Some Representations of Nongraded Divergence-Free Lie Algebras

0.41 MB·English

by  Ling Chen

Some Representations of Nongraded
Divergence-Free Lie Algebras ∗
Ling CHEN †
Beijing International Center for Mathematical Research, Peking University,
2
1 Beijing 100871, P. R. China
0
2

n
a
J
Abstract
5
1
Divergence-free Lie algebras are originated from the Lie algebras of volume-

preserving transformation groups. Xu constructed a certain nongraded generaliza-
]
T
tion, which may not contain any toral Cartan subalgebra. In this paper, we give a
R
complete classification of the generalized weight modules over these algebras with
.
h weight multiplicities less than or equal to one.
t
a Keywords: divergence-free Lie algebra, Cartan type, irreducible module, gener-
m alized weight module, classification.
[

1
1 Introduction
v
2
1
The Liealgebrasof volume-preserving transformationgroupsconsist ofdivergence-free
1
3 first-order differential operators. They are often called the special Lie algebras of Cartan
.
1 type, or divergence-free Lie algebras (see [23]). These algebras play important roles in
0
geometry, Lie groups and dynamics. Various graded generalizations of them were studied
2
1 byKac[5], Osborn[14], DokovicandZhao[4], andZhao[29]. BergenandPassman[1]gave
:
v a certain characterization of graded divergence-free Lie algebras. Kac also investigated
i suppersymmetric graded generalizations of them [6,7]. Motivated from his classification
X
of quadratic conformal algebras corresponding to certain Hamiltonian pairs in integrable
r
a
systems (cf. [27,28]), Xu [26] introduced nongraded generalizations of the Lie algebras of
Cartan type and determined their simplicity. An important feature of Xu’s algebras is
that they do not contain any toral Cartan subalgebras in generic case. The aim of this
work is to study certain representations of Xu’s nongraded divergence-free Lie algebras,
whose isomorphism classes were determined by Su and Xu [23].
The representations of the Lie algebras of Cartan type were vastly studied. Shen
[18–20] studied mixed product of graded modules over graded Lie algebras of Cartan type,
and obtained certain irreducible modules over a field with characteristic p. Penkov and
Serganova[15]gaveanexplicit descriptionofthesupportofanarbitraryirreducibleweight
module of the classical Witt algebra, as well as its subalgebra with constant divergence.
Rao [16,17] investigated the irreducibility of the weight modules over the derivation Lie
algebra of the algebra of Laurent polynomials virtually constructed by Shen [18]. Zhao
[31] determined the module structure of Shen’s mixed product over Xu’s nongraded Lie
algebras of Witt type (cf. [26]). Moreover, she [33] obtained a composition series for a
∗2000 Mathematics Subject Classification. Primary 17B10, 17B65.
†E-mail: [email protected](L. Chen)
family of modules with parameters over Xu’s nongraded Hamiltonian Lie algebras (cf.
[26]). These modules are constructed from finite-dimensional multiplicity-free irreducible
modules of symplectic Lie algebras by Shen’s mixed product.
Starting from Kac’s conjecture [8,9] on irreducible representations over the Virasoro
algebra (which can also be viewed as a central extension of the rank one classical Witt
algebra), Kaplansky [10,11] and Santharoubane [11] gave classifications of multiplicity-
one representations over classical Virasoro algebras. Later, Mathieu [12] classified the
Harish-Chandra modules over the Virasoro algebras, which confirms Kac’s conjecture.
Moreover, Su [21,22] generalized Kaplansky and Santharoubane’s result to multiplicity-
one modules over high rank Virasoro algebras and super-Virasoro algebras. Based on
their classifications of multiplicity-one representations over generalized Virasoro algebras
in [25], Zhao [30] classified the multiplicity-one representations over graded generalized
Witt algebras. Su and Zhou [24] then generalized Zhao’s result [30] to generalized weight
modules over the nongraded Witt algebras introduced by Xu [26].
Motivated by Kaplansky and Santharoubane’s works in [10,11], we [2] gave a complete
classification on multiplicity-one representations over graded generalized divergence-free
Lie algebras introduced by Dokovic and Zhao [4]. Based on our previous result, we give
in this paper a complete classification of the irreducible and indecomposable generalized
weight modules over the nongraded divergence-free Lie algebras introduced by Xu [26]
with weight multiplicities less than or equal to one.
Throughout this paper, we let F be an algebraically closed field of characteristic zero.
