The Weak Hopf Algebras Related to Generalized Kac-Moody Algebra
Wu Zhixiang∗
Mathematics Department,
Zhejiang University,Hangzhou, 310027, P.R.China
(Dated: October 19, 2006)
We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra G by
adding a new generator J satisfying J m = J m−1 for some integer m. We denote this algebra by
wUqτ (G). This algebra is a weak Hopf algebra if and only if m = 2. In general, it is a bialgebra, and
contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usually quantum envelope
algebra Uq (G) of a generalized Kac-Moody algebra G.
Keywords: generalized Kac-Moody algebras, irreducible representation, weak Hopf algebra
I.
INTRODUCTION
In his study of Monstrous moonshine3–5 , Borcherds introduced a new class of infinite dimensional Lie algebras
called generalized Kac-Moody algebras. These generalized Kac-Moody algebra have a contravariant bilinear form
which is almost positive definite. The fixed point algebra of any Kac-Moody algebra under a diagram automorphism
is a generalized Kac-Moody algebra. A generalized Kac-Moody algebra can be regarded as a Kac-Moody algebra
with imaginary simple roots. More explicitly, a generated Kac-Moody algebras is determined by a Borcherds-Cartan
matrix A = (aij )(i,j)∈I×I , where either aii = 2, or aii ≤ 0. If aii ≤ 0, then the index i is called imaginary, and the
corresponding simple root αi is called imaginary root. In this paper, the set {i ∈ I|aii = 2} is denoted by I + . Set
I im = I \ I + . The structure and the representation theory of generalized Kac-Moody algebras are very similar to
those of Kac-Moody algebras, and many basic facts about Kac-Moody algebras can be extended to generalized KacMoody algebras. For example, the Kac-Weyl formula about an irreducible representation over a Kac-Moody algebra
is generalized to a formula about an irreducible representation over a generalized Kac-Moody algebra as follows.
P
P
(−1)l(w)+|F | ew(λ+ρ−s(F ))
w∈W
P F ⊆T,F ⊥λ l(w)+|F | w(ρ−s(F ))
chV (λ) =
,
e
w∈W,F ⊆T (−1)
where T is the set of all imaginary simple roots, F runs all over finite subsets of T such that any two elements in F are
mutually perpendicular. We denote by s(F ) the sum of the roots in F . We call the above formula Borcherds-Kac-Weyl
formula.
On the other hand, many mathematicians are interested in generalization of Hopf algebras, of which importance
has been recognized in both mathematics and physics. One way to do this is to introduce a kind of weak coproduct
such that ∆(1) 6= 1 ⊗ 1 in Ref.[1]. The face algebras7 and generalized Kac algebras15 are examples of this class of
weak Hopf algebras. Li and Duplij have defined and studied another kind of weak Hopf algebras11 . A bialgebra
(H, µ, η, ∆, ε) is called a weak Hopf algebra if there is an anti-automorphism T such that T ∗ idH ∗ T = idH and
2
idH ∗ T ∗ idH = T , where idH is the identity map and ∗ is the convolution product. Hopf algebras, and left of right
Hopf12,13 algebras are weak Hopf algebras in this sense. In presented paper a weak Hopf algebra is always mean the
weak Hopf algebra in this sense. The weak quantized enveloping algebras of semi-simple Lie algebras are also weak
Hopf algebras14 . Our aim is to give more nontrivial examples of weak Hopf algebras. Thanks to the definition of
quantized enveloping algebra Uq (G) associated a generalized Kac-Moody algebra G defined in Ref. [7], we can also
replace the group G(Uq (G)) of grouplike elements by some regular monoid as in Ref. [14], we use a new generator
J instead of the projector in Ref.[14]. Our new generator J satisfies J m = J m−1 for some integer m ≥ 2. By this
way, we obtain a subclass of bialgebra wUq (G). This bialgebra contains a sub-bialgebra U τ (G), which is a weak Hopf
algebra in the sense of Ref. [11]. Moreover, the quotient algebra wUq (G)/(1 − J m ) is isomorphic to a sub-Hopf-algebra
of the quantized enveloping algebra Uq (G) as Hopf algebras. As in the case of the classic quantum group Uq (G), we
try to determine irreducible representation of wUq (G).
Finally, let us outline the structure of this paper. In Section II, we recall some basic facts related to the quantized
enveloping algebra of a generalized Kac-Moody algebra. In Section III, we give the definition of wUq (G). We study the
bialgebra structure of wUq (G) in Section IV. In the final section, we study the irreducible representation of wUq (G).
II.
NOTATIONS AND PRELIMINARIES
In this section, we fix notations and recall fundamental results about generalized Kac-Moody algebras.
Let I = {1, · · · , n} or the set of positive integers, and A = (aij )I×I , a Borcherds-Cartan matrix, i.e., it satisfies:
(1) aii = 2 or aii ≤ 0 for all i ∈ I,
(2) aij ≤ 0 for all i 6= j,
(3) aij ∈ Z,
(4) aij = 0 if only if aji = 0.
We say that an index i is real if aii = 2 and imaginary if aii ≤ 0. We denote I + = {i ∈ I|aii = 2} and I im = I − I + .
Kang considered the generalized Kac-Moody algebras associated with Borcherds-Cartan matrices with charge9
m = {(mi ∈ Z≥0 )|i ∈ I, mi = 1 f or i ∈ I}.
The charge mi is the multiplicity of the simple root corresponding to i ∈ I. In this paper, we follow [10], and assume
that mi = 1 for all i ∈ I. However, we do not lose generality by this hypothesis. Indeed, if we take BorcherdsCartan matrices with some of the rows and columns identical, then the generalized Kac-Moody algebras with charge
introduced in [9] can be recovered from the ones in present paper by identifying the hi s and di s( and hence the αi s)
corresponding to these identical rows and columns.
Moreover, we also assume that A is symmetrizable; that is, there is a diagonal matrix D = diag{si > 0|i ∈ I} such
that DA is a symmetric matrix.
3
Let P ˇ= (⊕i∈I Zhi ) ⊕ (⊕i∈I Zdi ) be a free abelian group generated by the set {hi , di |i ∈ I}. This free abelian group
is called the co-weight lattice of A. The element hi in Πˇ= {hi |i ∈ I} is called a simple co-weight. We call Πˇthe set
of all simple co-weights. The space H = Q ⊗Z P ˇover the rational number field Q is said to be a Cartan subalgebra.
The weight lattice is defined to be P := {λ ∈ H∗ |λ(P ˇ) ⊆ Z}, where H∗ is the dual space of the Cartan subalgebra
H = Q ⊗Z P ˇ. We denote by P + the set {λ ∈ P |λ(hi ) ≥ 0, f or every i ∈ I} of dominant integral weights.
Define αi , Λi ∈ H∗ by
αi (hj ) = aji ,
αi (dj ) = δij
Λi (hj ) = δij , Λi (dj ) = 0.
Then αi , i ∈ I are called simple roots of A. Let Π = {αi |i ∈ I} ⊂ P be the set of simple roots. The free abelian
P
group Q = ⊕i∈I Zαi is called the root lattice. Set Q+ = i∈I Z≥0 αi and Q− = −Q+ . For any α ∈ Q+ , we can write
Pn
α = k=1 αik for i1 , i2 , · · · , in ∈ I. We set ht(α) = n and call it the height of α.
Let (.|.) be the bilinear form on (⊕i (Qαi ⊕ QΛi )) × H∗ defined by
(αi |λ) = si λ(hi ), (Λi |λ) = si λ(di ).
Since it is symmetric on (⊕i (Qαi ⊕ QΛi )) × (⊕i (Qαi ⊕ QΛi )), one can extend this to a symmetric bilinear form on
H∗ . Then such a form is non-degenerated.
We always assume that K is a field of characteristic 0. Let q ∈ K and qi = q di . It is assumed that qi 6= ±1, 0 for
all i ∈ i. For an indeterminant ν and an integer m, let
[m]ν =
ν m − ν −m
, [m]!ν = [m]ν · · · [1]ν , [0]ν = 1,
ν − ν −1
and

