Ribbon Hopf algebra

A ribbon Hopf algebra

is a quasitriangular Hopf algebra which possess an invertible central element

more commonly known as the ribbon element, such that the following conditions hold: where

Note that the element u exists for any quasitriangular Hopf algebra, and

must always be central and satisfies

, so that all that is required is that it have a central square root with the above properties.

Here We assume that the underlying field

is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if

is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.