A fundamental representation of quantum generalized Kac-Moody algebras with one imaginary simple root

Jiangrong CHEN1, Zhonghua ZHAO2
1 School of Statistics, Capital University of Economics and Business, Beijing 100070, China
2 Department of Mathematics and Computer Science, School of Science,
Beijing University of Chemical Technology, Beijing 100029, China
c
Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract We consider the Borcherds-Cartan matrix obtained from a
symmetrizable generalized Cartan matrix by adding one imaginary simple root.
We extend the result of Gebert and Teschner [Lett. Math. Phys., 1994, 31:
327–334] to the quantum case. Moreover, we give a connection between the
irreducible dominant representations of quantum Kac-Moody algebras and those
of quantum generalized Kac-Moody algebras. As the result, a large class of
irreducible dominant representations of quantum generalized Kac-Moody
algebras were obtained from representations of quantum Kac-Moody algebras
through tensor algebras.
Keywords Quantum generalized Kac-Moody algebra, tensor algebra,
fundamental representation
MSC 20G42, 81R10
1 Introduction
Quantum groups were introduced independently by Drinfeld [5] and Jimbo [8]
in 1985, which are certain families of Hopf algebras that are deformations of
universal enveloping algebras of Kac-Moody algebras. More precisely, let U(g)
be the universal enveloping algebra of a symmetrizable Kac-Moody algebra
g. Then, to each generic parameter q, we can associate a Hopf algebra Uq(g),
called the quantum group, whose structure tends to that of U(g) as q approaches
1. For the basic structure of Uq(g), see [4,14]. Lusztig [13] showed that the
integrable highest weight modules over Uq(g) can be deformed to those over
Received December 19, 2013; accepted April 9, 2015
Corresponding author: Zhonghua ZHAO, E-mail: zhaozh@mail.buct.edu.cn
1042 Jiangrong CHEN, Zhonghua ZHAO
U(g) in such a way that the dimensions of weight spaces are invariant under
the deformation.
In another direction, Borcherds was led in his study of the Moonshine and
the Monster group to consider a new class of infinite-dimensional Lie algebras,
now called the generalized Kac-Moody algebras [2,3]. The structure and the
representation theory of generalized Kac-Moody algebras are similar to those
of Kac-Moody algebras, and many basic facts about Kac-Moody algebras can be
extended to generalized Kac-Moody algebras. But there are some differences,
too. For example, generalized Kac-Moody algebras may have imaginary simple
roots with norms at most 0 whose multiplicity can be greater than 1. Also,
they may have infinitely many simple roots. For more Details, please refer to
[9–11].
Kang [12] constructed the quantum group Uq(g) associated a generalized
Kac-Moody algebra g. For simplicity, call Uq(g) the quantum generalized Kac-
Moody algebra. Moreover, he showed, for a generic q, Verma modules and
irreducible highest weight modules over the universal enveloping algebra U(g) of
g with dominant integral weights can be obtained by deforming from those over
Uq(g). These results were extended to generalized Kac-Moody superalgebras by
Benkart et al. [1].
In this paper, we only consider a special class of Borcherds-Cartan matrices
with only one imaginary simple root. As in [6], we prove one of fundamental
representations over those quantum generalized Kac-Moody algebras is
isomorphic to the tensor algebra of an irreducible representation over the
corresponding quantum Kac-Moody algebras. Moreover, by this isomorphism,
we further show that a large class of irreducible dominant representations of
quantum generalized Kac-Moody algebras can be induced from representations
of quantum Kac-Moody algebras through tensor algebras.
2 Preliminaries
Let I be a finite or countably infinite index set. A real square matrix A =
(aij)i,j∈I is called a Borcherds-Cartan matrix if it satisfies
(1) aii = 2 or aii 0 for all i ∈ I,
(2) aij 0 if i = j,
(3) aij ∈ Z if aii = 2,
(4) aij = 0 if and only if aji = 0.
We say that an index i is real if aii = 2 and imaginary if aii 0. We denote
by
Ire = {i ∈ I | aii = 2}, Iim = {i ∈ I | aii 0},
the set of real indices and the set of imaginary indices, respectively.
In this paper, we assume that all the entries of A are integers and the
diagonal entries are even. Furthermore, we assume that A is symmetrizable;
A fundamental representation of quantum generalized Kac-Moody algebras 1043
that is, there is a diagonal matrix
D = diag(di ∈ Z>0 | i ∈ I)
such that DA is symmetric. In addition, we assume all entries of D are coprime
integers.
A Borcherds-Cartan datum (A, P∨, P,Π∨, Π) consists of
(i) a Borcherds-Cartan matrix A,
(ii) a free abelian group P∨, called the dual weight lattice,
(iii) Π∨ = {α∨
i
}i∈I , the set of simple coroots,
(iv) Π = {αi}i∈I ⊂ P = HomZ(P∨,Z), the set of simple roots,
satisfying the following properties:
(a) αi(α∨
j ) = aij for i, j ∈ I,
(b) Π is linearly independent,
(c) for any i ∈ I, there exists Λi ∈ P such that Λi(α∨
j ) = δij for i, j ∈ I.
The Q-vector space h = Q ⊗Z P∨ is called the Cartan subalgebra and we
call Λi the fundamental weights. The weight lattice P is also defined to be
P = {λ ∈ h
∗ | λ(P
∨) ⊂ Z} = HomZ(P
∨
,Z).
Each symmetrizable Borcherds-Cartan matrix gives rise to a Borcherds-
Cartan datum as follows. Take
P
∨ =