m

s

 =
[m]!ν
.
[s]!ν [m − s]!ν
ν
Definition The quantized enveloping generalized Kac-Moody algebra Uq (G) associated with a Borcherds-Cartan datum
(A, P ˇ,P, Πˇ, Π) is the associated algebra with unit 1 over a field K of characteristic 0, generated by the symbols ei ,
fi (i ∈ I) and P ˇ subject to the following defining relations:
0
0
q 0 = 1, q h q h = q h+h
q h ei q −h = q αi (h) ei ,
ei fj − fj ei = δij
ki − ki−1
,
qi − qi−1
∀h, h0 ∈ Pˇ,
q h fi q −h = q −αi (h) fi ,
where
ki = q si hi ,
4

1−aij
X
(−1)r 

ij −r
 e1−a
ej eri = 0
i
r
r=0

1−aij
X
1 − aij
(−1)r 
aii = 2, i 6= j,
if
aii = 2, i 6= j,
i
1 − aij
r
r=0
if

 fi1−aij −r fj fir = 0
i
ei ej − ej ei = fi fj − fj fi = 0,
if
aij = 0.
The quantum generalized Kac-Moody algebra Uq (G) has a Hopf algebra structure with the comultiplication ∆, the
counit ε, and antipode S defined by
∆(q h ) = q h ⊗ q h ,
∆(ei ) = ei ⊗ ki−1 + 1 ⊗ ei ,
∆(fi ) = ki ⊗ fi + fi ⊗ 1,
ε(q h ) = 1, ε(ei ) = ε(fi ) = 0,
S(q h ) = q −h , S(ei ) = −ei ki , S(fi ) = −ki−1 fi
for all h ∈ P ˇ and i ∈ I.
Let Uq+ (G) and Uq− (G) be the subalgebras of Uq (G) generated by elements ei and fi respectively, for i ∈ I, and let
Uq0 (G) be the subalgebra of Uq (G) generated by q h (h ∈ P ˇ ). Then we have the triangular decomposition [9,10]
Uq (G) = Uq− (G) ⊗ Uq0 (G) ⊗ Uq+ (G).
Finally, let us use Uq0 (G) to denote a subalgebra of Uq (G) generated by ei , fi , q h , where h ∈ ⊕i∈I Zsi hi ⊕ ⊕i∈I Zdi . It
is obvious that Uq0 (G) is a Hopf algebra.
III.
WEAK QUANTUM ALGEBRAS wUqT (G)
Let m be a fixed positive integer. To generalize the invertibility condition ki ki−1 = 1 in Uq (G), let us introduce
some new generators J, Ki and K̄i , which subject the following relations:
J m = J m−1 = Ki K̄i = K̄i Ki = Di D̄i = D̄i Di .
(III.1)
5
Moreover, we assume that J r 6= J r−1 for any r < m. Suppose Ki and K̄i are not zero divisors. Then
Ki J = JKi = Ki ,
K̄i J = J K̄i = K̄i .
(III.2)
Di J = JDi = Di ,
D̄i J = J D̄i = D̄i .
(III.3)
In this paper we always assume that (III.2) and (III.3) hold. We call an element Ei of type one if it satisfies
a
Kj Ei = qi ij Ei Kj ,
−aij
K̄j Ei = qi
Ei K̄j .
(III.4)
Similarly, if
−aij
K j Fi = q i
Fi K j ,
−aij
K̄j Fi = qi
Fi K̄j ,
(III.5)
then Fi is said to be type one. Suppose
Kj Ei K̄j = q aij Ei .
(III.6)
Then we say that Ei is type zero. Similarly, Fi is type zero if it satisfies the following:
Kj Fi K̄j = q −aij Fi .
(III.7)
Proposition III.1 Ei ( resp. Fi ) is type zero if and only if Ei is type one and Ei J m−1 = J m−1 Ei = Ei ( respectively,
Fi J m−1 = J m−1 Fi = Fi ).
Proof
If Ei is type zero, then we obtain from (III.6)
a
Kj Ei K̄j Kj = Kj Ei J m = qi ij Ei Kj .
On the other hand, since J m Kj = Kj ,
a
a
Kj Ei K̄j = Kj Ei J m K̄j = qi ij Ei Kj K̄j = qi ij Ei J m−1 .
So Ei = Ei J m−1 . Similarly, we can prove that Ei = J m−1 Ei . Then
a
Kj Ei = Kj Ei J m = qi ij Ei Kj ,
and
a
Ei K̄j = J m Ei K̄j = qi ij K̄j Ei .
That is, Ei is type one. On the other hand, if Ei is type one, and Ei J m = J m Ei = Ei , then
a
a
qi ij Ei = qi ij Ei Kj K̄j = Kj Ei K̄j .
Similarly,we can prove the statement about Fi is true. By now, we complete the proof.
6
The types of Ei and Fi are denoted by κi ,κ0 respectively. Let τ = ({κi }i∈I | {κ0i )}i∈I ). By now, we can give the
definition the weak quantum algebra of type τ as follows:
