---
tag: K49YFHYCHA
kind: concept
name: Ribbon Hopf algebra
msc: [16T05, 18M15, 57K16]
specializes: [NCRPWS9W5R]
sources:
  - url: https://en.wikipedia.org/wiki/Ribbon_Hopf_algebra
same_as:
  wikidata: Q7322260
  wikipedia: Ribbon Hopf algebra
---

## Summary

A ribbon Hopf algebra is a [[quasitriangular Hopf algebra|NCRPWS9W5R]] with a central invertible element, the ribbon element, that behaves like a full twist.
Its finite-dimensional modules form a ribbon category: a [[braided monoidal category|VT8SFWT0XW]] with duals and compatible twists.
In such a category every framed knot or link diagram coloured by modules gives a well-defined morphism.
So each ribbon Hopf algebra gives invariants of knots and links.
Suitable versions of the [[quantum enveloping algebras|1V8H34F6N9]] are ribbon, and the [[Jones polynomial|4SPC17FPXE]] arises this way from $U_q(\mathfrak{sl}_2)$.

## Explanations

### L1 human

Think of a ribbon, a flat band, rather than a thin string.
A ribbon can be twisted by a full turn, and that twist cannot be undone by sliding the band around.
In a ribbon Hopf algebra a special element $v$ records this full twist.
The braiding from the R-matrix records how two ribbons cross, and $v$ records how one ribbon twists.
Together they give exactly the bookkeeping needed to turn pictures of tangled bands into algebra.