i∈I
Zα
∨
i

⊕

i∈I
Zsi

,
and define αi and Λi by
αi(α
∨
j ) = aji, αi(sj) = δij ,
Λi(α
∨
j ) = δij , Λi(sj) = 0.
We also define ωi by
ωi(α
∨
j ) = 0, ωi(sj) = δij .
Then
P =

i∈I
ZΛi

⊕

i∈I
Zωi

.
We denote by P+ the set {λ ∈ P | λ(α∨
i ) 0 for every i ∈ I} of dominant
integral weights. The free abelian group
R :=

i∈I
Zαi
is called the root lattice. We set
R+ =

i∈I
Z 0αi, R− = −R+.
1044 Jiangrong CHEN, Zhonghua ZHAO
For α ∈ R+, we can write
α =
r
k=1
αik, i1, i2, . . . , ir ∈ I.
Let q be an indeterminate and set qi = qdi (i ∈ I). For an integer n ∈ Z, i ∈
Ire, define
[n] = qn − q−n
q − q−1 , [n]! =
n
k=1
[k],

m
n

=
[m]!
[n]![m − n]! .
[n]i = qn
i
− q
−n
i
qi − q
−1
i
, [n]i! =
n
k=1
[k]i,

m
n

i
=
[m]i!
[n]i![m − n]i! .
Definition 1 The quantum generalized Kac-Moody algebra Uq(g) associated
with a Borcherds-Cartan datum (A, P∨, P,Π∨, Π) is the associative algebra over
Q(q) with unit 1 generated by Ei, Fi (i ∈ I), and Kh (h ∈ P∨) subject to the
following defining relations:
KhKh = Kh+h , KhK−h = 1, h,h
∈ P
∨
,
KhEiK−h = qαi(h)Ei, KhFiK−h = q
−αi(h)Fi,
EiFj − FjEi = δi,j
K
i − K
−1
i
qi − q
−1
i
, Ki = Kdiα∨
i
,
1 −aij
r=0
(−1)r

1 − aij
r

i
E1−aij−r
i EjEr
i = 0, aii = 2, i = j,
1 −aij
r=0
(−1)r

1 − aij
r

i
F1−aij−r
i FjFr
i = 0, aii = 2, i = j,
EiEj − EjEi = 0, FiFj − FjFi = 0, aij = 0.
The quantum generalized Kac-Moody algebra Uq(g) has a Hopf algebra
structure with the comultiplication Δ, the counit ε, and the antipode S defined
by
Δ(Kh) = Kh ⊗Kh, Δ(Ei) = Ei ⊗ K
−1
i +1⊗Ei, Δ(Fi) = Fi ⊗1+ Ki ⊗Fi,
ε(Kh) = 1, ε(Ei) = ε(Fi) = 0,
S(Kh) = K−h, S(Ei) = −Ei Ki, S(Fi) = − K
−1
i Fi,
for h ∈ P∨ and i ∈ I ([1,7,12]).
Let U+q (g) and U−
q (g) be the Q(q)-subalgebras of Uq(g) generated by the
elements Ei and Fi, respectively, for i ∈ I, and let U0q
(g) be the Q(q)-subalgebra
A fundamental representation of quantum generalized Kac-Moody algebras 1045
of Uq(g) generated by Kh (h ∈ P∨), which is isomorphic to Q(q)[K
±1
i ,H
±1
i ].
Here, we set
K
±1
i = K±α∨
i
, H
±1
i = K±si .
Then we have the triangular decomposition ([1,7,12])
Uq(g) ∼=
U−
q (g) ⊗ U0q
(g) ⊗ U+q (g).
A Uq(g)-module V is called a weight module if it admits a weight space
decomposition
V =