Definition The type τ weak quantum algebra wUqτ (G) associated the generalized Kac-Moody algebra G is an associated algebra with unit 1 over a field K of characteristic 0, generated by J, Ei , Fi (i ∈ I) and Ki , Di (i ∈ I) subjecting
with the following defining relations:
J m = J m−1 = Ki K̄i = Di D̄i ,
Ki K̄j = K̄j Ki ,
Ki Kj = Kj Ki ,
K̄i K̄j = K̄j K̄i ,
(III.9)
Di D̄j = D̄j Di ,
Di Dj = Dj Di ,
D̄i D̄j = D̄j D̄i ,
(III.10)
Di K̄j = K̄j Di ,
Ki Dj = Dj Ki ,
D̄i Kj = Kj D̄i ,

(−1)r 
(III.12)
J D̄i = D̄i J = Di ,
J K̄i = K̄i J = K̄i ,
(III.13)
1−aij
X
r=0
1 − aij
r
r=0

(−1)r 
(III.11)
Ki J = JKi = Ki ,
Fi
are
type
Ei Fj − Fj Ei = δij
X
D̄i K̄j = K̄j D̄i ,
Di J = JDi = Di ,
Ei
1−aij
(III.8)
1 − aij
r
τ,
(III.14)
Ki − K̄i
,
qi − qi−1
(III.15)

 Ei1−aij −r Ej Eir = 0
if
aii = 2, i 6= j,
(III.16)
if
aii = 2, i 6= j,
(III.17)
i

 Fi1−aij −r Fj Fir = 0
i
Ei Ej − Ej Ei = Fi Fj − Fj Fi = 0,
if
aij = 0.
(III.18)
If m = 1, and the Borcherds-Cartan matrix A is symmetric, then wUqT (G) = Uq (G) provided that we identify Ki
with q hi , K̄i with q −hi , Di with q di and D̄i with q −di . If m = 2 and G is a semisimple Lie algebra, then wUqT (G) has
been defined and studied by Yang in Ref.[14]. Notice that the type zero was called type two by Yang.
Lemma III.1 J m is a center idempotent element of wUqτ (G).
7
Proof If Ei is type one, then J m Ei = Kj K̄j Ei = Ei Kj K̄j = Ei J m . Similarly, we can prove that Fi J m = J m Fi
provided Fi is type one. Hence this lemma follows from Proposition III.1.
In the following corollary, the subalgebra of Uq (G) generated by Ei , Fi , si hi , di (i ∈ I) is denoted by Uq0 (G). It is a
Hopf subalgebra of Uq (G).
Corollary III.1 (1) wUqτ (G) = wUqτ (G)J m ⊕ wUqτ (G)(1 − J m ) is a direct sum of algebras.
(2) wUqτ (G)/(1 − J) is isomorphic to the algebra Uq0 (G).
(3) wUqτ (G)J m is isomorphic to the algebra Uq0 (G).
Proof The proof of (1) and (2) is easy. To prove (3), let us define a map ψ from Uq0 (G) to wUqτ (G)J m as follows.
ψ(q si hi ) = Ki ,
ψ(q di ) = Di ,
ψ(ei ) = Ei J m ,
ψ(fi ) = Fi J m ,
ψ(1) = J m .
Then one can show that ψ is an algebra homomorphism. Similarly we can define an algebra homomorphism a map ϕ
from wUqτ (G)J m to Uq0 (G) as follows.
ϕ(Ki ) = q si hi ,
ϕ(Di ) = q di ,
ϕ(Ei J m ) = ei ,
ϕ(Fi J m ) = fi ,
ϕ(J m ) = 1.
It is easy to check that ϕψ = id and ψϕ = id. This proves (3).
Remark 1 By Proposition III.1, wUqτ (G)(1 − J m ) is generated by J r − J m−1 for 0 ≤ r ≤ m − 2, Ei (1 − J m ) and
Fi (1 − J m ), where Ei and Fi are type one. Since (III.15) holds in wUqτ (G), Ei (1 − J m )Fj (1 − J m ) = Fj (1 − J m )Ei (1 −
J m ). If all κi = κ̄i = 0, then wUqτ (G)(1 − J m ) is isomorphic to the group algebra K[Zm−2 ].
IV.
THE BIALGEBRA STRUCTURE OF wUqτ (G)
The algebras wUqτ (G)J m and wUqτ (G)(1 − J m ) are denoted by w and w̄ respectively in the following. By Corollary
III.1, w is isomorphic to the quantum group Uq (G) provided si = 1. Thus the comultiplication and counit of Uq (G)
transplant to the algebra wUqτ (G)J m , and wUqτ (G)J m becomes a Hopf algebra. Moreover, we can define three maps:
∆ : wUqτ (G) → wUqτ (G) ⊗ wUqτ (G),
ε : wUqτ (G) → K,
T : wUqτ (G) → wUqτ (G),
as follows:
∆(Ki ) = Ki ⊗ Ki ,
∆(K̄i ) = K̄i ⊗ K̄i ,
(IV.1)
∆(Di ) = Di ⊗ Di ,
∆(D̄i ) = D̄i ⊗ D̄i ,
(IV.2)
8
∆(J) = J ⊗ J
∆(Ei ) =
∆(Fi ) =