All the vector spaces are assumed over F. Denote by N the set of nonnegative integers
{0,1,2,···} andby Z theset ofpositive integers. Form,n ∈ Z, weshall use thenotation
+
m,n = {m,m+1,m+2,...,n} if m ≤ n; m,n = ∅ if m > n. (1.1)
For any positive integer n, an additive subgroup G of Fn is called nondegenerate if G
contains an F-basis of Fn. Let l , l and l be three nonnegative integers such that
1 2 3
l = l +l +l > 0. (1.2)
1 2 3
Take any nondegenerate additive subgroup Γ of Fl2+l3 and let Γ = {0} when l +l = 0.
2 3
Let A(l ,l ,l ;Γ) be the semi-group algebra F[Γ×Nl1+l2] with the basis
1 2 3
{xαti | α ∈ Γ,i ∈ Nl1+l2}. (1.3)
Define a commutative associative algebraic operation “·” on A(l ,l ,l ;Γ) by
1 2 3
xαti ·xβtj = xα+βti+j for α,β ∈ Γ, i,j ∈ Nl1+l2. (1.4)
Note that x0t0 is the identity element, which is denoted by 1 for convenience. Moreover,
we write ti and xα instead of x0ti and xαt0 for short. When the context is clear, we omit
the notation “·” in any associative algebra product.
For k ∈ Z and p ∈ 1,l, we denote
+
p
k = (0,··· ,0,k,0,··· ,0) ∈ Nl. (1.5)
[p]
For convenience, α ∈ Fl2+l3 is written as
α = (α ,α ,··· ,α ) with α = α = ··· = α = 0, (1.6)
1 2 l 1 2 l1
and i ∈ Nl1+l2 is written as
i = (i ,i ,··· ,i ) with i = i = ··· = i = 0. (1.7)
1 2 l l1+l2+1 l1+l2+2 l
Moreover, we denote
l1+l2
|i| = i for i ∈ Nl1+l2. (1.8)
X p
p=1
We define linear transformations {∂ ,∂ ,...,∂ } on A(l ,l ,l ;Γ) by
1 2 l 1 2 3
∂ (xαti) = α xαti +i xαti−1[p] for p ∈ 1,l, i ∈ Nl1+l2, α ∈ Γ, (1.9)
p p p
where ip = 0 and the monomial ipxαti−1[p] is treated as zero when i − 1[p] 6∈ Nl1+l2.
Then {∂ | p ∈ 1,l} are mutually commutative derivations of A(l ,l ,l ;Γ), and they are
p 1 2 3
F-linearly independent. Let
l
D = F∂ (1.10)
X i
i=1
and
W (l ,l ,l ;Γ) = A(l ,l ,l ;Γ)D. (1.11)
1 2 3 1 2 3
Then W (l ,l ,l ;Γ) is the simple Lie algebra of nongraded Witt type constructed by
1 2 3
Xu [26]. Its Lie bracket is given by
l l l
[ u ∂ , v ∂ ] = (u ∂ (v )−v ∂ (u ))∂ for u ,v ∈ A(l ,l ,l ;Γ). (1.12)
X p p X q q X p p q p p q q p p 1 2 3
p=1 q=1 p,q=1
Define the divergence on W (l ,l ,l ;Γ) by
1 2 3
l l
div∂ = ∂ (u ) for ∂ = u ∂ ∈ W (l ,l ,l ;Γ). (1.13)
X i i X i i 1 2 3
i=1 i=1
Let ρ ∈ Γ be any element. We define
D (u) = xρ(∂ (x−ρu)∂ −∂ (x−ρu)∂ ) (1.14)
p,q p q q p
for p,q ∈ 1,l and u ∈ A(l ,l ,l ;Γ). Then
1 2 3
S(l ,l ,l ;ρ,Γ) = Span{D (u) | p,q ∈ 1,l, u ∈ A(l ,l ,l ;Γ)} (1.15)
1 2 3 p,q 1 2 3
forms a Lie subalgebra of the Lie algebra W (l ,l ,l ;Γ). The Lie algebra S(l ,l ,l ;ρ,Γ)
1 2 3 1 2 3
was first introduced by Xu [26], as a nongraded generalization of graded simple Lie alge-
bras of Special type, and further studied by Su and Xu [23]. It was called a divergence-free
Lie algebra[23]inthesensethatS(l ,l ,l ;0,Γ)consistsofthedivergence-freeelementsof
1 2 3
W (l ,l ,l ;Γ). When l > 0, the algebra S(l ,l ,l ;ρ,Γ) does not contain any toral Car-
1 2 3 2 1 2 3
tan subalgebra. The special case S(0,0,l ;ρ,Γ) was introduced and studied by Dokovic
3
and Zhao [4]. In [2], we classified the multiplicity-one representations of the Lie algebras
S(0,0,l ;0,Γ) with l ≥ 3. In this paper, we further study the generalized multiplicity-
3 3
one representations of the Lie algebras S(l ,l ,l ,0;Γ) with l +l ≥ 3 and l +l ≥ 3.