μ∈P
Vμ,
where
Vμ := {v ∈ V | Khv = qμ(h)v, ∀ h ∈ P
∨}.
We call
wt(V ) := {μ ∈ P | Vμ = 0}
the set of weights of V.
Let O be the category consisting of weight modules V with finitedimensional
weight spaces such that
wt(V ) ⊂

s
j=1
(λj − R+)
for finitely many λ1, λ2, . . . , λs ∈ P.
The most interesting examples of Uq(g)-modules are the highest weight
modules in the category O defined below. A module V over Uq(g) is called
a highest weight module with highest weight λ ∈ P if there exists a non-zero
vector v ∈ V (called a highest weight vector ) such that
(i) V = Uq(g)v,
(ii) v ∈ Vλ,
(iii) Eiv = 0 for every i ∈ I.
Let J(λ) denote the left ideal of Uq(g) generated by Ei,Kh−qλ(h) (i ∈ I, h ∈
P∨), and set M(λ) = Uq(g)/J(λ). Then, via left multiplication, M(λ) becomes
a Uq(g)-module, which is called the Verma module. The basic properties of
Verma modules are summarized in the following propositions.
Proposition 1 [1,7,12] (a) The Verma module M(λ) is a highest weight
Uq(g)-module with highest weight λ and highest weight vector vλ := 1 + J(λ).
(b) The Verma module M(λ) is a free U−
q (g)-module generated by vλ.
(c) Every highest weight Uq(g)-module with highest weight λ is a quotient
of M(λ).
(d) The Verma module M(λ) contains a unique maximal submodule R(λ),
the corresponding quotient L(λ) = M(λ)/R(λ) is an irreducible highest weight
Uq(g)-module with highest weight λ.
1046 Jiangrong CHEN, Zhonghua ZHAO
Proposition 2 [1,7,12] Let λ ∈ P+ be a dominant integral weight.
(a) The highest weight vector vλ of L(λ) satisfies the following relations:

Fλ(h)+1
i vλ = 0, aii = 2,
Fivλ = 0, λ(α∨
i ) = 0.
(2.1)
(b) Conversely, let V be a highest weight Uq(g)-module with a highest
weight λ ∈ P+ and a highest weight vector v. If v satisfies the relations in (2.1),
then V is isomorphic to L(λ).
Proposition 3 [7] Let μ be a weight of L(λ) with λ ∈ P+, and let i ∈ Iim.
Then
(a) μ(α∨
i ) ∈ Z 0;
(b) if μ(α∨
i ) = 0, then L(λ)μ−αi = 0;
(c) if μ(α∨
i ) = 0, then Fi(L(λ)μ) = 0;
(d) if μ(α∨
i ) −aii, then Ei(L(λ)μ) = 0.
3 Mainresult
In this section, we consider the symmetric Borcherds-Cartan matrix of the
following shape:
A = (aij) =
⎛
⎜⎜⎜⎝
a00 a01 · · · a0n
a10
...
A
an0
⎞
⎟⎟⎟⎠
I×I
where a00 0, I = {0, 1, 2, . . . , n}, and A = (aij)n×n is a generalized Cartan
matrix.
Denote by Uq(A) (resp. Uq( A)) the corresponding quantum Kac-Moody
algebra (resp. quantum generalized Kac-Moody algebra). Take P∨ and P∨ as
the free abelian groups by
P
∨ =
n
i=1
Zα
∨
i

⊕
n
i=1
Zsi

⊂ P
∨ =
n
i=0
Zα
∨
i

⊕
n
i=0
Zsi

.
And let
P =
n
i=1
Zωi

⊕
n
i=1
ZΛi

, P =
n
i=0
Z ωi

⊕
n
i=0
Z Λi

,
where ωi,Λi, ωi, and Λi are defined as in Section 2. We may consider P as a
subset of P by the map ι : ωi → ωi, Λi → Λi. Hereafter, we will not distinguish
A fundamental representation of quantum generalized Kac-Moody algebras 1047
ωi,Λi and ωi, Λi for 1 i n. We also write Λ0, ω0 as long as there is no peril
of confusion.
For λ ∈ P+, let L(λ) be the irreducible highest weight Uq(A)-module with
highest weight λ and highest weight vector vλ. For λ ∈ P+, let L(λ) be the
irreducible highest weight Uq( A)-module with highest weight λ and highest
weight vector vλ.
Denote by T(L(λ)) the tensor algebra over L(λ), that is,
T(L(λ)) :=
+∞
n=0
Tn(L(λ)),
where
T0(L(λ)) = Q(q)1, Tn(L(λ)) = L(λ) ⊗ L(λ)⊗· · ·⊗L(λ)

