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