1 2 3 1 2 2 3
A linear transformation T of a vector space V is called locally finite if
dim(span {Tn(v) | n ∈ Z }) < ∞, ∀ v ∈ V; (1.16)
F +
it is called semi-simple, if for any v ∈ V, span {Tn(v) | n ∈ Z } has a F-basis consisting
F +
of eigenvectors of T; it is called locally nilpotent, if for any v ∈ V, there exists n ∈ Z
+
(may depend on v) such that Tn(v) = 0. A subspace U of EndV is called locally finite or
locally nilpotent, if each element of U is locally finite or locally nilpotent, respectively.
For convenience, we define
l
< ∂,β >= ∂(β) = β(∂) = a β (1.17)
X p p
p=l1+1
for ∂ = l a ∂ ∈ D and β = (β ,β ,··· ,β ) ∈ Fl2+l3. Let
Pp=1 p p 1 2 l
l1 l1+l2 l
D = F∂ , D = F∂ , D = F∂ . (1.18)
1 X p 2 X p 3 X p
p=1 p=l1+1 p=l1+l2+1
Observe that adD is locally nilpotent and adD is locally finite on S. Motivated from
1
this and Su and Zhou’s work [24], we are interested in considering S-modules V on which
D and D act locally nilpotently and locally finitely, respectively. For such a module V,
1
it has the decomposition
V = V , (1.19)
M β
β∈Fl2+l3
where
V = {v ∈ V | (∂ −β(∂))n(v) = 0 for ∂ ∈ D and some n ∈ Z }. (1.20)
β +
We set
V(n) = {v ∈ V | (∂ −β(∂))n+1(v) = 0 for ∂ ∈ D}, β ∈ Fl2+l3, n ≥ 0. (1.21)
β
(n) (0)
Clearly, V = V and V 6= {0} ⇐⇒ V 6= {0}.
A modβuleSVn∈oNf Sβ (l ,l ,l β,0;Γ) is calledβ a generalized weight module if it has the
1 2 3
decomposition (1.19). For β ∈ Fl2+l3, if V 6= {0}, then V is called the generalized weight
β β
(0)
space with weight β, and V is called the weight space with weight β. Moreover, we call
β
(0)
dimV the weight multiplicity corresponding to weight β.
β
The aim of this paper is to classify all the irreducible and indecomposable generalized
weight modules of S(l ,l ,l ,0;Γ) with weight multiplicities equal to or less than one.
1 2 3
Since our investigations are based on the previous result [2], we only consider the non-
graded divergence-free Lie algebra S(l ,l ,l ,0;Γ) with l +l ≥ 3. When l +l = 1, the
1 2 3 2 3 1 2
Lie algebra S(l ,l ,l ,0;Γ) does not contain the whole vector space D, which means we
1 2 3
don’t get the generalized weight module. The case l +l = 2 will be studied later due to
1 2
a technical difficulty. The main result of this paper is the following theorem:
Theorem 1.1 Suppose l + l ≥ 3 and l + l ≥ 3. Assume that V = V is a
1 2 2 3 Lβ∈Fl2+l3 β
generalized weight module of S(l ,l ,l ,0;Γ) with dimV(0) ≤ 1 for all β ∈ Fl2+l3. If V
1 2 3 β
is irreducible or indecomposable, then V is either isomorphic to the 1-dimensional trivial
module Fv , or isomorphic to the module A for appropriate µ ∈ Fl2+l3, where A is the
0 µ µ
module with F-basis {v | α ∈ Γ, i ∈ Nl1+l2} and the following action:
α,i
D (xαti).v
p,q β,j
= (xαti(α ∂ −α ∂ )+i xαti−1[p]∂ −i xαti−1[q]∂ ).v
p q q p p q q p β,j
= (α (β +µ )−α (β +µ ))v +(j α −i (β +µ ))v
p q q q p p β+α,i+j q p q p p β+α,i+j−1[q]
+(i (β +µ )−j α )v +(i j −i j )v (1.22)
p q q p q β+α,i+j−1[p] p q q p β+α,i+j−1[p]−1[q]
for α,β ∈ Γ, i,j ∈ Nl1+l2 and p,q ∈ 1,l with p 6= q.