n
, n 1.
The following theorem is our main result.
Theorem 1 Keep the notations above. Let L(λ) be the irreducible highest
weight Uq(A)-module with highest weight λ defined by
λ(α
∨
i ) = −a0i, λ(si) = 0, 1 i n,
and with highest-weight vector vλ. Then the tensor algebra T(L(λ)) over L(λ)
is a Uq( A)-module isomorphic to the highest-weight module L(Λ0) of Uq( A),
where Λ0 ∈ P+ is defined by Λ0(α∨
i ) = δ0i, 0 i n.
Proof For convention, in the following, the indices i, j, k run from 1 to n unless
stated otherwise. The generators Ei, Fi,K
±1
i ,H
±1
i of Uq(A) act trivially on the
1-dimensional subspace Q(q)1 and as highest weight representation on L(λ). We
extend those actions to the tensor algebra T(L(λ)) by the comultiplication in
the Hopf algebra Uq(A). First, the generators K0,H0 act diagonally as follows:
K0(1) = q, K0(vλ) = q1−a00vλ, H0(1) = 1, H0(vλ) = q
−1vλ,
K0(Fkϕ) = q
−a0kFkK0(ϕ), H0(Fkϕ) = FkH0(ϕ), ϕ∈ L(λ),
K0(Φ ⊗ Ψ) = q−1K0(Φ) ⊗K0(Ψ),
H0(Φ ⊗ Ψ) = H0(Φ) ⊗ H0(Ψ),
Φ,Ψ ∈ T(L(λ)),
F0(1) = vλ, F0(Ψ) = vλ ⊗ Ψ, Ψ ∈ T(L(λ)),
E0(1) = 0, E0(vλ) = 1.
Since
Tn(L(λ)) = U 0
q (A)(vλ ⊗ Tn−1(L(λ))), ∀ n 1,
1048 Jiangrong CHEN, Zhonghua ZHAO
we can inductively define the action as follows: for Ψ ∈ Tn(L(λ)), n 0,
E0(Fi(Ψ)) = Fi(E0(Ψ)), E0(vλ ⊗ Ψ) = K0 − K
−1
0
q − q−1 (Ψ) + vλ ⊗ E0(Ψ).
First, we observe that the generators K0 and H0 are defined to act diagonally
on the tensor algebra. So, for h ∈ P∨, the generators Kh commute with each
other. Second, all the relations involving the quantum Kac-Moody generators
Ei, Fi,K
±1
i ,H
±1
i are valid by assumption. Hence, we need to check the following
relations on T(L(λ)) :
E0Fi − FiE0 = 0, EiF0 − F0Ei = 0, E0F0 − F0E0 = K0 − K
−1
0
q − q−1 ,
K0E0K
−1
0 = qa00F0, K0F0K
−1
0 = q
−a00F0,
KiE0K
−1
i = qa0iE0, KiF0K
−1
i = q
−a0iF0,
K0EiK
−1
0 = qa0iEi, K0FiK
−1
0 = q
−a0iFi,
H0EiH
−1
0 = Ei, H0FiH
−1
0 = Fi,
H0E0H
−1
0 = qE0, H0F0H
−1
0 = q
−1F0.
Take ψ ∈ L(λ), Fk ∈ U−
q (A) and Ψ ∈ Tn(L(λ)), Φ ∈ Tm(L(λ)). Then the
following first six relations can be checked immediately on the tensor algebra:
(E0F0 − F0E0)(Ψ) = K0 − K
−1
0
q − q−1 (Ψ), (E0Fi − FiE0)(Ψ) = 0,
(EiF0 − F0Ei)(Ψ) = Ei(vλ ⊗ Ψ) − vλ ⊗ Ei(Ψ) = 0,
KiF0K
−1
i (Ψ) = Ki(vλ ⊗ K
−1
i (Ψ)) = q
−a0iF0(Ψ),
K0F0K
−1
0 (Ψ) = K0(vλ ⊗ K
−1
0 (Ψ)) = q
−1K0vλ ⊗Ψ = q
−a00F0(Ψ),
H0F0H
−1
0 (Ψ) = H0(vλ ⊗ H
−1
0 (Ψ)) = H0vλ ⊗Ψ = q
−1F0(Ψ).
Finally, we check the remaining eight relations on T(L(λ)) :
K0FiK
−1
0 (1) = q
−1K0(Fi1) = q
−1−a0iFiK0(1) = q
−a0iFi(1),
K0FiK
−1
0 (vλ) = K0Fi(qa00−1vλ) = qa00−a0i−1FiK0vλ = q
−a0i(Fivλ),
K0FiK
−1
0 (Fkψ) = K0Fiqa0kFkK
−1
0 ψ
= qa0kK0(FiFkK
−1
0 ψ)
= qa0k−a0iFiK0(FkK
−1
0 ψ)
= qa0k−a0iFi(K0FkK
−1
0 ψ)
= q
−a0iFi(Fkψ) (by induction),
A fundamental representation of quantum generalized Kac-Moody algebras 1049
K0FiK
−1
0 (Ψ ⊗ Φ)
= qK0Δ(Fi)(K
−1
0 Ψ ⊗ K
−1
0 Φ)
= qK0(FiK
−1
0 Ψ ⊗K
−1
0 Φ+KiK
−1
0 Ψ ⊗ FiK
−1
0 Φ)
= K0FiK
−1
0 Ψ ⊗Ψ+K0KiK
−1
0 Ψ ⊗ K0FiK
−1
0 Φ (by induction)
= q
−a0i(FiΨ ⊗Φ+KiΨ ⊗ FiΦ)
= q
−a0iFi(Ψ ⊗ Φ).
Similarly, the equality H0FiH
−1
0 = Fi holds on T(L(λ)).
K0E0K
−1
0 (1) = 0 = qa00E0(1),
K0E0K
−1
0 (vλ) = K0E0(qa00−1vλ) = qa00−1K0(1) = qa00E0(vλ),
K0E0K
−1
0 (Fkψ) = K0E0(qa0kFkK
−1
0 (ψ))
= qa0kK0(FkE0K
−1
0 (ψ))
= FkK0(E0K
−1
0 (ψ))
= Fk(K0E0K
−1
0 (ψ)) (by induction)
= Fkqa00E0(ψ)
= qa00E0(Fkψ),
K0E0K
−1
0 (vλ ⊗ Ψ)
= qa00K0E0(vλ ⊗ K
−1
0 Ψ)
= qa00K0
K0 − K
−1
0
q − q−1 K
−1
0 Ψ+vλ ⊗ E0K
−1
0 Ψ