 1 ⊗ Ei + E i ⊗ K i ,
Ei
(IV.3)
is type one
 J m−1 ⊗ E + E ⊗ K ,
i
i
i
Ei is type zero,

 Fi ⊗ 1 + K̄i ⊗ Fi ,
is
Fi
 F ⊗ J m−1 + K̄ ⊗ F ,
i
i
i
ε(Ki ) = ε(K̄i ) = 1,
type one
(IV.4)
(IV.5)
Fi is type zero,
ε(Di ) = ε(D̄i ) = 1,
ε(J) = 1,
ε(Ei ) = ε(Fi ) = 0,
(IV.6)
(IV.7)
while the map T is defined as follows:
T (1) = 1,
T (Ki ) = K̄i ,
T (K̄i ) = Ki ,
(IV.8)
T (J) = J,
T (Di ) = D̄i ,
T (D̄i ) = Di ,
(IV.9)
T (Ei ) = −Ei K̄i ,
T (Fi ) = −Ki Fi .
Then we extend them to the whole wUqτ (G). Thus we obtain the following Lemma.
Lemma IV.1 wUqτ (G) is a bialgebra with comultiplication ∆ and counit ε.
Proof It can be shown by direct calculation that the following relations hold.
∆(Ki )∆(K̄j ) = ∆(K̄j )∆(Ki ),
∆(Di )∆(D̄j ) = ∆(D̄j )∆(Di ),
∆(J m−1 ) = ∆(Ki )∆(K̄i ) = ∆(Di )∆(D̄i ),
∆(JKi ) = ∆(Ki ),
∆(J K̄i ) = ∆(K̄i ),
∆(JDi ) = ∆(Di ),
∆(J D̄i ) = ∆(D̄i ),
ε(Ki K̄j ) = ε(Ki )ε(K̄j ),
ε(Di D̄j ) = ε(Di )ε(D̄j ),
ε(JKi ) = ε(Ki ),
ε(J K̄j ) = ε(K̄j ),
ε(J D̄i ) = ε(Di ),
ε(J D̄j ) = ε(D̄j ),
(IV.10)
9
a
ε(Kj )ε(Ei ) = qi ij ε(Ei )ε(Kj ),
a
ε(Fi )ε(K̄j ) = qi ij ε(Fj )ε(K̄j ),
ε(Ei )ε(Fj ) − ε(Fj )ε(Ei ) = δij
ε(Ki ) − ε(K̄i )
.
qi − qi−1
If Ei is type one, then
∆(Kj )∆(Ei ) = (Kj ⊗ Kj )((1 ⊗ Ei + Ei ⊗ Ki )
= K j ⊗ K j E i + K j Ei ⊗ K j K i
a
= qi ij ∆(Ei )∆(Kj ).
If Ei is type zero, then
∆(Kj )∆(Ei )∆(K̄j ) = (Kj ⊗ Kj )(J m−1 ⊗ Ei + Ei ⊗ Ki )(K̄j ⊗ K̄j )
= Kj K̄j ⊗ Kj Ei K̄j + Kj Ei K̄j ⊗ Kj Ki K̄j
a
= qi ij ∆(Ei ).
Next we prove that
∆(Ei )∆(Fj ) − ∆(Fj )∆(Ei ) = δij
∆(Ki ) − ∆(K̄i )
.
qi − qi−1
For any integers 0 ≤ r, s ≤ m, if J m−r Fj = Fj J m−r and J m−s Ei = Ei J m−s , then
(J m−r ⊗ Ei + Ei ⊗ Ki )(Fj ⊗ J m−s + K̄j ⊗ Fj )
−(Fj ⊗ J m−s + K̄j ⊗ Fj )(J m−r ⊗ Ei + Ei ⊗ Ki )
= J m−r Fj ⊗ Ei J m−s + K̄j ⊗ Ei Fj + Ei Fj ⊗ Ki + Ei K̄j ⊗ Ki Fj
−Fj J m−r ⊗ J m−s Ei − Fj Ei ⊗ Ki − K̄j ⊗ Fj Ei − K̄j Ei ⊗ Fj Ki
= K̄j ⊗ (Ei Fj − Fj Ei ) + (Ei Fj − Fj Ei ) ⊗ Ki
)−∆(K̄i )
.
= δij ∆(Kqi −q
−1
i
i
Thus (VI.11) holds for all i, j.
Finally, we prove that ∆ satisfies the Quantum Serre relations, i.e., the relations from (III.16) to (III.18).
a
From aij = 0, we obtain Ei Ej = Ej Ei , and Ki Ej = qj ji Ej Ki = Ej Ki . Hence
(J m−r ⊗ Ei + Ei ⊗ Ki )(J m−s ⊗ Ej + Ej ⊗ Kj )
−(J m−s ⊗ Ej + Ej ⊗ Kj )(J m−r ⊗ Ei + Ei ⊗ Ki )
= J 2m−r−s ⊗ Ei Ej + J m−r Ej ⊗ Ei Kj + Ei J m−s ⊗ Ki Ej + Ei Ej ⊗ Ki Kj
−J 2m−r−s ⊗ Ej Ei − J m−s Ei ⊗ Ej Ki − Ej J m−r ⊗ Kj Ei − Ej Ei ⊗ Kj Ki
= J m−r Ej ⊗ Ei Kj + Ei J m−s ⊗ Ki Ej − J m−s Ei ⊗ Ej Ki − Ej J m−r ⊗ Kj Ei
= 0,
for any 1 ≤ r, s ≤ m. So
∆(Ei )∆(Ej ) − ∆(Ej )∆(Ei ) = 0.
(IV.11)
10
Similarly, we can prove
∆(Fi )∆(Fj ) − ∆(Fj )∆(Fi ) = 0.
By now we have proven that ∆ satisfies the relation (III.18).
To prove that ∆ satisfies the relation (III.16), we must consider the following cases:
(1) Both Ei and Ej are type one.
(2) Only one of the Ei and Ej is type one.
(3) Both Ei and Ej are type zero.
For the case (3). Since Ei is type zero,
∆(Ei ) = J m ⊗ Ei + Ei ⊗ Ki
= (J m ⊗ 1)(1 ⊗ Ei + Ei ⊗ Ki ) .
= (1 ⊗ Ei + Ei ⊗ Ki )(J m ⊗ 1)
Let us introduce new notations. (1 ⊗ Ei + Ei ⊗ Ki ) is denoted by ∆0 (Ei ). Set s = 1 − aij . Then

