A ribbon Hopf algebra ( A , m , Δ , u , ε , S , R , ν ) is a quasitriangular Hopf algebra which possess an invertible central element ν more commonly known as the ribbon element, such that the following conditions hold:
ν 2 = u S ( u ) , S ( ν ) = ν , ε ( ν ) = 1 Δ ( ν ) = ( R 21 R 12 ) − 1 ( ν ⊗ ν ) where u = m ( S ⊗ id ) ( R 21 ) . Note that the element u exists for any quasitriangular Hopf algebra, and u S ( u ) must always be central and satisfies S ( u S ( u ) ) = u S ( u ) , ε ( u S ( u ) ) = 1 , Δ ( u S ( u ) ) = ( R 21 R 12 ) − 2 ( u S ( u ) ⊗ u S ( u ) ) , so that all that is required is that it have a central square root with the above properties.
Here
A is a vector space
m is the multiplication map
m : A ⊗ A → A Δ is the co-product map
Δ : A → A ⊗ A u is the unit operator
u : C → A ε is the co-unit operator
ε : A → C S is the antipode
S : A → A R is a universal R matrix
We assume that the underlying field K is C
