Classification of simple weight modules for super-Virasoro algebra with a finite-dimensional weight space

Xiufu ZHANG
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
c
Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract There are two extensions of Virasoro algebra with particular
importance in superstring theory: the Ramond algebra and the Neveu-Schwarz
algebra, which are Z2-graded extensions of the Virasoro algebra. In this paper,
we show that the support of a simple weight module over the Ramond algebra
with an infinite-dimensional weight space coincides with the weight lattice and
that all intersections of non-trivial weight spaces and odd part or even part
of the module are infinite-dimensional. This result together with the one that
we have obtained over the Neveu-Schwarz algebra generalizes the result for the
Virasoro algebra to the super-Virasoro algebras.
Keywords Super-Virasoro algebra, Ramond algebra, weight module, Harish-
Chandra module
MSC 17B10, 17B65, 17B68
1 Introduction
A simple weight module over the Virasoro algebra is called a Harish-Chandra
module, if all weight spaces are finite-dimensional. The classification of
irreducible Harish-Chandra modules over the Virasoro algebra was obtained
in [11], it can also be derived from the results in [10,15]. Mazorchuk and
Zhao [12] proved that the support of a simple weight module over the Virasoro
algebra with an infinite-dimensional weight space coincides with the weight
lattice and that all non-trivial weight spaces of the module are infinitedimensional.
So the irreducible modules with a finite-dimensional weight space
over the Virasoro algebra are classified. For the last few years, the study on
irreducible weight modules with infinite-dimensional weight spaces and non-
Received May 8, 2014; accepted March 9, 2015
E-mail: xfzhang@jsnu.edu.cn
1234 Xiufu ZHANG
weight modules over the Virasoro algebra has been very active (see [2,4,5,7–
9,13,17,18,21]). The irreducibility problem for the tensor products over the
Virasoro algebra was a long-standing problem (see [18]). The tensor product
of highest weight modules with intermediate series modules over the Virasoro
algebra were the first such simple Virasoro modules with infinite-dimensional
weight spaces [18]. The tensor product of intermediate series modules over the
Virasoro algebra never gives simple modules [21]. In [2], the irreducibility for
the tensor product of highest weight modules with intermediate series modules
over the Virasoro algebra was completely determined by using Feigin-Fuchs’
Theorem (see [3] or [1]) on singular vectors of Verma modules and the so called
shifting technique. In the process of the study in [2], the main result in [12]
plays an important rule. Moreover, the result in [12] is generalized to twisted
Schr¨odinger-Virasoro algebra by Li and Su [6], twisted Heisenberg-Virasoro by
Shen and Su [14].
In mathematical physics, a super-Virasoro algebra is an extension of the
Virasoro algebra to a Lie superalgebra. There are two extensions with particular
importance in superstring theory: the Ramond algebra and the Neveu-Schwarz
algebra. Both algebras have N = 1 supersymmetry with even part given by the
Virasoro algebra. They describe the symmetries of a superstring in two different
sectors, called the Ramond sector and the Neveu-Schwarz sector. The odd part
of the algebra has basis Gr, where r is either an integer (the Ramond case), or
half an odd integer (the Neveu-Schwarz case). Concretely, the super-Virasoro
algebra is the Lie superalgebra
SV (θ) = SV (θ)0
⊕ SV (θ)1,
where θ = 1/2 or 0 according as SV (θ) is the Neveu-Schwarz algebra or the
Ramond algebra, SV (θ)0 has a basis {Ln, c | n ∈ Z} and SV (θ)1 has a basis
{Gr | r ∈ θ + Z}, with the commutation relations
[Lm, Ln] = (n − m)Ln+m + δm+n,0
m3 − m
12 c,
[Lm,Gr] =

r − m
2

Gm+r,
[Gr,Gs] = 2Lr+s − 1
3 δr+s,0

r2 − 1
4

c,
[SV0(θ), c] = 0 = [SV1(θ), c],
for m, n ∈ Z, r,s ∈ θ + Z. It is obvious that the Neveu-Schwarz algebra is
1
2
Z-graded and Ramond algebra is Z-graded. Define
SV (θ)[n] = CLn ⊕ CGn ⊕ δn,0c,
where n ∈ 1
2
Z (if θ = 1/2) or n ∈ Z (if θ = 0). Let
SV (θ)+ =

