# What kind of topological invariants can you get from just Hopf algebras?

In A minus sign that used to annoy me but now I know why it is there, Peter Tingley shows how to build knot invariants from the representations of the $$U_q(\mathfrak{sl}_2)$$ quantum group by comparing it to a certain algebra of ribbons. This is to say the representation theory of this quantum group is "tangled" in some way.

It is known that symmetric functions (characters of the symmetric group $$S_n$$) form a Hopf algebra. Is there a corresponding topological structure?

I'm just wondering if there's a diagrammatic way to look at representations of $$S_n$$. In general, does Hopf algebra structure of an algebra, imply there exists a diagrammatic way of looking at its representation theory?

More examples of Hopf algebras in combinatorics.

• Can you be more specific about what king of topological structure you're hoping for? Like are you looking for a space whose cohomology is this Hopf algebra, or something like that? Nov 11, 2011 at 23:11
• If I only knew what I was looking for... from a quantum group you can get a knot invaraint b/c there is a functor from representations of quantum groups to a tangle category. I am guessing that for $S_n$ you lose the crossings or something like that. I'm just wondering if there's a diagrammatic way to look at representations of $S_n$. In general, does Hopf algebra structure of a ring, imply there exists a diagrammatic way of looking at its representation theory? Nov 12, 2011 at 17:42
• You should add the adjective "quasi-triangular" in front of "Hopf algebra." This condition exactly ensures that the category of representations is braided monoidal. Nov 12, 2011 at 17:48
• @John: the functor goes the other direction (from a tangle category to the category of representations). Nov 12, 2011 at 21:25
• I mean it could be argued that what you are looking for is Dijkgraaf Witten theory math.ucr.edu/home/baez/qg-winter2005/w05week06.pdf for a finite group. It is a 2d TQFT which tells a lot about the representation theory of G. For 3d you can get look at Chern-Simons theory for finite groups... Nov 12, 2011 at 23:20

The category of finite-dimensional representations of a Hopf algebra has the structure of a monoidal category with duals. The comultiplication induces a monoidal structure, the tensor product of vector spaces. The counit induces a monoidal identity, the ground field. And the antipode induces a duality structure, the vector space dual. Thus, diagrammatically, you can think of intertwiners of representations of a Hopf algebra as being oriented curves with boundary embedded in a strip $\mathbb{R}\times\left[0,1\right]$. The boundary points would be colored by representations, with induced orientations $+$ or $-$ indicating whether we are considering the representation $V$ or its dual $V^\star$. Intertwiners would color the component curves with their orientation determining their source and target endpoints. A quasi-triangular Hopf algebra has an element $R\in H\otimes H$ which induces a braiding on its category of representations. A ribbon Hopf algebra is quasi-triangular and has an element $\theta\in H$ which induces a twist, thus making its category of representations a ribbon category, allowing one to notate intertwiners diagrammatically as colored, framed, oriented tangles.