= qa00
K0 − K
−1
0
q − q−1 Ψ+q
−1K0vλ ⊗ K0E0K
−1
0 Ψ

(by induction)
= qa00
K0 − K
−1
0
q − q−1 Ψ+vλ ⊗ E0Ψ

= qa00E0(vλ ⊗ Ψ).
From the proof of the equality of K0FiK
−1
0 = q−a0iFi acting on T(L(λ)),
we can deduce that for any Ψ ∈ Tn(L(λ)),
K0Fi(Ψ) = q
−a0iFiK0(Ψ),
K0E0K
−1
0 (Fkvλ ⊗ Ψ)
= K0E0K
−1
0 (Fk(vλ ⊗ Ψ) − Kkvλ ⊗ FkΨ)
= K0E0K
−1
0 Fk(vλ ⊗ Ψ) − K0E0K
−1
0 (Kkvλ ⊗ FkΨ)
= FkK0E0K
−1
0 (vλ ⊗ Ψ) − K0E0K
−1
0 (Kkvλ ⊗ FkΨ)
= qa00FkE0(vλ ⊗ Ψ) − qa00E0(Kkvλ ⊗ FkΨ) (by induction)
= qa00E0(Fkvλ ⊗ Ψ).
1050 Jiangrong CHEN, Zhonghua ZHAO
Similarly, the equality H0E0H
−1
0 = qE0 holds on T(L(λ)).
KiE0K
−1
i (1) = 0 = qa0iE0(1),
KiE0K
−1
i (vλ) = KiE0(qa0ivλ) = qa0iKi(1) = qa0iE0(vλ),
KiE0K
−1
i (Fkψ) = KiE0(K
−1
i Fkψ)
= KiE0qaki(FkK
−1
i ψ)
= qakiKiFkE0K
−1
i ψ
= FkKiE0K
−1
i ψ
= Fkqa0iE0ψ (by induction)
= qa0iE0(Fkψ),
KiE0K
−1
i (vλ ⊗ Ψ) = qa0iKiE0(vλ ⊗ K
−1
i Ψ)
= qa0iKi
K0 − K
−1
0
q − q−1 K
−1
i Ψ+vλ ⊗ E0K
−1
i Ψ