n>0
SV (θ)[n], SV(θ)− =

n<0
SV (θ)[n],
Simple weight modules for the super-Virasoro algebra 1235
and
h = CL0 ⊕ Cc.
Then SV (θ) has a triangular decomposition:
SV (θ) =
⎧⎨
⎩
NS− ⊕ h ⊕ NS+, θ=
1
2,
R− ⊕ h ⊕ R+ ⊕ CG0, θ= 0,
where NS = SV (1/2) is the Neveu-Schwarz algebra and R = SV (0) is the
Ramond algebra.
The subalgebra h is called the Cartan subalgebra of SV (θ). An hdiagonalizable
SV (θ)-module is usually called a weight module. If M is a weight
module, then M can be written as a direct sum of its weight spaces,
M = ⊕Mλ,
where
Mλ = {v ∈ M | L0v = λ(L0)v, cv = λ(c)v}.
We call {λ | Mλ = 0} the support of M and is denoted by supp(M). A
weight module is called a Harish-Chandra module if each weight space is finitedimensional.
For an SV (θ)-module M, it has a Z2-grading
M = M0 ⊕M1
with
SViMj
⊂ Mi+j , i, j ∈ Z2.
Suppose that M is a simple weight R-module. Then c acts on M by a scalar
which is called the central charge. By abuse of notation, we use c to denote the
central charge, too.
The Harish-Chandra modules over the super-Virasoro algebras were
classified by Su [16]. The classification of the simple weight modules with a
finite-dimensional weight space over the Neveu-Schwarz algebra was given in
[20], and it played an important role in the study of tensor product weight
representations of the Neveu-Schwarz algebra in [19].
In this paper, the classification of the simple weight modules with a finitedimensional
weight space over the Ramond algebra is given. The proof for the
case of Ramond algebra is very different from the one over the Neveu-Schwarz
algebra since these two algebras have different gradations. We expect that this
result will be useful to the study of weight modules with infinite-dimensional
weight spaces over the Ramond algebra, such as the tensor product weight
modules. Our main result is as follows.
Theorem 1 Let
M = M0
⊕M1
1236 Xiufu ZHANG
be a simple weight R-module. If there exist λ ∈ C and i ∈ Z2 such that
dimMλ
i = +∞,
then
supp(M) = λ + Z.
Moreover,
dimMλ+k
0
= +∞, dimMλ+k
1
= +∞, ∀ k ∈ Z.
Combining with [16, Theorem 4.2], we get the following theorem
immediately.
Theorem 2 Let M be a simple weight R-module. Assume that there exist
λ ∈ C and i ∈ Z2 such that
0 < dimMλ
i < +∞.
Then M is a Harish-Chandra module. Consequently, M is either a highest or
lowest weight module, or else a module of the intermediate series.
Theorem 2 and [20, Theorem 2] complete the classification of simple weight
modules with a finite-dimensional weight space over super-Virasoro algebra.
Theorem 3 Let M be a simple weight module over the super-Virasoro
algebras. Assume that there exist λ ∈ C and i ∈ Z2 such that
0 < dimMλ
i < +∞.
Then M is a Harish-Chandra module. Consequently, M is either a highest or
lowest weight module, or else a module of the intermediate series.
By Theorem 3, we have the following two corollaries.
A simple weight moduleM over the super-Virasoro algebras is called pointed
provided that there exist λ ∈ C and i ∈ Z such that
dimMλ
i = 1.
Corollary 1 Every simple pointed module over the super-Virasoro algebras is
a Harish-Chandra module.
A weight moduleM over the super-Virasoro algebras is called mixed module
if there exist λ ∈ C, i, j ∈ Z2, and k ∈ Z such that
dimMλ
i = +∞, dimMλ+k
j
< +∞.
Corollary 2 There are no simple mixed modules over the super-Virasoro
algebras.
Simple weight modules for the super-Virasoro algebra 1237
2 Proof of Theorem 1
Note that each of the sets
{L−1,G−1, L1,G1}, {G−1,G0,G1}, {L−1,G0, L1}
can generate R. We have the following fact immediately.
Principal Fact If there exist μ ∈ C and 0 = v ∈ Mμ such that one of the
following holds:
L1v = G1v = 0, L−1v = G−1v = 0, G0v = G1v = 0,
G0v = G−1v = 0, G0v = L1v = 0, G0v = L−1v = 0,
then M is a Harish-Chandra R-module.
Let
M = M0
⊕M1
be a simple weight R-module. Suppose that there exist λ ∈ C and i ∈ Z2 such
that
dimMλ
i = +∞.
Lemma 1 For any μ ∈ {λ + k | k ∈ Z},
dimMμ = +∞.
Proof Assume that there exists μ ∈ {λ + k | k ∈ Z} such that
dimMμ < +∞.
Without loss of generality, we may assume
μ = λ + 1,
i.e.,
dimMλ+1 < +∞.
Let V denote the intersection of the kernels of the linear maps
G1 : Mλ
i
→ Mλ+1
i+1
and
L1 : Mλ
i
→ Mλ+1
i
.
Since
dimMλ
i = +∞, dimMλ+1 < +∞,
we see that
dimV = +∞.
1238 Xiufu ZHANG
By the Principal Fact, M is a Harish-Chandra module, a contradiction.
Lemma 2 If there exist j ∈ Z2 and μ ∈ {λ + k | k ∈ Z} such that
dimMμ
j
< +∞,
then
μ +
1
24 c = 0.
In particular, there exists at most one of {Mμ
j
| μ ∈ λ + Z, j ∈ Z2} with finite
dimension.
Proof For any μ ∈ {λ + n | n ∈ Z}, by Lemma 1, if
dimMμ
j
< +∞,
then
dimMμ
j+1
= +∞.
Let V denote the kernel of the linear map
G0 : Mμ
j+1
→ Mμ
j
.
Then
dimV = +∞
and there exists 0 = v ∈ V such that G0v = 0. By identity
[G0,G0] = 2L0 +
1
12 c,
we have
2L0 +
1
12 c