This paper is organized as follows. In Section 2, we give some properties of the
Lie algebra S(l ,l ,l ,0;Γ) and the S-module A . Moreover, we review our earlier
1 2 3 µ
classification theorem on the multiplicity-one representations over the graded divergence-
free Lie algebras (cf. [2]). In Sections 3 and 4, we prove the main theorem.
2 Some properties
In this section, we would like to digress to give some properties of the Lie algebra
S(l ,l ,l ,0;Γ)andtheS-moduleA , whichareusefulfortheproofofthemaintheorem.
1 2 3 µ
We always assume that l +l ≥ 3 and l +l ≥ 3 in the present and the following sections.
1 2 2 3
For S(l ,l ,l ,0;Γ)-module A , it is easy to verify:
1 2 3 µ
Proposition 2.1 (1) The module A is irreducible if and only if µ 6∈ Γ.
µ
(2) When µ ∈ Γ, A ≃ A is indecomposable, and it has the trivial submodule Fv .
µ 0 −µ,0
The quotient module A′ = A /Fv is simple.
µ µ −µ,0
(3) The possible isomorphisms between modules of type A are: A ≃ A iff µ−η ∈ Γ.
µ µ η
We would like to remark that, the simple quotient module V = A′ = A /Fv is a
0 0 0,0
generalized weight module of S(l ,l ,l ,0;Γ) with weight set Γ; and we have dimV(0) =
1 2 3 0
(0)
l +l , dimV = 1 for all β ∈ Γ\{0}. The following conclusion will be frequently used
1 2 β
later:
Lemma 2.2 Let T be a linear transformation on a vector space U, and let U be a
1
subspace of U such that T(U ) ⊂ U . Suppose that u , u , ··· , u are eigenvectors of T
1 1 1 2 n
n
corresponding to different eigenvalues. If u ∈ U , then u , u , ··· , u ∈ U .
Pj=1 j 1 1 2 n 1
Proposition 2.3 The Lie algebra S(l ,l ,l ,0;Γ) is generated by
1 2 3
{D (xα),D (tj) | p,q ∈ 1,l, α ∈ Γ\{0}, j ∈ Nl1+l2 with |j| ≤ 2}. (2.1)
p,q p,q
Proof. Denote by K the Lie subalgebra of S(l ,l ,l ,0;Γ) which is generated by the
1 2 3
set (2.1). It suffices to prove K = S(l ,l ,l ,0;Γ). By (1.9) and (1.14), we have
1 2 3
D (xαti) = xαti(α ∂ −α ∂ )+i xαti−1[p]∂ −i xαti−1[q]∂ (2.2)
p,q p q q p p q q p
for p,q ∈ 1,l, α ∈ Γ and i ∈ Nl1+l2. Then we proceed our proof in two steps.
Step 1. D (xαti) ∈ K for any p,q ∈ 1,l, α ∈ Γ\{0} and i ∈ Nl1+l2.
p,q
We prove this step by induction on |i|. By (2.1), we know that this holds when |i| = 0.
Assume that
D (xαti) ∈ K for any p,q ∈ 1,l, α ∈ Γ\{0}, i ∈ Nl1+l2 with |i| ≤ k, (2.3)
p,q
where k ≥ 0. It suffices to prove Dp,q(xαti+1[r]) ∈ K for all p,q ∈ 1,l, α ∈ Γ\{0},
r ∈ 1,l +l and i ∈ Nl1+l2 with |i| = k.
1 2
Fix any α ∈ Γ\{0} and i ∈ Nl1+l2 with |i| = k. Choose s ∈ 1,l such that α 6= 0. Then
s
for any p,q ∈ 1,l and r ∈ 1,l +l \{s}, we have
1 2
[D (xαti),D (t2[r])]
p,q r,s
= 2[xαti(α ∂ −α ∂ )+i xαti−1[p]∂ −i xαti−1[q]∂ ,t1[r]∂ ]
p q q p p q q p s
= 2(δ D (xαti)−δ D (xαti)−i D (xαti+1[r]−1[s])−α D (xαti+1[r])).(2.4)
p,r s,q q,r s,p s p,q s p,q
Thus by (2.4) and the induction hypothesis (2.3), we find
D (xαti+1[r]) ∈ K for p,q ∈ 1,l, r ∈ 1,l +l \{s}. (2.5)
p,q 1 2
If s 6∈ 1,l1 +l2, then it is done. Consider the case that s ∈ 1,l1 +l2. If αs′ 6= 0 for
some s′ ∈ 1,l\{s}, then with s replaced by s′ in the above arguments, we can get
Dp,q(xαti+1[s]) ∈ K for p,q ∈ 1,l. Otherwise, we have αr = 0 for all r ∈ 1,l\{s}.
