Representation theory in braided monoidal categories

The crux of what I wish to know is what results from representation theory, a subject usually framed within the category $$\text{Vect}_\mathbb{k}$$, follow in more general braided monoidal categories? I would be very satisfied with references to texts that cover this.

I'll try to be more specific. Let $$\mathcal{C}$$ be monoidal, abelian, complete under arbitrary countable biproducts, and enriched over $$\text{Vect}_\mathbb{C}$$. An algebra in $$\mathcal{C}$$ is an object $$A$$ with morphisms $$m:A\otimes A\rightarrow A, u:1\rightarrow A$$ satisfying suitable axioms, and a left $$A$$-module in $$\mathcal{C}$$ is an object $$V$$ with morphism $$a_V:A\otimes V\rightarrow V$$, again satifying some conditions. If $$\mathcal{C}$$ has a braiding $$\psi$$, we can define commutative algebras as those algebras such that $$m\psi=m$$.

For instance take $$\mathcal{C}$$ to be $$H\text{-Mod}$$, the category of (finite-dimensional) modules over quasitriangular Hopf algebra $$H$$ (the quasitriangular structure on $$H$$ makes $$H\text{-Mod}$$ braided monoidal). Recall the following classical (i.e. in $$\text{Vect}_\mathbb{C}$$) result from representation theory: for commutative algebra $$A$$, every simple finite-dim'l $$A$$-module is $$1$$-dim'l. Is there an analogue of this statement for braided commutative algebras in $$H\text{-Mod}$$?

• Probably you are aware of this textbook?: www-math.mit.edu/~etingof/egnobookfinal.pdf Aug 8 '20 at 14:47
• That's a great book, but I don't think it tackles my question, unless I missed something (quite probable)? Aug 8 '20 at 14:58
• Yes, you're probably right, I don't know the specifics, but in general it addresses "studying representation theory in symmetric/braided monoidal categories" Aug 8 '20 at 15:01
• If we consder the special case of $H=\mathbb{CZ}_2$, i.e. the group hopf algebra with its non-trivial quasitriangular structure then we get the super-commutative superalgebras (that is $\mathbb{Z}_2$-graded-commutative assoc algebras). I think we can get an analogous statement but the reps have to have dim $\leq 2$. Aug 9 '20 at 1:25
• I'm confused by "arbitrary countable biproducts". I've seen "biproduct" used to mean "simultaneously a product and coproduct", like what you get with finite products and coproducts in any abelian category. But even in $\text{Vect}_k$, countable products and coproducts are generally different. So what are countable biproducts here? Aug 16 '20 at 2:34

When the underlying category is a quantum group at a root of unity, understanding commutative algebras has been studied extensively by Ocneanu and others under the name “quantum subgroups.” For an explanation in the algebraic language you’ve phrased this question in for the special case of quantum SU(2), see Ostrik-Kirillov. There’s a very nice ADE classification there, which was originally studied in subfactor language by Jones, Ocneanu, etc.

As for your specific question, that basically never happens outside of the case of Vec. Just think about the case where the algebra is trivial, then the category of modules will again be the original category.

I will try to provide an answer for a particular case of your last question: Let us consider (following my comment above) the case of $$H=\mathbb{CZ}_2$$ i.e. the group hopf algebra equipped with its non-trivial quasitriangular structure, that is the $$R$$-matrix $$R=\frac{1}{2}(1\otimes 1+1\otimes g+g\otimes 1-g\otimes g)$$.
Its category of representations $$\mathbb{CZ}_2\text{-Mod}$$, consists of the $$\mathbb{Z}_2$$-graded vector spaces, commonly refered to as super-vector spaces, with morphisms the even linear maps and an algebra $$A$$ in the category $$\mathbb{CZ}_2\text{-Mod}$$ is a $$\mathbb{Z}_2$$-graded associative algebra (or: associative superalgebra). The braiding of the category is provided by the $$R$$-matrix.

Consider a f.d. super-module $${}_{A}M$$ and the super version of the interwiner, that is an homogeneous linear map $$g\in \mathcal{E}nd_\beta(M)$$,of degree $$\beta(=0,1)$$ which super-commutes with the $$A$$-action on $$M$$, that is: $$g(a\cdot m)=(-1)^{\beta\gamma}a\cdot g(m)$$ for all $$a\in A_\gamma$$, $$\gamma=0,1$$, $$m\in M$$. Notice that if $$\beta=0$$ then $$g$$ is a usual super module morphism (that is an even super module homomorphism) while if $$\beta=1$$ then we get $$\mathbb{C}$$-linear, odd maps which are antilinear in the $$A$$-action. The set of all super-interwiners forms a superalgebra $$\mathcal{E}nd(M)=\mathcal{E}nd_0(M)\oplus\mathcal{E}nd_1(M)$$

In case $${}_{A}M$$ is simple (in the super sense, i.e. it contains no proper $$\mathbb{Z}_2$$-graded submodules), we also have a super-version ($$\mathbb{Z}_2$$-graded version) of Schur's lemma:

$$\mathcal{E}nd_0(M)=\mathbb{C}\cdot Id$$ and $$\mathcal{E}nd_1(M)=\mathbb{C}\cdot\theta$$, where $$\theta=0$$ or $$\theta$$ is an odd linear map with $$\theta^2=Id$$

Considering the usual definitions for the super-center $$Z_s(A)$$, the super-commutative superalgebra $$A$$ and the super version of Schur's lemma, we get that:

If $$A$$ is a superalgebra, $${}_{A}M$$ is a f.d. simple super-module and $$z_\xi$$ an homogeneous element in the supercenter $$Z_s(A)$$ then there exists some $$\lambda=\lambda(z_\xi)\in\mathbb{C}$$ such that either $$z_\xi\cdot m=\lambda m$$ (if $$\xi=0$$ i.e. $$z$$ is even) or $$z_\xi\cdot m=\lambda\theta(m)$$ (if $$\xi=1$$ i.e. if $$z$$ is odd), for all $$m\in M$$.

Finally, using the last lemma, we can get that:

Any f.d., simple super-module $${}_{A}M$$ over the super-commutative superalgebra $$A$$ is at most $$2$$-dimensional.

P.S.: If you want similar results for more general braided monoidal categories (let say over the category of modules of some more general quasitriangular hopf algebra -even for some other group hopf algebra), i think we will need a braided generalization of Schur's lemma.
I am not aware if something like that exists in the literature (and i would be very interested to know if somebody else knows of such results).