Applied Mathematics, 2010, 1, 230-233
doi:10.4236/am.2010.13028 Published Online September 2010 (http://www.SciRP.org/journal/am)
Ribbon Element on Co-Frobenius Quasitriangular
Hopf Algebras
Guohua Liu
Department of Mathematics, Southeast University, Nanjing, China
Received June 5, 2010; revised July 23, 2010; accepted July 29, 2010
Abstract
Let (H, R) be a co-Frobenius quasitriangular Hopf algebra with antipode S. Denote the set of group-like ele-
ments in H by G (H). In this paper, we find a necessary and sufficient condition for (H, R) to have a ribbon
element. The condition gives a connection with the order of G (H) and the order of S2.
Keywords: Co-Frobenius Hopf Algebra, Ribbon Element
1. Introduction
A Hopf algebra H is called co-Frobenius if H is either
left or right co-Frobenius as a coalgebra, i.e., if there
exists a left or right H* monomorphism from H to H*. It
turns out that H is co-Frobenius if and only if H has
nonzero integrals [1,2]; in particular every finite dimen-
sional Hopf algebra is co-Frobenius. Among the proper-
ties of finite dimensional Hopf algebras that hold for all
co-Frobenius Hopf algebras are the bijectivity of the an-
tipode, a bijective correspondence between the group-like
elements of the Hopf algebra and the one dimensional
ideals of the dual algebra, the existence of a distinguish-
ed group-like element, and a reasonable theory of Galois
extensions.
The class of infinite dimensional co-Frobenius Hopf
algebras includes cosemisimple Hopf algebras, such as
the group algebra of an infinite group. Tensoring such a
Hopf algebra H with a finite dimensional Hopf algebra K,
yields an infinite dimensional Hopf algebra with non-zero
integral obtained by tensoring the integrals of H and K.
As well, a recent example of Van Daele [3] gives an in-
finite dimensional co-Frobenius Hopf algebra without
normal Hopf subalgebra.
The topological motivation for this paper is supported
by the fact that ribbon Hopf algebras (Hopf algebra with
a distinguished ribbon element) can be used to construct
invariants of framed links embedded in three dimen-
sional space [4]. And the same structure can be used to
produce invariant of three dimensional manifolds. These
three dimensional manifolds are represented by surgery
on framed links, and their invariants are special cases of
invariants for the links. In the case of quantum group
SLq(2), these invariants have been intensively investi-
gated by Reshetukhin and Turaev [5], Kirby [6], and
others.
In this paper, we give a necessary and sufficient con-
dition for the co-Frobenius quasitriangular Hopf algebra
to have a ribbon element. Based on the ideals and results
of Beattie, Bulacu and others [7-9], we generalize the
results of Kauffman and Radford [10] to co-Frobenius
quasitriangular Hopf algebras. We find the group-like
elements

and g which play a special role in the the-
ory of ribbon Hopf algebras. Our main result is Theorem
5, which states that a co-Frobenius quasitriangular Hopf
algebra (,)
H
R (()GH has odd order) has a ribbon
element if and only if, 2
S has odd order.
Throughout this paper, H will denote a co-Frobenius
Hopf algebra over a field k. All maps are assumed to be
k-linear. We use the Sweedler-type notation for the com-
ultiplication maps

(h) = 12 hhfor all h  H. As usual,
the H*-bimodule structure on H and the H-bimodule
structure on H* are given by


12 3
lhmmhhlh

 


 
hm lmmlmh

 
for all ,,hlm H

and ,lm H


. The antipode of H is
denoted by S with composition inverse 1 S. The set of
group-like elements in H is denoted by G(H) and the
group-likes of 0
H
, namely the set of algebra maps from
H to k, by 0 ()GH .
Let H be a co-Frobenius Hopf algebra over a field k.
Recall that a Hopf algebra H is co-Frobenius if rat
H
,

