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

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:

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)].

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 \[ \text{length} (X^{\otimes n})\leq N^n\text{ for any integer }n\geq 0. \]

- (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 \(\mathcal{R}ep (G,\varepsilon)\).

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)