Ps
r
r=0 (−1)
s


 ∆(Ei )s−r ∆(Ej )∆(Ei )r
r
 
s
P
s
= (J m ⊗ 1)(s+1) r=0 (−1)r   ∆0 (Ei )s−r ∆0 (Ej )∆0 (Ei )r
r
= 0,
by the discussion in [8,pp67–68]. Hence ∆ satisfies (III.16) in this case. For the other cases, the argument is more or
less the same as case (3).
Similarly, we can prove that ∆ satisfies (III.17). Therefore ∆ and ε can be extended to an algebra morphism from
wUqτ (G) to wUqτ (G) ⊗ wUqτ (G), and from wUqτ G) to K, respectively.
It is easy to prove that
(∆ ⊗ 1)∆(X) = (1 ⊗ ∆)∆(X),
(IV.12)
(ε ⊗ 1)∆(X) = (1 ⊗ ε)∆(X)
(IV.13)
for any X = Ei , Fi , Ki K̄i , Di D̄i , J. Since ∆, ε are algebra morphisms, (IV.12) and (IV.13) hold for any X ∈ wUqτ (G).
By now we have completed the proof.
Next we prove that the map T , defined by (IV.8),(IV.9),(IV.10), is a weak antipode of the subalgebra of a subbialgebra of the bialgebra wUqτ (G) generated by Ei , Fi , Ki , K̄i , Di , D̄i , J m . First we prove that T can be extended to an
anti-automorphism of wUqτ (G). It is easy to prove the following relations are true.
T (Ki )T (K̄j ) = T (K̄j )T (Ki ),
T (Di )T (D̄j ) = T (D̄j )T (Di ),
11
T (Di )T (K̄j ) = T (K̄j )T (Di ),
T (D̄i )T (K̄j ) = T (K̄j )T (D̄i ),
T (Ki )T (D̄j ) = T (D̄j )T (Ki ),
T (J)T (K̄i ) = T (K̄i )
T (J)T (D̄i ) = T (D̄i )
T (Ei )T (Ej ) = T (Ej )T (Ei ),
T (J)T (Ki ) = T (Ki )
T (J)T (Di ) = T (Di ).
T (Fi )T (Fj ) = T (Fj )T (Fi ),
if
aij = 0.
If Ei is type one, then
a
a
T (Ei )T (Kj ) = −Ei K̄i K̄j = −qi ij K̄j Ei K̄i = qi ij T (Kj )T (Ei ).
If Ei is type zero, then
a
a
T (K̄j )T (Ei )T (Kj ) = −Kj Ei Ki K̄j = −qi ij Ei Ki = qi ij T (Ei ).
Similarly, we can prove
−aij
T (Fi )T (Kj ) = qi
T (Kj )T (Fi )
if Fi is type one, and
−aij
T (K̄j )T (Fi )T (Kj ) = qi
T (Fi )
if Fi is type zero. Moreover,
T (Fj )T (Ei ) − T (Ei )T (Fj ) = Kj (Fj Ei )K̄i − Ei K̄i Kj Fj
−ajj aij
qi Fj Ei Kj K̄i
= qj
−ajj aji
qj Fj Ei Kj K̄i
− qj
i −Ki
= δij qK̄−q
−1 Kj K̄i
i
=
i
)−T (K̄i )
δij T (Kqi −q
−1
i
i
Similarly to [14, p.8], we can prove the following anti-relations to the quantum Serre relations hold.
s
X