= qa0i
Ki − K
−1
i
q − q−1 Ψ+Kivλ ⊗ KiE0K
−1
i Ψ

= qa0iE0(vλ ⊗ Ψ) (by induction),
KiE0K
−1
i (Fkvλ ⊗ Ψ)
= KiE0K
−1
i (Fk(vλ ⊗ Ψ) − Kkvλ ⊗ FkΨ)
= FkKiE0K
−1
i (vλ ⊗ Ψ) − KiE0(K
−1
i Kkvλ ⊗ K
−1
i FkΨ)
= qa0iE0Fk(vλ ⊗ Ψ) − qa0i−a0kKiE0(vλ ⊗ K
−1
i FkΨ)
= qa0iE0Fk(vλ ⊗ Ψ) − qa0i−a0kKi
K0 − K
−1
0
q − q−1 K
−1
i FkΨ+vλ ⊗ E0K
−1
i FkΨ

= qa0iE0Fk(vλ ⊗ Ψ) − qa0i−a0k
K0 − K
−1
0
q − q−1 FkΨ+Kivλ ⊗ KiE0K
−1
i FkΨ

(by induction)
= qa0iE0Fk(vλ ⊗ Ψ) − qa0i−a0kE0(vλ ⊗ FkΨ)
= qa0iE0(Fkvλ ⊗ Ψ),
K0EiK
−1
0 (1) = 0 = qa0iEi(1),
K0EiK
−1
0 (vλ) = K0(Eiqa00−1vλ) = 0 = qa0iEivλ,
K0EiK
−1
0 (Fkψ) = K0Eiqa0kFkK
−1
0 ψ = qa0kK0EiFkK
−1
0 (ψ).
If k = i, then
qa0kK0EiFkK
−1
0 (ψ) = qa0kK0FkEiK
−1
0 ψ
= qa0k−a0kFkK0EiK
−1
0 ψ
= qa0iEi(Fkψ).
A fundamental representation of quantum generalized Kac-Moody algebras 1051
If k = i, then
qa0kK0EiFkK
−1
0 (ψ) = qa0kK0EkFkK
−1
0 ψ
= qa0kK0

FkEk − Kk − K
−1
k
q − q−1

K
−1
0 ψ
= FkK0EkK
−1
0 ψ − qa0kK0
Kk − K
−1
k
q − q−1 K
−1
0 ψ
= qa0k

FkEk − Kk − K
−1
k
q − q−1

(ψ)
= qa0kEk(Fkψ),
K0EiK
−1
0 (Ψ ⊗ Φ) = qK0Δ(Ei)(K
−1
0 Ψ ⊗ K
−1
0 Φ)
= qK0(EiK
−1
0 Ψ ⊗K
−1
i K
−1
0 Φ+K
−1
0 Ψ ⊗ EiK
−1
0 Φ)
= K0EiK
−1
0 Ψ ⊗ K0K
−1
i K
−1
0 Ψ+K0K
−1
0 Ψ ⊗ K0EiK
−1
0 Φ
= qa0i(EiΨ ⊗ K
−1
i Φ+Ψ⊗ EiΦ)
= qa0iEi(Ψ ⊗ Φ) (by induction).
Similarly, the equality H0EiH
−1
0 = Ei also holds on T(L(λ)).
Having proved that T(L(λ)) as a Uq( A)-module, now we need to prove that
T(L(λ)) is indeed isomorphic to the irreducible hightest weight module L(Λ0).
Suppose that g is the generalized Kac-Moody algebra associated with the
Borcherds-Cartan matrix A. Then we have the triangular decomposition
g
= n− ⊕ h ⊕ n+.
Here, n− is the subalgebra obtained from the quotient of the free algebra ˆn−
generated by f0, f1, . . . , fn by the ideal generated by (adfi)1−aij fj if aii = 2, j =
i and fifj − fjfi if aij = 0.
Denote the highest weight vector of L(Λ0) by v0. Define a linear map
ρ
: Uq(ˆn−)v0 → T(L(λ))
by
ρ
(Fi1Fi2
· · · Finv0) = Fi1Fi2
· · · Fin (1),
where i1, i2, . . . , in ∈ {0, 1, . . . , n}. To prove that ρ induces a well-defined
Uq( A)-module homomorphism
ρ
: Uq( n−)v0 → T(L(λ)),
we need to check the Serre relations are valid. That is, for 1 i n, 0 j
n, j = i, we need to prove
ρ