v = 0,
which means
μ +
1
24 c = 0.
Lemma 3 (i) For 0 = x ∈ Mμ−1
j+1
, if G1x = 0, then
2(L1G0 + G1)G0x = 0.
Moreover,
L1x = 0.
(ii) For 0 = y ∈ Mμ+1
j+1
, if G−1y = 0, then
2(L−1G0 − G−1)G0y = 0.
Simple weight modules for the super-Virasoro algebra 1239
Moreover,
L−1y = 0.
Proof It can be checked directly by using Lemma 2.
Now, Theorem 1 follows from the following lemma.
Lemma 4 There is no μ ∈ {λ + k | k ∈ Z} and j ∈ Z2 such that
dimMμ
j
< +∞.
Proof Suppose that this is not the case. Then by Lemma 2, we can only
assume that
dimMμ
j
< +∞, dimMν
i = +∞
for ν = μ or i = j. Define
K = ker(G0 : Mμ
j+1
→ Mμ
j
) ∩ ker(G1L−1 : Mμ
j+1
→ Mμ
j
)
∩ ker(G−1L1 : Mμ
j+1
→ Mμ
j
) ∩ ker(G−1L1L1L−1 : Mμ
j+1
→ Mμ
j
).
Since
dimMμ
j
< +∞, dimMμ
j+1
= +∞,
K is a vector subspace of finite codimension in Mμ
j+1
. In order not to get a
direct contradiction by the Principal Fact, we assume that L−1v and L1v are
all nonzero for any 0 = v ∈ K. Thus,
dimL−1K = +∞, dimL1K = +∞.
Moreover,
dimker(G1 : L−1K → Mμ
j
) = +∞, dimker(G−1 : L1K → Mμ
j
) = +∞.
Choose an arbitrary nonzero x ∈ ker(G1 : L−1K → Mμ
j
), i.e.,
G1x = 0. (1)
In order not to reach a contradiction by Principal Fact, we must assume L1x =
0. Since
G0L1x =
1
2 G1x + L1G0x = L1G0L−1v = L1G−1v = 0,
we have L1x ∈ K and, in order not to get a contradiction by the Principal Fact,
we must assume
y := L1L1x = 0.
Since
G−1y = G−1L1L1x = G−1L1L1L−1v, v ∈ K ⊂ Mμ
j+1
,
1240 Xiufu ZHANG
we have
G−1y = 0 (2)
by the definition of K.
By (1) and Lemma 3 (i), we have
L1x = 0. (3)
By (2) and Lemma 3 (ii), we have
L−1y = 0. (4)
Denote the universal enveloping algebra of R by U(R). Then U(R) has a
natural Z2-graded structure induced from R:
U(R) = U(R)0
⊕ U(R)1.
Let U(R−) and U(R+) denote the subalgebras of U(R) generated by the
subalgebras R− and R+ of R, respectively. Define
N = U(R−)x ⊕ U(R−)G0x ⊕ U(R+)y ⊕ U(R+)G0y ⊂ M.
Since both U(R−) and U(R+) are stable under the action of L0 and both x
and y are eigenvectors for L0, we derive that N decomposes into a direct sum
of weight spaces which are obvious finite-dimensional.
If N is a submodule of M, then we can get a contradiction which means
that Theorem 1 holds. So our next goal is show that N is stable under the
action of R.
Since {G−1,G0,G1} is a set of generators of R, we need only to show
G0U(R−)x,G0U(R+)y,G1U(R−)x,G−1U(R+)y ⊂ N.
and
G0U(R−)G0x,G0U(R+)G0y,G1U(R−)G0x,G−1U(R+)G0y ⊂ N.
For any a ∈ U(R−)i, a ∈ U(R+)j , by the Poincar´e-Birkhoff-Witt theorem,
there exist
ak, ak,l ∈ U(R−)i+1, bk ∈ U(R−)i
and
a