G. H. LIU
231
the unique maximal rational submodule of
H

, is non-
zero, or, equivalently, if the space of left or right inte-
grals for H, denoted by *
H
l
and *
H
r
 respectively, is
nonzero. It was shown in [9] that H contains a distin-
guished group-like element g, which is also called the
modular element of H, such that for all *
H
l
 

and
hH:
 
1
12
hhhg 

 and 21
Sg g 

 
For either a nonzero left or right integral for H in
rat
H
, there are bijective maps from H to rat
H
 given
by
()hh and ()hh .
Let

denote the generalized Frobenius automor-
phism of H defined in [7], that is, for *
H
l
 ,

is the
algebra automorphism of H defined by
()hh

  , for all hH.
Then the algebra map
H


 is called the
modular element for H in
H
, and
 
2
21
hhhS 

, for all hH.
Recall that, for a Hopf algebra H and
1212
RRRrrH H, then (H, R) is called
quasitriangular if for all hH,




 

 

 
121122
121122
21 12
1;
2;
3,;
41,1.
cop
QTRRRrR r
QTRRR rrR
QThRRhforall hH
QTRRR R 

 
 

Set

21
,().uSRRcuSu
By the result of Drinfeld [11] or Radford [12], 2 S is
an inner automorphism induced by u and
1
()Su, i.e.
21
()Sa uau

, and 21 () ()()SaSu aSu

,
for all aH.
c is called the Casimir element of (H, R), and
21
()Sa uau

 implies that c is in the center of H.
If (H, R) is quasitriangular, Beattie and Bulauc [13]
introduced two group homomorphisms from 0 ()GH to
G (H) given by
  

1212 1
121
:,:aRRbSRR
RR





 

They showed that ,()ab GH

 and 1 ab
 

() ()GH ZH

. Now set
11
11
(),
g
bbhbg
 


 .
By [13], we have11 () ()uS uSuugg


.
By [12] and [13], we have (g,

denote the modular
elements),
2
cuh.
Since c is central, 212 () ,Sauaucu h

, implies that
 


2
42121
2
21 11
Sa uSauuau
uacuh ah

 


We say that
H


is a quasi-ribbon element of (H, R)
if the following conditions are satisfied:




 
2
1
21 12
.1 ;
.2 ;
.3 1;
.4 ,
Rc
RS
R
RRR









 
Drinfeld observed that u satisfies the last condition. A
quasi-ribbon element in the center of H is called a ribbon
element, and in this case (H, R, v) is called a ribbon Hopf
algebra [14]. The reader is referred to [14] for a detailed
discussion of ribbon Hopf algebras and their relationship
to links and three-manifolds.
2. Ribbon Hopf Algebra
Let (H, R) be a finite dimensional quasitriangular Hopf
algebra. In [10], the authors found a necessary and suffi-
cient condition for the existence of ribbon elements on
(H, R). The purpose of this section is to generalize their
result to co-Frobenius quasitriangular Hopf algebras. We
find that most of the results in [10] also hold for co-
Frobenius quasitriangular Hopf algebras.
Lemma 1. Suppose that (H, R) is a co-Frobenius qua-
sitriangular Hopf algebra over a field k, H contains a
distinguished group-like element g and that v
H

is a
quasi-ribbon element of H. Let


1121
11
,,
g
bbhbguSRR
 


 
and set 1 lu


. Then:
1) 2;lh

2) ().lGH

Proof. 1) By (R.2) (S(v) = v), we have 2()S



.
Thus u and v commute by 21 ()Suau 

 for all aH

.
By 2 (.1)Rc 

and 2 cuh we have 22 uh  .
Thus 2 lh

.
2)

G. H. LIU
232



 



11
1
1
11
21 1221 12
11
luu
u
RRu uRR
uull











 
 

Theorem 2. Suppose that (H, R) is a co-Frobenius
quasitriangular Hopf algebra over a field k, and let u
and v be as above. Then:
1) luldefines a one-one correspondence between
2
{() }lGHl h{quasi-ribbon elements of (H, R)};
2) Suppose that ()lGHand 2 lh, Then v = ul is
a ribbon element of (H,R) if and only if 21 ()Salal


for all aH.
Proof. 1) Recall that u commutes with the group-like
elements of H. Thus v = ul = lu. Using 2
()uh cwe see
that 2222 uluh c  , so (R.1) holds for v. Now,
 