(−1)r 

 T (Ei )r T (Ej )T (Ei )s−r = 0,
if
aii = 2,
r
r=0
s
X
s

(−1)r 
r=0
s

 T (Fi )r T (Fj )T (Fi )s−r = 0 if
aii = 2,
r
where s = 1 − aij .
From the above discussion, we get that T is an anti-automorphism of wUqτ (G). Let U τ be a subalgebra of wUqτ (G)
generated by Ki , K̄i , Di , D̄i , Ei , Fi , J m .
12
Theorem IV.2 T is a weak antipode of U τ (G) and U τ (G) is a weak Hopf algebra.
Proof It is easy to verified the following relations hold
(id ∗ T ∗ id)(X) = X,
(T ∗ id ∗ T )(X) = T (X),
for X = Ki , K̄i , Di , D̄i , Ei , Fi , J m .
Since
id ∗ T ∗ id = (µ ⊗ 1)µ(id ⊗ T ⊗ id)(∆ ⊗ 1)∆,
id ∗ T ∗ id is a linear automorphism of wUqτ (G). To prove (id ∗ T ∗ id)(X) = X, for any X ∈ U τ (G), we only need
prove that
(id ∗ T ∗ id)(xy) = xy
(IV.14)
provided that (id ∗ T ∗ id)(x) = x, and y is one of the generators Ki , K̄i , Di , D̄i , Ei , Fi , J m . Suppose (∆ ⊗ 1)∆(x) =
P
P
x(1) ⊗ x(2) ⊗ x(3) . Then (∆ ⊗ 1)∆(xJ m ) = x(1) J m ⊗ x(2) J m ⊗ x(3) J m and hence
id ∗ T ∗ id(xJ m ) =
X
x(1) T (x(2) )x(3) J m = xJ m .
Similarly
id ∗ T ∗ id(xEi ) =
=
P
x(1) J m T (x(2) J m )x(3) Ei +
P
P
x(1) Ei T (x(2) Ki )x(3) Ki
P
x(1) T (x(2) )x(3) Ei − x(1) Ei K̄i T (x(2) )x(3) Ki +
P
x(1) Ei K̄i T (x(2) )x(3) Ki
P
x(1) J m T (x(2) Ei )x(3) Ki +
= xEi ,
if Ei is type zero. We can prove (IV.14) is true for other generators of U τ (G). So id ∗ T ∗ id(x) = x for any x ∈ U τ (G)
by induction.
Similarly, we can prove T ∗ id ∗ T (x) = T (x) for any x ∈ U τ (G). So T is a weak antipode of U τ (G), and U τ (G) is a
weak Hopf algebra.
Corollary IV.1 wUqτ (G) is a weak Hopf algebra if and only if m = 2. Moreover, if m = 2, then wUqτ (G) is a
noncommutative and noncocommutative weak Hopf algebra with the weak antipode T , but not a Hopf algebra.
Proof Since m = 2, J m−1 = J m = J. Thus U τ (G) = wUqτ (G). So it is a weak Hopf algebra. Suppose it is a Hopf
algebra with antipode S. Then J m = S(J m )J m = 1. On the other hand, since J m = J m−1 , J m−1 (J − 1) = 0 implies
that J = 1. This is impossible. So wUqτ (G) is not a Hopf algebra.
If wUqτ (G) is a weak Hopf algebra, then id ∗ T ∗ id(J) = J 3 = J. From this and J m = J m−1 , we can obtain J 2 = J.
Thus m = 2 by our assumption.
13
Corollary IV.2 U τ (G) is a noncommutative and noncocommutative weak Hopf algebra with the weak antipode T , but
not a Hopf algebra. Moreover, wUqτ (G)J m = U τ (G)J m is isomorphic to Uq0 (G) as Hopf algebras.
Proof It follows from Corollary III.1, Theorem IV.2.
Let H be a coalgebra. The set of group-like elements of H is denoted by G(H) in the next proposition.
Proposition IV.1 G(U τ (G)) = G(U τ (G)J m ) ∪ {1}.
Proof If g ∈ G(U τ (G)), then g = gJ m + g(1 − J m ). Let g1 = gJ m , g2 = g(1 − J m ). Then g ⊗ g = ∆(g) =
g1 ⊗ g1 + g1 ⊗ g2 + g2 ⊗ g1 + g2 ⊗ g2 . Since ∆(g1 ) = g1 ⊗ g1 is a group-like element, ∆(g2 ) = g1 ⊗ g2 + g2 ⊗ g1 + g2 ⊗ g2 .
So
(1 ⊗ ∆)∆(g2 ) = g1 ⊗ g1 ⊗ g2 + g1 ⊗ g2 ⊗ g1
+ g1 ⊗ g2 ⊗ g2 + g2 ⊗ g1 ⊗ g1
+ g2 ⊗ g1 ⊗ g2 + g2 ⊗ g2 ⊗ g1
+ g2 ⊗ g2 ⊗ g2 .
Then