Pick r ∈ 1,l +l \{s}. Expressions (2.1) and (2.5) give
1 2
K ∋ [D (xαti+1[r]),D (t2[s])]
p,q s,r
= −2((i +1)D (xαti+1[s])+δ D (xαti+1[r])−δ D (xαti+1[r])) (2.6)
r p,q p,s q,r q,s p,r
for p,q ∈ 1,l. So this and (2.5) indicate
D (xαti+1[s]) ∈ K for p,q ∈ 1,l. (2.7)
p,q
To summarize, (2.5) and (2.7) show
D (xαti+1[r]) ∈ K (2.8)
p,q
for p,q ∈ 1,l, α ∈ Γ\{0}, r ∈ 1,l +l and i ∈ Nl1+l2 with |i| = k, which completes the
1 2
proof of the step.
Step 2. D (ti) ∈ K for any p,q ∈ 1,l and i ∈ Nl1+l2.
p,q
We shall prove this step by induction on |i|. When |i| = 0, D (ti) = 0 for any
p,q
p,q ∈ 1,l. Assume that
D (tj) ∈ K for any p,q ∈ 1,l, j ∈ Nl1+l2 with |j| ≤ k, (2.9)
p,q
where k ≥ 0. It suffices to prove D (ti) ∈ K for all p,q ∈ 1,l and i ∈ Nl1+l2 with
p,q
|i| = k +1. Suppose p 6= q and |i| = k +1 in the following. We give the proof in several
cases.
Case 1. p,q ∈ l +l +1,l.
1 2
In this case, D (ti) = 0.
p,q
Case 2. p,q ∈ 1,l +l .
1 2
Pick r ∈ l +1,l\{p,q}. Choose α ∈ Γ\{0} such that α 6= 0. Then (2.2) and Step 1
1 r
indicate
K ∋ [D (xαti),D (x−α)]
r,q r,p
= [xαti(α ∂ −α ∂ )+i xαti−1[r]∂ −i xαti−1[q]∂ ,x−α(α ∂ −α ∂ )]
r q q r r q q r p r r p
= α2D (ti)+α α D (ti)+α α D (ti)+α i D (ti−1[r])
r p,q r p q,r r q r,p p r q,r
+α i D (ti−1[p]). (2.10)
r p r,q
By the induction hypothesis (2.9), we get
α2D (ti)+α α D (ti)+α α D (ti) ∈ K . (2.11)
r p,q r p q,r r q r,p
Since
[D (t1[p]+1[q]),D (ti)] = (i −i )D (ti), (2.12)
p,q p,q q p p,q
[D (t1[p]+1[q]),D (ti)] = (i −i −1)D (ti), (2.13)
p,q q,r q p q,r
[D (t1[p]+1[q]),D (ti)] = (i −i +1)D (ti), (2.14)
p,q r,p q p r,p
we have D (ti) ∈ K by (2.1) and Lemma 2.2.
p,q
Case 3. p ∈ 1,l +l , q ∈ l +l +1,l.
1 2 1 2
Pick s ∈ 1,l +l \{p}. Case 2 tells that D (ti) ∈ K . So by (2.1) and (2.2), we have
1 2 p,s
1
K ∋ [D (ti), D (t2[s])] = i ti−1[p]∂ = D (ti). (2.15)
p,s s,q p q p,q
2
Case 4. p ∈ l +l +1,l, q ∈ 1,l +l .
1 2 1 2
We have D (ti) = −D (ti) ∈ K by Case 3.
p,q q,p
So this step holds.
Thus, from (1.15) we see that K = S(l ,l ,l ,0;Γ). So this proposition holds. ✷
1 2 3
We also have:
Corallary 2.4 The set
{D (xαti) | p,q ∈ 1,l, α ∈ Γ\{0},i ∈ Nl1+l2} (2.16)
p,q
generates the Lie algebra S(l ,l ,l ,0;Γ).
1 2 3
Proof. Denote by K ′ the Lie subalgebra of S(l ,l ,l ,0;Γ) which is generated by the
1 2 3
set (2.16). By Proposition 2.3, it suffices to prove D (ti) ∈ K ′ for all p,q ∈ 1,l and
p,q
i ∈ Nl1+l2 with |i| ≤ 2. Notice that SpanF{Dp,q(t1[r]) | p,q ∈ 1,l, r ∈ 1,l1 +l2} = D. So
first, we need to prove D ⊆ K ′.