b= 1−aij
n=0
(−1)n

b
n

Fn
i FjFb−n
i v0

= 0.
1052 Jiangrong CHEN, Zhonghua ZHAO
We only consider j = 0. By definition, it suffices to show
ρ

b
n=0
(−1)n

b
n

Fn
i F0Fb−n
i v0

= 0.
Since Fi (1 i n) acts trivially on 1, we have
ρ

b
n=0
(−1)n

b
n

Fn
i F0Fb−n
i v0

= (−1)1−ai0F1−ai0
i F0(1).
By the action of F0, we have
F1−ai0
i F0(1) = F1−ai0
i (vλ) = F1−ai0
i vλ = F
1+λ(α∨
i )
i vλ = 0.
So we obtain
ρ
: Uq( n−)v0 → T(L(λ)).
Since L(Λ0) is obtained from the Verma module M(Λ0) modulo the subspace
generated by the primitive vectors F
Λ0(αi)+1
i v0 (1 i n), we have
ρ
(F
Λ0(α∨
i )+1
i v0) = F
Λ0(α∨
i )+1
i (1) = 0.
Finally, we obtain
ρ: L(Λ0) → T(L(λ)),
which is injective since L(Λ0) is a simple Uq( A)-module. Note that any element
of Tn(L(λ)) is of the form
u1vλ ⊗ u2vλ ⊗· · ·⊗unvλ,
where
ui = Fi,1Fi,2 · · · Fi,ti, Fi,1, Fi,2, . . . , Fi,ti
∈ {F0, F1, . . . , Fn}.
First, we take
ρ(Fn,1Fn,2 · · · Fn,tnF0v0) = Fn,1Fn,2 · · · Fn,tnvλ = unvλ.
To obtain
un−1vλ ⊗ unvλ = Fn−1,1Fn−1,2 · · · Fn−1,tn−1vλ ⊗ unvλ,
we take
ρ

Fn−1,tn−1F0Fn,1Fn,2 · · · Fn,tnF0 − q
λ(α∨
n−1,tn−1
)
F0Fn,1Fn,2 · · · Fn,tnF0

(v0)
= Fn−1,tn−1vλ ⊗ unvλ.
A fundamental representation of quantum generalized Kac-Moody algebras 1053
Inductively, we can find an element u ∈ L(Λ0) such that
ρ(u) = u1vλ ⊗· · ·⊗unvλ,
and ρ is surjective. This completes the proof.
Remark 1 Theorem 1 can be generalized to the symmetrizable Borcherds-
Cartan matrices.
Taking
A =
⎛
⎜⎜⎜⎝
0 0 · · · 0
0...
A
0
⎞
⎟⎟⎟⎠
I×I
,
we will show that certain irreducible representations of Uq( A) can be deduced
from the irreducible representations of Uq(A).
Since
Uq( A) ∼=
Uq(A) ⊗
Q(q) [E0, F0],
we can view Uq(A) as a subalgebra of Uq( A). Given a Uq(A)-module M, we
write Ind AA
M (or IndM) for the induced module Uq( A)⊗
Uq(A)M. Thus, we may
consider Ind as a functor from the category of Uq(A)-modules to the category
of Uq( A)-modules. It is known that this functor is left adjoint to the restriction
functor Res AA
(or Res) going in the other direction.
Corollary 1 Keep the notations above. Then we have the following
statements.
(1) For any μ ∈ P+, define an action of Uq( A) on L(μ) as follows:
E0(Fi1Fi2
· · · Fir vμ) = F0(Fi1Fi2
· · · Firvμ) = 0,
K0(Fi1Fi2
· · · Fir vμ) = H0(Fi1Fi2
· · · Firvμ) = Fi1Fi2
· · · Fir vμ,
for any Fi1, Fi2, . . . , Fir
∈ U−
q ( A). Then L(μ), as a Uq( A)-module, is
isomorphic to L(μ).
(2) For any μ ∈ P+, there is a unique Uq( A)-module monomorphism
φ: L( Λ0 + μ) → L (Λ0) ⊗ L (μ) ∼→
T(L(λ0)) ⊗ L(μ),
where λ0 ∈ P+ satisfies λ0(α∨
i ) = 0 (1 i n) and T(L(λ0)) is the
corresponding tensor algebra. Moreover, for any
u = Fi1Fi2
· · · Fir vΛ0+μ ∈ L( Λ0 + μ),
we have
φ(u) =