 (T ∗ id ∗ T )(g2 ) = T (g2 )g2 T (g2 ) = T (g2 ),
 (id ∗ T ∗ id)(g ) = g T (g )g = g .
2
2
2
2
(IV.15)
2
Because U τ (G)(1 − J m ) is generated by Ei (1 − J m ), Fj (1 − J m ), 1 − J m and T (Ei (1 − J m )) = T (Fj (1 − J m )) = 0,
T (g2 ) = k(1 − J m ) for some k ∈ K. From (IV.15), we obtain the following:

 k 2 g2 = k 2 (1 − J m )2 g2 = k(1 − J m ),
 kg 2 (1 − J m ) = kg 2 = g .
2
2
2
If k = 0, then g = g1 ∈ G(U τ (G)). If k 6= 0, then g2 = k1 (1 − J m ). Thus
1
1
1
1
(1 ⊗ 1 − J m ⊗ J m ) = g1 ⊗ (1 − J m ) + (1 − J m ) ⊗ g1 + 2 (1 − J m ) ⊗ (1 − J m ).
k
k
k
k
Multiplying by k(J m ⊗ 1) on the both sides of the above equation, we get
J m ⊗ 1 − J m ⊗ J m = g1 ⊗ (1 − J m ).
Similarly, we have
1 ⊗ J m − J m ⊗ J m = (1 − J m ) ⊗ g1 .
Then
1 ⊗ 1 − J m ⊗ J m = J m ⊗ 1 + 1 ⊗ J m − 2J m ⊗ J m +
1
(1 − J m ) ⊗ (1 − J m ).
k
Hence
(1 − J m ) ⊗ (1 − J m ) =
1
(1 − J m ) ⊗ (1 − J m ).
k
(IV.16)
14
Consequently, k = 1. Notice that the set of group-like elements of U τ (G)J m is a monoid generated by Ki , K̄i , Di , D̄i
and J m , and the elements from this monoid are linearly independent over K. So we get g1 = J m from 1 ⊗ J m − J m ⊗
J m = (1 − J m ) ⊗ J m = (1 − J m ) ⊗ g1 . Hence g = 1.
Proposition IV.2 Suppose ϕ is an automorphism of the bi-algebra wUqτ (G). Then ϕ(J) = J and the restriction of
ϕ on w ( resp. w̄ )is an isomorphism of w (respect. w̄).
Proof From J m = J m−1 , we obtain ϕ(J)m = ϕ(J)m−1 . Thus ϕ(J)m J m = ϕ(J)m−1 J m is a group-like element
in wUqτ (G) ⊆ Uq (G). Hence ϕ(J)J m = J m . Suppose ϕ(J) = J m + x, where x = ϕ(J)(1 − J m ) ∈ w̄. Since
Pm−2
T ϕ(J) = ϕ(T (J)) = ϕ(J), T x = x. Consequently, x = i=0 ki (J i − J m ). Then
∆(ϕ(J)) = (J m + x) ⊗ (J m + x)
Pm−2
= J m ⊗ J m + i=0 ki (J i ⊗ J i )
Pm−2
− i=0 ki (J m ⊗ J m )
Pm−2
= J m ⊗ J m + i=0 ki (J i ⊗ J m )
Pm−2
Pm−2
Pm−2
+ i=0 ki (J m ⊗ J i ) + i=0 ki J i ⊗ j=0 kj J j
(IV.17)
From (IV.17), we obtain that ϕ(J) = J m + (J r − J m ) = J r for some 0 ≤ r ≤ m − 2. If r ≥ 2, then ϕ(J m−2 − J m−1 ) =
J r(m−2) − J r(m−1) = 0. This is contradict to the fact that ϕ is an isomorphism. So r = 1. Since ϕ(J) = J,
ϕ(w) = ϕ(wJ m ) = ϕ(w)J m = w. Consequently, the restriction of ϕ on w is a Hopf algebra isomorphism of w.
Similarly, we can prove the restriction of ϕ on w̄ is an algebra isomorphism of w̄.
Similarly to Ref. 14, we can prove the following:
Corollary IV.3 Suppose G is a semi-simple Lie algebra. Then the automorphism group of the bialgebra of wUqτ (G) is
the semi-direct product of N and H, where H is the group of diagram automorphism, and N is the group of diagonal
automorphism and it is a normal subgroup of the automorphism group of wUqτ (G).
Remark
If G is a semisimple Lie algebra, then U τ (G) is isomorphic to the weak quantum algebra defined by Yang
in Ref. [14].
V.
THE REPRESENTATIONS OF wUqτ (G)
In this section, we try to determine the irreducible representations of wUqτ (G). Suppose V is a simple module over
the bi-algebra wUqτ (G). Then V = J m V ⊕ (1 − J m )V . Since both J m V and (1 − J m )V are modules over wUqτ (G),
either V = J m V or V = (1 − J m )V .
If V = J m V , then Jv = v for any v ∈ V . Suppose v ∈ V satisfying Ki v = λi v, then λi 6= 0. In this case,
Ki K̄i Ki v = λ2i K̄i v = λi v. So K̄i v =
1
λi v.
If V = (1 − J m )V , then J m v = 0 v ∈ V , Ki v = Ki J m v = 0 and K̄i v = K̄i J m v = 0 for any v ∈ V .
By now we have completed the proof of the following proposition.
15
Proposition V.1 Let V be simple wUqτ (G)-module. Then either Jv = v for all v ∈ V , or J m v = 0 for any v ∈ V .
Suppose
 there exists an i ∈ I such that Ki v = λi v for some nonzero vector v. Then K̄i v = λ̄i v for λi , where
 λ−1 if λi 6= 0;
i
λ̄i =
Moreover, λi 6= 0 if and only if Jv = v.
0
if λi = 0.
Suppose V = J m V . Then V can be viewed as a module over wUqτ (G)/(1 − J) by Proposition V.1. Notice that
wUqτ (G)/(1−J) is isomorphic to Uq0 (G) by Corollary III.1(2). In this case, V has been studied by Kang9 . For example,
the limit of highest weight simple module is a highest weight simple module over the generalized Kac-Moody algebra G
with the same weight λ. Then this simple module is unique determined by its formally Borcherd-Kac-Weyl character
formula ( see Section I).
Suppose V = (1 − J m )V . Then J m V = 0 and Ki V = K̄i V = 0 for any i ∈ I by Proposition V.1. Similarly, we can
prove that Di V = D̄i V = 0 for all i ∈ I. Hence Ei Fj V = Fj Ei V for all i, j ∈ I. Moreover, V can be viewed as a
module over w̄. Recall that w̄ is generated by J r − J m (0 ≤ r ≤ m − 2), Ei (1 − J m ), and Fj (1 − J m ), where Ei , Fj
are type one. Hence Ei (1 − J m )Fj (1 − J m )V = Fj (1 − J m )Ei (1 − J m )V for all i, j ∈ I. In the following, we try to
determine the structure of V in some special case.
Proposition V.2 For any wUqτ (G) module V , if V = (1 − J m )V , then JV = 0.
Proof Consider the subalgebra B of w̄ generated by J i −J m for 0 ≤ r ≤ m−2. Since (J i −J m )(J j −J m ) = J r −J m ,
where r ≡ i + jmod(m − 2), B is isomorphic to K[Zm−2 ]. Because charK = 0, B is a semi-simple algebra. Hence V
is a semi-simple module over B. Let S be a simple B-module contained in V and v be a nonzero element of S. Then
there exists r such that J r v = 0 and J r−1 v 6= 0. So S is equal to a vector space spanned by v, Jv, · · · J r−1 v over K.
Since KJ r−1 v is a nonzero module of S, S = KJ r−1 v and JS = 0. Consequently JV = 0.
The above proposition also characterizes the simple w̄ module when all Ei , Fj are type zero. In general case, let
Xi = Ei (1 − J m ), Yi = Fi (1 − J m ). Then every simple module V over w̄ is a module over the algebra generated by
{Xi , Yj | for i ∈ I1 ,J ∈ I2 }, where I1 = {i ∈ I|Ei is type one}, I2 = {j ∈ I|Fj is type one}. The generators Xi , Yj
satisfy the following relation:
Xi Yj = Yj Xi ,