Fix some r ∈ l +1,l. Choose α ∈ Γ such that α 6= 0. Pick p ∈ 1,l +l \{r}. Then
1 r 1 2
for any q ∈ 1,l\{r,p}, we have
K ′ ∋ [D (xαt1[p]),D (x−α)]
r,q r,p
= [xαt1[p](α ∂ −α ∂ ),x−α(α ∂ −α ∂ )]
r q q r p r r p
= α (α ∂ −α ∂ ) (2.17)
r r q q r
by (2.16). Namely, we obtain
α ∂ −α ∂ ∈ K ′ for q ∈ 1,l\{r,p}. (2.18)
r q q r
By picking another p ∈ 1,l +l \{r}, we can get
1 2
α ∂ −α ∂ ∈ K ′ for q ∈ 1,l\{r}. (2.19)
r q q r
Take r′ ∈ l1 +1,l\{r}. Choose β ∈ Γ such that βr 6= 0 and αrβr′ −αr′βr 6= 0. With α
replaced by β in (2.17)–(2.19), we can get that βr∂r′−βr′∂r ∈ K ′. Since (2.19) also gives
αr∂r′ −αr′∂r ∈ K ′, we have
1
∂r = (βr(αr∂r′ −αr′∂r)−αr(βr∂r′ −βr′∂r)) ∈ K ′. (2.20)
αrβr′ −αr′βr
Thus, from (2.19) and (2.20) we deduce
D ⊆ K ′. (2.21)
Second, we want to prove t1[p]∂q ∈ K ′ for any p ∈ 1,l1 +l2 and q ∈ 1,l\{p}.
Fix any p ∈ 1,l +l . Pick r ∈ l +1,l\{p}. Choose α ∈ Γ such that α 6= 0. Then
1 2 1 r
for any q ∈ 1,l\{r,p}, we have
K ′ ∋ [D (xαt2[p]),D (x−α)]
r,q r,p
= [xαt2[p](α ∂ −α ∂ ),x−α(α ∂ −α ∂ )]
r q q r p r r p
= 2α t1[p](α ∂ −α ∂ ) (2.22)
r r q q r
by (2.16). Namely,
t1[p](α ∂ −α ∂ ) ∈ K ′ for q ∈ 1,l\{r,p}. (2.23)
r q q r
Pick r′ ∈ l1 +1,l\{r,p}. Choose β ∈ Γ such that βr 6= 0 and αrβr′ −αr′βr 6= 0. Likewise,
we can get t1[p](βr∂r′ −βr′∂r) ∈ K ′. Since (2.23) gives t1[p](αr∂r′ −αr′∂r) ∈ K ′, we have
1
t1[p]∂r = (βrt1[p](αr∂r′ −αr′∂r)−αrt1[p](βr∂r′ −βr′∂r)) ∈ K ′. (2.24)
αrβr′ −αr′βr
So (2.23) and (2.24) show
t1[p]∂ ∈ K ′ for all p ∈ 1,l +l and q ∈ 1,l\{p}. (2.25)
q 1 2
Third, we want to prove that t1[q]∂q −t1[p]∂p ∈ K ′ for any p,q ∈ 1,l1 +l2.
Fix any p,q ∈ 1,l +l . We further suppose that p 6= q. Pick r ∈ 1,l\{p,q}. Choose
1 2
α ∈ Γ such that α 6= 0. Then (2.16) indicates
r
K ′ ∋ [D (xαt1[p]+1[q]),D (x−α)]
r,q r,p
= [xαt1[p]+1[q](α ∂ −α ∂ )−xαt1[p]∂ ,x−α(α ∂ −α ∂ )]
r q q r r p r r p
= α2(t1[q]∂ −t1[p]∂ )+α α t1[p]∂ −α α t1[q]∂ −α ∂ . (2.26)
r q p r p r r q r r r
So by (2.21), (2.25) and (2.26), we have
t1[q]∂ −t1[p]∂ ∈ K ′ for all p,q ∈ 1,l +l . (2.27)
q p 1 2
In summary, (2.21), (2.25) and (2.27) give
{D (ti) | p,q ∈ 1,l, i ∈ Nl1+l2 with |i| ≤ 2} ⊆ K ′. (2.28)
p,q
✷
So the corollary follows from (2.16), (2.28) and Proposition 2.3.
Last, for convenience of the reader, we shall review below the multiplicity-one repre-
sentations over the graded divergence-free Lie algebras [2].