1
SSulSlSulSu 

 ,
by 2 lh we have 11 llh

and hu = S(u), which fol-
lows from 2()uh c uSu , Therefore
 
111
SlSu lhulhhuluul



.
Thus, (R.2) holds for v. Note that (R.3) is immediate
since () 1()ul

  . Also
  

1
21 12
1
21 12
ululR Ruull
RR




 

and (R.4) holds for v. The proof of (1) is finished by
Lemma 1.
2) If 211111 ,() ()()ulSauaulall al 

 .
On the other hand, 211 ()Salal uau


 implies va =
av for all aH.
Corollary 3. Suppose that (H, R) is a co-Frobenius
quasitriangular Hopf algebra. Let g and a be the dis-
tinguished group-like elements of H and
H
, respec-
tively, and let h be as above. Then:
1) If h has odd order, or if g and a have odd order,
Then (H, R) has a quasi-ribbon element;
2) If G (H) has odd order, Then (H, R) has a unique
quasi-ribbon element.
Proof. 1) By 1a
b commuting with all ()aGH

,
and b


 is a group homomorphism from 0 ()GH
to G (H). We have that h has a square root in G(H),
which must be unique if G (H) has odd order. Therefore
the corollary follows by part 1 of the above Theorem.
Proposition 4. Suppose that (H, R) is a co-Frobenius
quasitriangular Hopf algebra. Let g and a be the distin-
guished group-like elements of H and
H
, respectively,
and let h be as above. Then if either
1) If h and 2 S, or;
2) If g, a and 2 S have odd order. Then (H, R) has a
ribbon element.
Proof. First, condition (2) implies condition (1). Sup-
pose that h and 2 S have odd order. Let l be the unique
square root of h having odd order. Define map
():aH H

 by 1
()()a baba 

.
Then 12 1 ()( )lh 

 and 1 ()l 
 have odd order.
Recall that 41 1 ()( )Sa hahh 

 . Since ()lGH

,
2
S and 1
()l 
 are two elements of odd order whose
squares are equal. Consequently, 21 ()Sl 

, and the
proposition follows by part (2) of Theorem 2.
Theorem 5. Suppose that (H, R) is a co-Frobenius
quasitriangular Hopf algebra with antipode S over a
field k and assume that G (H) has odd order, Then (H, R)
has a ribbon element (which is necessarily unique) if and
only if 2 S has odd order.
Proof. If (H, R) has a ribbon element, then there exists
an ()
x
GH

such that 21
()Sa xax

 for all aH

by Theorem 2. Since x has odd order it follows that 2 S
does also.
Conversely, suppose that 2 S has odd order. Since h
has odd order, it follows that (H, R) has a ribbon element
by Proposition 4. This completes our proof.
When H is unimodular, We note that 1
hg

 since
a


in this case, by Theorem 2, the existence of rib-
bon (or quasi-ribbon) elements is determined by square
roots of g.
Proposition 6. Suppose that (H, R) is a co-Frobenius
quasitriangular Hopf algebra with antipode S over a
field k, Suppose further that H is unimodular and let g be
the distinguished group-like element of H. Then:
1) (H, R) has a quasi-ribbon element if and only if
2
lg

for some ()lGH

;
2) (H, R) has a ribbon element if and only if 2 lg

for some ()lGH

which satisfies 21 ()Sa lal

 for
all aH

.
Proposition 7. Suppose that (H, R) is a co-Frobenius
quasitriangular Hopf algebra with antipode S over a
field k, suppose further that H and
H
 are both uni-
modular Then:
1) u is a quasi-ribbon element of (H, R);
2) u is a ribbon element of (H, R) if and only if
2
SI

.
3. Acknowledgements
The author is supported by the NSFC project 10826037.
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