P1−aij
r=0
(−1)r 
1 − aij
r

P1−aij
r=0
(−1)r 
1 − aij
r

 Xi1−aij −r Xj Xir = 0
if
aii = 2, i 6= j,
if
aii = 2, i 6= j,
i
 Yi1−aij −r Yj Yir = 0
i
Xi Xj − Xj Xi = Yi Yj − Yj Yi = 0,
if
aij = 0.
This simple module V satisfies JV = 0. From the above discussion, we obtain the following result.
Corollary V.1 If aij = 0 for any i ∈ (I1 ∪ I2 ) ∩ I + , then every simple module over w̄ is isomorphic to w̄/M , where
M is a maximal ideal of w̄.
16
By Corollary V.1, the only simple over w̄ is K[x]/(p(x)) if |I1 ∪ I2 | = 1, where p(x) is an irreducible polynomial
in K[x]. Suppose K is an algebraically closed field. If aij = 0 for any i ∈ (I1 ∪ I2 ) ∩ I + , and |I1 ∪ I2 | = n, then
the simple module V over w̄ is isomorphic to K[Xi , Yj |i ∈ I1 , Jj ∈ I2 ]/({Xi − ai , Yj − bj |i ∈ I1 , j ∈ I2 }) for some
((ai )i∈I1 , (bj )j∈I2 ) ∈ Kn .
ACKNOWLEDGMENT
The author is sponsored by SRF for ROCS,SEM. He is also sponsored by the Scientific Research Foundation of
Zhejiang Provincial Education Department. The author would like to thank the referee for his/her suggestion and
help.
∗
Electronic address: wzx@zju.edu.cn
1
Aizawwa, N. and Isaac, P.S., Weak Hopf algebras corresponding to Uq [sln ], J. Math. Phys. 44, 5250-5267(2003).
2
Böhm,G.,Nil,F.,and Szlachányi,K. Weak Hopf algebras I,Integral theory and C ∗ -structure, J.Algebra,221(1998),385-438.
3
R.E. Borcherds, Generalised Kac-Moody algebras, J. Algebra 115(1988), 501-512.
4
R.E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math., 109(1992), 405-444.
5
R.E. Borcherds,Vertex algebras, Kac-Moody algebras and the monster,Proc. Nat. Acad. Sci. USA,83(1986),3068-3071.
6
Chin,W. and Musson,I.M., Corrigenda, the coradical filtration for quantized enveloping algebras, J. Lond.Math. Soc.
61(2000),319-320.
7
Hayashi,T., An algebra related to the fusion rules of Wess-Zumino-Witten models, Lett. Math. Phs. 22, 291-296(1991)
8
Jantzen, J.C., Lectures on Quantum Group (American Mathematical Society, Providence, RI, 1995) vol.6.
9
S.J.Kang, Quantum deformations of generalised Kac-Moody algebras and their modules, J. of Algebra 175(1995), 1041-1066.
10
K. Jeong, S.-J.Kang, M. Kashiwara, Crystal bases for quantum generalised Kac-Moody algebras and their modules, Proc.
London Math.Soc.(3)90(2205),395-438.
11
Li,F. and Duplij,S.,Weak Hopf algebras and singular solutions of quantum Yang-Baxter equation, Commun. Math.
Phys.225(2002),191-217.
12
Nichols,W.D. and Taft,E.J. The left antipodes od a left Hopf algebra, Contemp.Math. 13(Ametrica Mathematical Society,Providence,RI,1982).
13
Sweedler,M.E., Hopf Algebrss(Benjamin,New York,1960).
14
Yang, S. ,Weak Hopf algebras corresponding to Cartan matrices,J.Math.Phys.46(2005),1-18.
15
Yamanouchi,T.,Duality for generalized Kacalgebras and a characterization of finite groupoid algebras, J.algebra,163(1994),950.