- Source: Ribbon Hopf algebra
A ribbon Hopf algebra
(
A
,
∇
,
η
,
Δ
,
ε
,
S
,
R
,
ν
)
{\displaystyle (A,\nabla ,\eta ,\Delta ,\varepsilon ,S,{\mathcal {R}},\nu )}
is a quasitriangular Hopf algebra which possess an invertible central element
ν
{\displaystyle \nu }
more commonly known as the ribbon element, such that the following conditions hold:
ν
2
=
u
S
(
u
)
,
S
(
ν
)
=
ν
,
ε
(
ν
)
=
1
{\displaystyle \nu ^{2}=uS(u),\;S(\nu )=\nu ,\;\varepsilon (\nu )=1}
Δ
(
ν
)
=
(
R
21
R
12
)
−
1
(
ν
⊗
ν
)
{\displaystyle \Delta (\nu )=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-1}(\nu \otimes \nu )}
where
u
=
∇
(
S
⊗
id
)
(
R
21
)
{\displaystyle u=\nabla (S\otimes {\text{id}})({\mathcal {R}}_{21})}
. Note that the element u exists for any quasitriangular Hopf algebra, and
u
S
(
u
)
{\displaystyle uS(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
)
)
{\displaystyle S(uS(u))=uS(u),\varepsilon (uS(u))=1,\Delta (uS(u))=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-2}(uS(u)\otimes uS(u))}
, so that all that is required is that it have a central square root with the above properties.
Here
A
{\displaystyle A}
is a vector space
∇
{\displaystyle \nabla }
is the multiplication map
∇
:
A
⊗
A
→
A
{\displaystyle \nabla :A\otimes A\rightarrow A}
Δ
{\displaystyle \Delta }
is the co-product map
Δ
:
A
→
A
⊗
A
{\displaystyle \Delta :A\rightarrow A\otimes A}
η
{\displaystyle \eta }
is the unit operator
η
:
C
→
A
{\displaystyle \eta :\mathbb {C} \rightarrow A}
ε
{\displaystyle \varepsilon }
is the co-unit operator
ε
:
A
→
C
{\displaystyle \varepsilon :A\rightarrow \mathbb {C} }
S
{\displaystyle S}
is the antipode
S
:
A
→
A
{\displaystyle S:A\rightarrow A}
R
{\displaystyle {\mathcal {R}}}
is a universal R matrix
We assume that the underlying field
K
{\displaystyle K}
is
C
{\displaystyle \mathbb {C} }
If
A
{\displaystyle A}
is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if
A
{\displaystyle A}
is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.
See also
Quasitriangular Hopf algebra
Quasi-triangular quasi-Hopf algebra
References
Altschuler, D.; Coste, A. (1992). "Quasi-quantum groups, knots, three-manifolds and topological field theory". Commun. Math. Phys. 150 (1): 83–107. arXiv:hep-th/9202047. Bibcode:1992CMaPh.150...83A. doi:10.1007/bf02096567.
Chari, V. C.; Pressley, A. (1994). A Guide to Quantum Groups. Cambridge University Press. ISBN 0-521-55884-0.
Drinfeld, Vladimir (1989). "Quasi-Hopf algebras". Leningrad Math J. 1: 1419–1457.
Majid, Shahn (1995). Foundations of Quantum Group Theory. Cambridge University Press.