(δ1,δ2,...,δr)∈{0,1}r
Kδ1
i1
F1−δ1
i1
Kδ2
i2
F1−δ2
i2
· · ·Kδr
ir F1−δr
ir (1) ⊗ Fδ1
i1
Fδ2
i2
· · · Fδr
ir vμ.
1054 Jiangrong CHEN, Zhonghua ZHAO
(3) For λ ∈ P+, a0 ∈ Z 0, b0 ∈ Z, we have
Res L(λ + a0Λ0 + b0ω0) ∼=
⎧⎪⎨
⎪⎩
L(λ), a0 = 0,
+∞
0
L(λ), a0 ∈ Z 1.
Proof (1) Since all the other relations can be checked trivially on L(λ), we
only need to check the following two relations on L(μ):
K0EiK
−1
0 = Ei, K0FiK
−1
0 = Fi.
It is easy to check that this indeed defines a Uq( A)-module structure on L(μ).
Since
K0vμ = vμ = qμ(α∨
0 )vμ, Eivμ = 0, i ∈ I,
there is a unique Uq( A)-module homomorphism
θ : L(μ) → L(μ),
vμ
→ vμ.
Obviously, it is surjective. Since L(μ) is irreducible, it is also injective. Thus,
θ is an isomorphism.
(2) For any h ∈ P∨ and i ∈ I, we have
Kh( vΛ0
⊗ vμ) = q(Λ0+μ)(h) vΛ0
⊗ vμ, Ei( vΛ0
⊗ vμ) = 0.
There is a unique Uq( A)-module homomorphism
π : L( Λ0 + μ) → L (Λ0) ⊗ L (μ),
vΛ0+μ
→ vΛ0
⊗ vμ.
Take
φ = ρ ◦ π : L( Λ0 + μ) → T(L(λ0)) ⊗ L(μ).
Then the result follows.
(3) Denote by v λ the highest weight vector of L(λ + a 0Λ0 + b0ω0). Then
L(λ + a 0Λ0 + b0ω0) = Uq( A)v λ.
Note that all the elements of L(λ + a 0Λ0 + b0ω0) are of the form Fi1Fi2
· · · Firv λ,
where i1, i2, . . . , ir ∈ {0, 1, 2, . . . , n}. Since a0i = 0, we have
Fi1Fi2
· · · Fir v λ = Fj1Fj2
· · · FjtFm
0 v λ
A fundamental representation of quantum generalized Kac-Moody algebras 1055
for some m ∈ Z 0, j1, j2, . . . , jt ∈ {1, 2, . . . , n}. If
a0 = 0, (λ + a0Λ0 + b0ω0)(α
∨
0) = 0,
then we have
F0v λ = 0.
In this case, we have v λ
∈ (ResL(λ + a 0Λ0 + b0ω0))λ and all the non-zero
elements of L(λ + a 0Λ0 + b0ω0) must be of the form Fi1Fi2
· · · Firv λ, where
i1, i2, . . . , ir ∈ {1, 2, . . . , n}. So we have a non-zero map
θ : M(λ) → ResL(λ + a 0Λ0 + b0ω0),
vλ
→ v λ.
It is easy to show that θ is surjective as Uq(A)-modules and
L(λ) ∼=
M(λ)/ ker θ
∼=
ResL(λ + a 0Λ0 + b0ω0).
If
a0 = 0, (λ + a0Λ0 + b0ω0)(α
∨
0) = a0,
then the non-empty set {v λ , F0v λ , F2
0 v λ , . . .} is infinite and
(Res L(λ + a0Λ0 + b0ω0))λ = {v λ, F0v λ , F2
0 v λ , . . .}.
There is a non-zero map ψ defined by
ψ: ⊕+∞
0 M(λ) → ResL(λ + a 0Λ0 + b0ω0),
(vλ, vλ, . . .)
→ v λ + F0v λ + F2
0 v λ + · · · .
It is easy to check that ψ is surjective as Uq(A)-modules and
⊕+∞
0 M(λ)/ ker ψ
∼=
⊕+∞
0 L(λ) ∼=
ResL(λ + a 0Λ0 + b0ω0).
Remark 2 Corollary 1 (1) can be generalized to those A obtained from A by
adding several imaginary simple roots with all the added entries being zero.
Acknowledgements The authors would like to thank the referees for many helpful
suggestions and express their sincere gratitude to Professor Bangming Deng for many
valuable discussion. The second author (Zhao) was partially supported by the National
Natural Science Foundation of China (Grant No. 11226063) and the Fundamental Research
Funds for the Central Universities.
1056 Jiangrong CHEN, Zhonghua ZHAO
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