On half-ribbon categories

Publication status
In Progress
Work type
Article

There is an excellent question on mathoverflow asking whether there is category theory with a half-twist as graphical calculus. It turns out there is. Basically, the half-twist is a square root of the twist in a balanced braided category. The most general treatment I know of is Jeff Egger's article On Involutive Monoidal Categories.

There are two important special cases: Quantum groups,  or, more precisely, deformed universal enveloping algebras of simple complex Lie algebras. Noah Snyder and Peter Tingley have written about this. The other special case are categories in which every object is naturally self-dual, such as real inner product vector spaces (Hilbert spaces don't work!). Peter Selinger has a preprint about it.

The deepest insight about half-twists for me was that it gives raise to a braiding. You may know that the twist of the tensor product of two objects is the twist of the two individual objects and the square of the braiding:

$$\theta_{X \otimes Y} = c_{Y,X}c_{X,Y}(\theta_X \otimes \theta Y)$$

Now the twist has a square root, $\theta_X = \varsigma_X^2$. Luckily, the identity above also has a square root!

$$\varsigma_{X \otimes Y} = c_{X,Y}(\varsigma_X \otimes \varsigma_Y)$$

This means that the half-twist $\varsigma_X$ contains at least the information about the braiding and the twist!

So far the known material. There are a lot of gaps, which I'm attempting to close. More specifically, the following seems to be missing from the literature:

  • The relation between the different works on half-twists.
  • What happens in full generality when we have ribbon structures and not just balanced structures? How do half-twists behave in the presence of dualities?
  • What happens if the category has a dagger structure? For example in categories of modules of Hopf algebras, what if the Hopf algebra is actually a *-algebra? See also my question on mathoverflow.
  • How can Noah Snyder and Peter Tingley's work be generalised to arbitrary Hopf algebras (not necessarily $U_q\mathfrak{g}$)?

And a lot of new things have come up in the process. A few of them are:

  • Something similar to the Drinfel'd centre of a monoidal category. The Drinfel'd centre of a monoidal category $\mathcal{C}$ in a sense puts all possible braidings on it. The result is a braided category with lots of nice properties (e.g. when $\mathcal{C}$ is spherical fusion, then the Drinfel'd centre is modular). It seems there is a way of putting all possible half-twists on an involutive monoidal category.
  • Given a category with a twist and an involutive structure, it is possible create a new category that contains all half-twists that square to the given twist.
  • There is some extra structure which which it is possible to do unoriented ribbon graphs like the Möbius band.
  • When considering fusion categories, there are interesting relations between involutive structures and Frobenius-Schur indicators.
  • All known modular categories of rank < 5 seem to admit very naturalhalf-twists.

Half-twists are very fruitful, and they have spawned two further ongoing pieces of work for me:

  • Noncommutative geometry seems to be related to half-twists (this originates from an idea by John Barrett).
  • There is a common generalisation of involutive monoidal categories and dagger categories.

There are some open questions that I'm still wondering about:

  • The braiding is obviously a higher-categorical structure. Are half-twists higher structures as well?
  • Given an involutive structure, there may be zero, one or many half-twists for it. Given a half-twist, there is exactly one involutive structure. More generally, if you take a natural isomorphism from the identity functor to another functor, then the other functor is completey fixed. Is there something deeper to this observation?
  • How to classify the inequivalent involutive structures on a given fusion category?