Catégories tensorielles. (Tensor categories). (French) Zbl 1005.18009

Deligne, P.

Mosc. Math. J. 2, No. 2, 227-248 (2002).

Let \(k\) be an algebraically closed field of characteristic zero. A \(k\)-tensor category \({\mathcal A}\) is an essentially small, \(k\)-linear abelian category with an associative tensor product, a unit object 1 and some additional properties such as rigidity (existence of dual objects) and \(\text{End(\textbf 1})\cong k\).
Such a \(k\)-tensor category is said to be “of finite \(\otimes\)-generation” if there is an object \(X\) from which any object can be obtained by an iteration process of forming direct sums, tensor products, duals, sub-objects, or quotient objects. For example, the category \(\mathcal{R}ep(G)\) of finite-dimensional linear representations of an affine group scheme \(G\) over \(k\) is a \(k\)-tensor category, which is of finite \(\otimes\)-generation if and only if \(G\) is a linear algebraic group.
In the paper under review, the author proves a super-mathematics analogue to the characterization of categories of representations of affine group schemes. More precisely, he turns to the case of a super affine group scheme \(G\) over \(k\) and considers the \(k\)-tensor category \(\mathcal{R}ep(G,\varepsilon)\) of finite-dimensional super-representations \((V, \rho)\) of \(G\) where \(\varepsilon\in G(k)\) is a distinguished element of order at most 2 and \(\rho(\varepsilon)\) is the parity automorphism of \(V\).
\(\mathcal{R}ep(G,\varepsilon)\) is a \(k\)-tensor category of finite \(\otimes\)-generation if and only if \(G\) is of finite type over \(k\), and the author’s (super) characterization theorem states the following:
Let \({\mathcal A}\) be a \(k\)-tensor category of finite \(\otimes\)-generation.
Then \({\mathcal A}\) is of the form \(\mathcal{R}ep(G,\varepsilon)\) if and only if any object \(X\) of \({\mathcal A}\) satisfies the following conditions:

(a)

There is a Schur functor that annihilates \(X\).

(b)

The tensor powers of \(X\) are of finite length and there is a constant \(N\) such that \[ \text{length} (X^{\otimes n})\leq N^n\text{ for any integer }n\geq 0. \]

This main theorem implies two corollaries:

(A)

Let \({\mathcal A}\) be a \(k\)-tensor category of finite \(\otimes\)-generation, in which every object is of finite length. If there is only a finite number of isomorphism classes of simple objects in \({\mathcal A}\), then \({\mathcal A}\) is of the form \(\mathcal{R}ep (G,\varepsilon)\).

(B)

If, on the other hand, \({\mathcal A}\) is semisimple, then there is a finite group \(G\) and an element \(\varepsilon\in G\) of order at most 2 such that \({\mathcal A}\) is \(\otimes\)-equivalent to the category \(\mathcal{R}ep (G,\varepsilon)\).

As for the idea and method of proof, the author introduces and analyzes the so-called “super fibre functors” from a \(k\)-tensor category \({\mathcal A}\) to a tensor category of \(R\)-super modules with respect to a commutative super \(k\)-algebra \(R\). It turns out that the existence of such fibre functors is related to the conditions of the author’s main theorem, and that earlier results of the author himself [P. Deligne, “Catégories tannakiennes”, in: The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 111-195 (1990; Zbl 0729.14010)] give the link between the existence of such functors and categorical equivalences to categories of representations.
An analogue of the (super) characterization theorem proved in this paper under review has been obtained by P. Etingof and S. Gelaki for finite-dimensional semisimple Hopf algebras and their categories of modules [Math. Res. Lett. 5, No. 1-2, 191-197 (1998; Zbl 0907.16016)].

Reviewer: Werner Kleinert (Berlin)

MSC:

18D99	Categorical structures
20C99	Representation theory of groups

Keywords:

tensor categories; super group schemes; Tannaka duality; fibre functor; super algebras; super fibre functors; super groups; Schur functor; categories of representations

Citations:

Zbl 0727.14010; Zbl 0907.16016; Zbl 0729.14010