By (1.9) and (1.17), we have
∂(xα) = ∂(α)xα for α ∈ Γ and ∂ ∈ D; (2.29)
(D +D )(α) = {0} with α ∈ Fl2+l3, iff α = 0; (2.30)
2 3
∂(Γ) = {0} with ∂ ∈ D +D , iff ∂ = 0. (2.31)
2 3
Denote
kerβ = {∂ ∈ D | ∂(β) = 0} for β ∈ Fl2+l3. (2.32)
Then the main classification theorem in [2] is:
Proposition 2.5 Suppose l ≥ 3. Assume that V = V is a Γ-graded S(0,0,l,0;Γ)-
Lθ∈Γ θ
module with dimV ≤ 1 for θ ∈ Γ. If V is irreducible or indecomposable, then V is
θ
isomorphic to one of the following modules for appropriate µ ∈ Fl and η ∈ Fl\{0}:
(i)the trivial module Fv , (ii)M′ , (iii)A , (iv)B , (2.33)
0 µ η η
where M , A and B are S(0,0,l,0;Γ)-moduleswith basis{v | α ∈ Γ} andthe following
µ η η α
actions:
M : xα∂.v = ∂(β +µ)v for α ∈ Γ\{0},β ∈ Γ,∂ ∈ kerα; (2.34)
µ β α+β
A : xα∂.v = ∂(β)v for α,β ∈ Γ\{0},∂ ∈ kerα,
η β α+β
xα∂.v = ∂(η)v for α ∈ Γ\{0},∂ ∈ kerα; (2.35)
0 α
B : xα∂.v = ∂(β)v for α ∈ Γ\{0},β ∈ Γ\{−α},∂ ∈ kerα,
η β α+β
xα∂.v = ∂(η)v for α ∈ Γ\{0},∂ ∈ kerα, (2.36)
−α 0
and M′ is denoted as the only nontrivial irreducible quotient module of M .
µ µ
3 Proof of the main theorem (I)
Suppose that V is a generalized weight module of S(l ,l ,l ,0;Γ) with dimV(0) ≤ 1
1 2 3 β
for all β ∈ Fl2+l3. Let
V(µ) = V for µ ∈ Fl2+l3. (3.1)
M α+µ
α∈Γ
Then V(µ) is a submodule of V, and V is a direct sum of V(µ) for different µ ∈ Fl2+l3.
So, if V is irreducible or indecomposable, we must have V = V(µ) for some µ ∈ Fl2+l3.
By Proposition 2.3, we only need to derive the action of the set (2.1) on V = V(µ).
Except the trivial case in Lemma 3.1, we should determine a basis of V = V(µ), and then
derive the action of the set (2.1) on the basis. For the nontrivial cases, we shall first give
a general discussion for any µ ∈ Fl2+l3, then prove the main theorem for µ ∈ Fl2+l3\Γ at
the end of this section, and last prove the main theorem for µ ∈ Γ in the next section.
(0) (0)
Lemma 3.1 If V 6= {0} for some σ ∈ Γ, and V = {0} for all ρ ∈ Γ\{σ}, then
σ+µ ρ+µ
σ +µ = 0, and V = V(0) is a 1-dimensional trivial S(l ,l ,l ,0;Γ)-module.
0 1 2 3
Proof. By (3.1), we have V = V . Therefore,
σ+µ
D (xγti).V = {0} for all p,q ∈ 1,l, γ ∈ Γ\{0},i ∈ Nl1+l2 (3.2)
p,q σ+µ
by (1.21). Moreover, Corollary 2.4 gives rise to
S(l ,l ,l ,0;Γ).V = {0}. (3.3)
1 2 3 σ+µ
So V = V is a direct sum of trivial submodules. Since V is irreducible or inde-
σ+µ
composable, we must have V = V(0) as a trivial S-module. As one result, we get
σ+µ
∂.V(0) = {0} for all ∂ ∈ D + D , which implies σ + µ = 0. So V = V(0), and it is a
σ+µ 2 3 0
trivial S(l ,l ,l ,0;Γ)-module. This completes the proof of the lemma. ✷
1 2 3
In the rest of the paper, we shall always assume that
(0) (0)
V 6= {0} and V 6= {0} for some σ,ρ ∈ Γ with σ 6= ρ. (3.4)
σ+µ ρ+µ
(0) (0) (0)
Lemma 3.2 If V 6= {0} and V 6= {0} for some σ,ρ ∈ Γ with σ 6= ρ, then V 6=
σ+µ ρ+µ γ+µ
(0)
{0} for all γ ∈ Γ\{−µ}. Moreover, there exist nonzero v ∈ V with γ ∈ Γ\{−µ},
γ,0 γ+µ
such that
D (xα).v = xα(α ∂ −α ∂ ).v = (α (γ +µ )−α (γ +µ ))v (3.5)
p,q γ,0 p q q p γ,0 p q q q p p α+γ,0
for p,q ∈ 1,l, α ∈ Γ\{0} and γ ∈ Γ\{−µ}.
