### Question about Tensor Categories

#### Posted by Urs Schreiber

*Hendryk Pfeiffer asked me to forward the following question to the Café. *

Dear $n$-category people,

I have a question about tensor categories on which I would appreciate comments and references. As probably several people are interested in this, I decided to ask this question here.

The short version of my question is:

Are there examples of $k$-linear additive spherical categories that are non-degenerate, but not semisimple?

In more detail:

I am interested in $k$-linear additive spherical categories where $k$ is a field. The notion of a spherical category was defined in

[1] J. W. Barrett, B. W. Westbury, Spherical categories, Adv Math 143 (1999) 357, hep-th/9310164

A spherical category is a pivotal category in which left- and right-traces agree. A pivotal category is, roughly speaking, a monoidal category in which each object X has a specified dual $X^*$ and in which $X^{**}$ is naturally isomorphic to $X$. The details are in

[2] P. Freyd, D. N. Yetter, Coherence theorems via knot theory, J Pure Appl Alg 78 (1992) 49

In a $k$-linear spherical category, the trace defines bilinear maps

$\begin{aligned} \mathrm{tr}: Hom(X,Y) \otimes Hom(Y,X) &\to k \\ f \otimes g &\mapsto \mathrm{tr}_Y (f\circ g) \end{aligned}$

A $k$-linear spherical category is called non-degenerate if all these traces are non-degenerate, i.e. if for any $f:X \to Y$ the following holds:

$\mathrm{tr}_Y (f\circ g)=0 \;\mathrm{for}\; \mathrm{all}\; g:Y \to X \; \mathrm{implies} f=0 \,.$

As usual, an object $X$ is called simple if $\mathrm{Hom}(X,X)=k$, and the category is called [finite] semisimple if every object is isomorphic to a finite direct sum of simple objects [and if the set of isomorphism classes of simple objects is finite].

The following implication is known to hold:

If a $k$-linear additive spherical category is finite semisimple, then it is non-degenerate.

In

[3] J. W. Barrett, B. W. Westbury, Invariants of piecewise-linear 3-manifolds, Trans AMS 348 (1996) 3997, hep-th/9311155

the property of being non-degenerate is part of the definition of semisimple. With the more standard definitions I used above, however, one has to prove this. The proof is completely analogous to the one for ribbon categories in Section II.4.2 of

[4] V. G. Turaev, Quantum invariant of knots and 3-manifolds, de Gruyter, 1994.

In order to understand why the converse implication fails, I am interested in learning about examples of k-linear additive spherical categories that are

(1) non-degenerate and not finite semisimple

(2) non-degenerate and not semisimple

(3) non-degenerate, not semisimple, and which are of the form A-mod for some finite-dimensional k-algebra A

I should be grateful for any sort of comments.

Hendryk Pfeiffer

## Re: Question about Tensor Categories

I feel like (3) cannot be possible. If you look at the endomorphism induced by an element of the Jacobson radical on any representation, there’s no way that could have non-zero trace, since it induces the 0 map on the associated graded by any filtration with simple quotients.

More generally, this should imply that if your category is non-degenerate, then the endomorphism rings of all objects with a finite composition series must have semi-simple endomorphism rings. I think this should imply that all objects of finite length must be semi-simple (if you have an object of finite length $M$ that isn’t semi-simple, then there’s a non-split injection of a simple object $N\to M$. Consider the endomorphism algebra $\mathrm{End}(M\oplus N)$. The inclusion of $N$ into $M$ considered as an element of this algebra should be in the Jacobson radical. Q.E.D.).

Of course, if you’re willing to leave Artinian categories, you might get some more interesting stuff.