k, a

k,l
∈ U(R+)j+1, b

k
∈ U(R+)j
such that
G0a =

k
ak + (−1)iaG0,
G1a = (−1)iaG1 +

k,l
ak,lLk0
cl +

k
bkG0,
Simple weight modules for the super-Virasoro algebra 1241
G0a
=

k
a

k + (−1)ja

G0,
G−1a
= (−1)ja

G−1 +

k,l
a

k,lLk0
cl +

k
b

kG0.
By using Lemma 2 and (1)–(4), we can check that the following holds:
G0ax =

k
akx + (−1)iaG0x ⊂ N,
G1ax = (−1)iaG1x +

k,l
ak,lLk0
clx +

k
bkG0x
=

k,l
ak,l(μ − 1)kclx +

k
bkG0x
⊂ N,
G0a(G0x) =

k
ak(G0x) + (−1)iaG20
x =

k
ak(G0x) − (−1)iax ⊂ N,
G1a(G0x) = (−1)iaG1G0x +

k,l
ak,lLk0
clG0x +

k
bkG20
x
=

k,l
ak,l(μ − 1)kclG0x −

k
bkx
⊂ N,
G0a

y =

k
a

ky + (−1)ja

G0y ⊂ N,
G−1a

y = (−1)ja

G−1y +

k,l
a

k,lLk0
cly +

k
a

kG0y
=

k,l
a

k,l(μ + 1)kcly +

k
a

kG0y
⊂ N,
G0a

G0y =

k
a

kG0y + (−1)ja

G20
y =

k
a

kG0y + (−1)ja

y ⊂ N,
G−1a

G0y = (−1)ja

G−1G0y +

k,l
a

k,lLk0
clG0y +

k
a

kG20
y
= 2(−1)ja

L−1y − (−1)ja

G0G−1y +

k,l
a

k,l(μ + 1)clG0y +

k
a

ky
=

k,l
a

k,l(μ + 1)clG0y +

k
a

ky
⊂ N.
This completes the proof of Lemma 4 and then of Theorem 1.
1242 Xiufu ZHANG
Acknowledgements The research presented in this paper was carried out during the
visit of author to Wilfrid Laurier University, Canada. The author thanks Wilfrid Laurier
University for hospitality. The author would like to thank Prof. K. Zhao for stimulating
discussion. The author also thanks the referees for good suggestions. This work was supported
in part by the National Natural Science Foundation of China (Grant Nos. 11271165, 11471333)
and the Natural Science Foundation for Colleges and Universities in Jiangsu Province (No.
14KJB110006).
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