXiv.org|Jan 25, 2004

Algebraic structures and combinatorial models1 references3 citations

TL;DR

This paper provides an exposition of P. Deligne's theorem on tensor categories over an algebraically closed field of characteristic zero, proving that any finitely generated, rigid, symmetric monoidal abelian category arises as the representation category of an affine supergroup scheme. The proof constructs a fiber functor via Schur functors and algebraic geometry techniques, establishing a supergroup-theoretic classification of such categories.

ABSTRACT

These notes give an exposition of Deligne's theorem on the existense of super fiber functor.

Motivation & Objective

• To provide a self-contained, accessible exposition of Deligne's classification theorem for tensor categories, tailored for representation theorists.
• To establish that any finitely generated, rigid, symmetric monoidal abelian category over an algebraically closed field of characteristic zero is equivalent to the category of representations of an affine supergroup scheme.
• To demonstrate the existence of a fiber functor (specifically, a super fiber functor) for such categories using algebraic and categorical techniques.
• To clarify the role of Schur functors in detecting the superdimension and ensuring the existence of a fiber functor over a suitable algebraic base.
• To bridge the gap between abstract tensor categories and concrete representation-theoretic realizations via supergroup schemes.

Proposed method

• Use of Schur functors $ S_\lambda(X) $ to analyze the structure of tensor powers $ X^{\otimes n} $, leveraging the action of the symmetric group $ S_n $.
• Construction of a universal algebra $ R $ via inductive limits of tensor products of algebras to ensure exactness and splitting of short exact sequences after base change.
• Definition of a functor $ \omega: \mathcal{A} \to R\text{-mod} $ via $ \omega(X) = \rho(X \otimes A) $, where $ \rho $ extracts Hom spaces to the unit and its superpartner.
• Leveraging Hilbert's Nullstellensatz to lift solutions from a finitely generated $ R $-algebra to the base field $ k $, ensuring existence of a fiber functor over $ k $.
• Reduction to the semisimple case with finitely many simple objects to simplify the construction of tensor structure on the fiber functor.
• Use of Littlewood-Richardson coefficients to describe decomposition rules for Schur functors on direct sums, enabling control over superdimension.

Experimental results

Research questions

• RQ1 Under what conditions does a tensor category over a field of characteristic zero admit a fiber functor to super vector spaces?
• RQ2 How can Schur functors be used to detect the superdimension of objects in a tensor category?
• RQ3 What algebraic structure underlies the existence of a fiber functor for a finitely generated tensor category?
• RQ4 Can a fiber functor over a finitely generated algebra be descended to a fiber functor over the base field?
• RQ5 How does the existence of a fiber functor relate to the representation theory of affine supergroup schemes?

Key findings

• Any finitely generated, rigid, symmetric monoidal abelian category over an algebraically closed field of characteristic zero is equivalent to the category of finite-dimensional representations of an affine supergroup scheme.
• The existence of a fiber functor over a suitable supercommutative algebra $ R $ is guaranteed by the vanishing of certain Schur functors on all objects.
• After base change to a universal algebra $ A $, all short exact sequences in the category split, and all objects become direct sums of the unit and its superpartner, ensuring well-defined superdimension.
• The functor $ \omega(X) = \rho(X \otimes A) $ defines an exact, tensor functor to $ R\text{-mod} $, where $ R = \rho(A) $, with $ R $ a supercommutative $ k $-algebra.
• Using Hilbert's Nullstellensatz, a solution over $ R $ lifts to a solution over the base field $ k $, yielding a super fiber functor.
• In the semisimple case with finitely many simple objects, the tensor structure on the fiber functor exists over $ k $, as the associated system of equations has a solution in a finitely generated $ k $-algebra.