Hopf algebra related to monoidal category

Question

Recently, I heard that

Braided rigid monoidal category corresponds to a quasi-triangular hopf algebra.

I know in braided condition gives hexagonal equations and monoidal category gives pentagon/triangular equation by construction.

In terms of Hopf algebra, quasi-triangular structure gives $R$ matrix which satisfies quantum Yang-Baxter equation and from its structure it definitely related with braiding in braided monoidal category and seemingly related to Hexagonal equation. [This is also well described in https://math.stackexchange.com/questions/1223528/motivation-behind-quasitriangular-hopf-algebra]

Then how about pentagonal equations?

Navively, I can easily think of that triangular equations in monodial categories are related to algebra/co-algebra structure of Hopf algebra.

But I am not sure about the analogy of pentagon equation in monoidal category in Hopf algebra. It seems antipode maps some how related but I am not sure how one identify them.

Can you give me some relation (in terms of construction equations) between monoidal category and Hopf algebra?

Answer 1

As you correctly state the $R$-matrix is associated with the braiding and the QYB-equation to the hexagon.

In general, if $H$ is a bialgebra then $\operatorname{Rep}(H)$ is monoidal, with the tensor product representation given by the coproduct $h. (u\otimes w ) := \Delta(h).(v\otimes w)$.

In fact it's almost strict in the sense that the associators are given by the canonical isomorphisms $(U\otimes V) \otimes W \cong U\otimes (V\otimes W)$ of the underlying vector spaces. In particular, the pentagon (and triangle) equations are not so interesting in the case of a bialgebra. This will be generalised by quasi-bialgebras (see later).

If $H$ has an antipode $S$ (i.e. it's a Hopf algebra), then $\operatorname{Rep}(H)$ (fin. dim. reps) becomes a rigid monoidal category as linear duals $V^*$ carry an $H$-action from the antipode, i.e. $h.\varphi(-):= \varphi(S(h).-)$. In particular, (to your question) antipodes are not related to the pentagon axioms but rather to the rigidity.

The $R$-matrix gives the braided structure (which I won't repeat).

Now, to get more general (braided tensor) categories one can introduce quasi-(triangular/Hopf) bialgebras. For instance a quasi-bialgebra H is an algebra with comultiplication $\Delta$, counit $\epsilon$ and an associator $\Phi \in H^{\otimes 3}$ satisfying some modified bialgebra conditions (where $\Phi$ appears), see Prop. 5.12.4 EGNO. The category $\operatorname{Rep}(H)$ is monoidal but associators are given by $$\alpha_{U,V,W}(u\otimes v \otimes w) := \Phi.(u\otimes v \otimes w).$$

For instance, the pentagon equation amounts to the quasi-bialgebra relation: \begin{equation} (\Phi\otimes 1) (\mathrm{id}\otimes \Delta \otimes \mathrm{id})(\Phi) (1 \otimes \Phi) = (\mathrm{id}\otimes \mathrm{id}\otimes \Delta)(\Phi) (\Delta\otimes \mathrm{id} \otimes \mathrm{id})(\Phi)~. \end{equation}

Similarly, the notion generalizes (triangular) Hopf algebras to (quasi)-triangular Hopf algebra.

Remark: A nice motivation for all of the above are usually (quasi)-fiber functors $\mathcal{C} \rightarrow \operatorname{Vect}$ and Tannakian reconstructions (also contained in EGNO).