Proof. Since
S := Span {D (xα) | p,q ∈ l +1,l, α ∈ Γ\{0}} ≃ S(0,0,l +l ,0;Γ), (3.6)
0 F p,q 1 2 3
we can see V(0) = V(0) as a Γ-graded S -module with dimV(0) ≥ 2 and dimV(0) ≤
Lα∈Γ α+µ 0 α+µ
1 for all α ∈ Γ. So first of all, we need to prove:
Claim 1. S .V(0) 6= {0}.
0
Suppose S .V(0) = {0}, namely, V(0) is a direct sum of some trivial S -submodules,
0 0
then we will get a contradiction.
(0) (0) (0)
Since V 6= {0} and V 6= {0} for some σ,ρ ∈ Γ with σ 6= ρ, we have V 6= {0}
σ+µ ρ+µ θ+µ
(0)
for some θ ∈ Γ\{−µ}. Fix θ ∈ Γ\{−µ} satisfying V 6= {0}.
θ+µ
¯
Let α¯ = α + µ for all α ∈ Γ. Since θ 6= 0, we can choose r ∈ l +1,l such that
1
¯
θ 6= 0. Pick p ∈ 1,l +l \{r} and s ∈ l +1,l\{r,p}. Choose α ∈ Γ such that α 6= 0
r 1 2 1 r
¯ ¯
and θ α −θ α 6= 0. Moreover, we let
r s s r
∂ = θ¯ ∂ −θ¯ ∂ , ∂′ = α ∂ −α ∂ , ∂′′ = (θ¯ −α )∂ −(θ¯ −α )∂ . (3.7)
r s s r r s s r r r s s s r
From (1.21) it can be deduced that
t1[p]∂.V(0) ⊆ V(0) , t1[p]∂′′.V(0) ⊆ V(0) . (3.8)
θ+µ θ+µ θ−α+µ θ−α+µ
(0)
Take 0 6= v ∈ V . So on one hand, we have
θ θ+µ
∂′.v = ∂′(θ¯)v = −(θ¯ α −θ¯ α )v 6= 0 (3.9)
θ θ r s s r θ
by (1.21). On the other hand, (3.8) and S .V(0) = {0} give
0
xαt1[p]∂′.v
θ
1
= − [xα∂′,t1[p]∂].v
θ
∂(α)
1
= − (xα∂′.(t1[p]∂.v )−t1[p]∂.xα∂′.v )
¯ ¯ θ θ
θ α −θ α
r s s r
= 0 (3.10)
and
1
xαt1[p]∂′.V(0) = − [xα∂′,t1[p]∂′′].V(0)
θ−α+µ θ¯ α −θ¯ α θ−α+µ
r s s r
= {0}, (3.11)
which further implies
∂′.v
θ
1
= − [xαt1[p]∂′,x−α(α ∂ −α ∂ )].v
r p p r θ
α
r
1
= − (xαt1[p]∂′.(x−α(α ∂ −α ∂ ).v )−x−α(α ∂ −α ∂ ).(xαt1[p]∂′.v ))
r p p r θ r p p r θ
α
r
= 0, (3.12)
where in the third line of (3.12), we have x−α(α ∂ − α ∂ ).v ∈ V(0) . Since (3.12)
r p p r θ θ−α+µ
contradicts (3.9), we must have S .V(0) 6= {0}. Thus this claim holds.
0
Note that V(0) = V(0) is a Γ-graded S -submodule with dimV(0) ≥ 2 and
Lα∈Γ α+µ 0
dimV(0) ≤ 1 for all α ∈ Γ. Moreover, Claim 1 shows that V(0) is not a direct sum of
α+µ
some trivial S -submodules. So Proposition 2.5 implies that, there exists η ∈ Fl2+l3 such
0
(0) (0)
that V 6= {0} for all γ ∈ Γ\{−η}, and that we can choose nonzero v ∈ V with
γ+µ γ,0 γ+µ
γ ∈ Γ\{−η} such that
D (xα).v = xα(α ∂ −α ∂ ).v = (α (γ +η )−α (γ +η ))v (3.13)
p,q γ,0 p q q p γ,0 p q q q p p α+γ,0
for p,q ∈ l +1,l, α ∈ Γ\{0} and γ ∈ Γ\{−η,−η −α}.
1