Classification of irreducible weight modules over $W$-algebra W(2,2)

0.13 MB·English

by  Dong Liu

Classification of irreducible weight modules
over W-algebra W(2, 2) ∗
Dong Liu, Gao Shoulan
8
0
Department of Mathematics, Huzhou Teachers College
0
2 Zhejiang Huzhou, 313000, China

n
Linsheng Zhu
a
J
Department of Mathematics, Changshu Institute of Technology

8
1 Jiangsu Changshu, 215500, China

]
T
R
. Abstract. We show that the support of an irreducible weight module over the W-algebra
h
at W(2,2), which has an infinite dimensional weight space, coincides with the weight lattice and that
m
all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain
[
that every irreducible weight module over the the W-algebra W(2,2), having a nontrivial finite

2
v dimensional weight space, is a Harish-Chandra module (and hence is either an irreducible highest
3
or lowest weight module or an irreducible module of the intermediate series).
0
6
2 Key Words: The the W-algebra W(2,2), weight modules, support
.
1
0 Mathematics Subject Classification (2000): 17B56; 17B68.
8
0
1. Introduction
:
v
i The W-algebra W(2,2) was introduced in [ZD] for the study the classification of vertex
X
r operator algebras generated by weight 2 vectors.
a
Definition 1.1 The W-algebra L = W(2,2) is a Lie algebra over C (the field of complex
numbers) with the basis
{x ,I(n),C,C |n ∈ Z}
n 1
and the Lie bracket given by
n3 −n
[x ,x ] = (m−n)x +δ C, (1.1)
n m n+m n,−m
12
n3 −n
[x ,I(m)] = (m−n)I(n+m)+δ C , (1.2)
n n,−m 1
12
∗
E-mail: [email protected]
1
[I(n),I(m)] = 0, (1.3)
[L,C] = [L,C ] = 0. (1.4)
1
TheW-algebraW(2,2)canberealizedfromthesemi-product oftheVirasoroalgebraVir
andtheVir-moduleA oftheintermediateseriesin[OR]. Infact,letW = C{x | m ∈ Z}
0,−1 m
be the Witt algebra (non-central Virasoro algebra) and V = C{I(m) | n ∈ Z} be a W-
module with the action x · I(n) = (n − m)I(m + n), then W(2,2) is just the universal
m
central extension of the Lie algebra W ⋉V (see [OR] and [GJP]). The W-algebra W(2,2)
studied in [ZD] is the restriction for C = C of W(2,2) in our paper.
1
The W-algebra W(2,2) can be also realized from the so-called loop-Virasoro algebra (see
[GLZ]). Let C[t,t−1]be theLaurents polynomialring over C, then the loop-Virasoroalgebra
V˜L is the universal central extension of the loop algebra Vir ⊗ C[t1,t−1] and W(2,2) =
V˜L/C[t2].
The W-algebra W(2,2) is an extension of the Virasoro algebra and is similar to the
twisted Heisenberg-Virasoro algebra (see [ADKP]). However, unlike the case of the later,
the action of I(0) in W(2,2) is not simisimple, so its representation theory is very different
from that of the twisted Heisenberg-Virasoro algebra in a fundamental way.
Next we recallthedefinitions of Z-gradedL-modules. IfL-moduleV = ⊕m∈ZVm satisfies
L ·V ⊂ V , ∀m,n ∈ Z, (3.2)
m n m+n
then V is called a Z-graded L-module, and V is called a homogeneous subspace of V with
m
degree m ∈ Z.
AZ-gradedmoduleV iscalled quasi-finiteif allhomogeneoussubspaces arefinitedimen-
sional; a uniformly bounded module if there exists a number n ∈ N such that all dimensions
of the homogeneous subspaces are ≤ n; a module of the intermediate series if n = 1.
For any L-module V and λ ∈ C, set V := v ∈ V x v = λv , which we generally call
λ (cid:8) (cid:12) 0 (cid:9)
(cid:12)
the weight space of V corresponding the weight λ.
An L-module V is called a weight module if V is the sum of all its weight spaces. For a
weight module V we define
Supp(V) := λ ∈ C V 6= 0 ,
(cid:8) (cid:12) λ (cid:9)
(cid:12)
which is generally called the weight set (or the support) of V.
A nontrivial weight L-module V is called a weight module of intermediate series if V is
indecomposable and any weight spaces of V is one dimensional.
2
A weight L-module V is called a highest (resp. lowest) weight module with highest
weight (resp. highest weight) λ ∈ C, if there exists a nonzero weight vector v ∈ V such
λ
that
1) V is generated by v as L-module;
2) L v = 0 (resp. L v = 0).
+ −
Remark. For a highest (lowest) vector v we always suppose that I v = c v for some c ∈ C
0 0 0
although the action of I is not semisimple.
0
Obviously, if M is an irreducible weight L -module, then there exists λ ∈ C such that
Supp(M) ⊂ λ+Z. So M is a Z-graded module.
If, in addition, all weight spaces M of a weight L -module M are finite dimensional, the
λ
module is called a Harish-Chandra module. Clearly a highest (lowest) weight module is a
Harish-Chandra module.
Let U := U(L) be the universal enveloping algebra of L. For any λ,c ∈ C, let
I(λ,c,c ,c ) be the left ideal of U generated by the elements
0 1
(cid:8)xi,I(i) (cid:12) i ∈ N(cid:9)[(cid:8)x0 −λ·1,C −c·1,I0 −c0 ·1,C1 −c1 ·1(cid:9).
(cid:12)
Then the Verma module with the highest weight λ over L is defined as
M(λ,c,c ,c ) := U/I(λ,c,c ,c ).
0 1 0 1
It is clear that M(λ,c,c ,c ) is a highest weight module over L and contains a unique
0 1
maximal submodule. Let V(λ,c,c ,c ) be the unique irreducible quotient of M(λ,c,c ,c ).
0 1 0 1
The following result was given in [ZD].
Theorem 1.2 [ZD] The Verma module M(λ,c,c ,c ) is irreducible if and only if m2−1c +
0 1 12 1
2c 6= 0 for any nonzero integer m.
0
The classification of Harish-Chandra modules over the W-algebra W(2,2) was given in
[LLZ].
Theorem 1.3 [LLZ] A Harish-Chandra module L-module V is a highest weight module or
lowest weight module or a module of the intermediate series.
An irreducible weight module M is called a pointed module if there exists a weight λ ∈ C
such that dimV = 1. Xu posted the following in [X]:
λ
3
Problem 1.1 Is any irreducible pointed module over the Virasoro algebra a Harish-chandra
module?
An irreducible weight module M is called a mixed module if there exist λ ∈ C and i ∈ Z
such that dimV = ∞ and dimV < ∞. The following conjecture was posted in [M]:
λ λ+i
Conjecture 1.2 There are no irreducible mixed module over the Virasoro algebra.
MazorchukandZhao[MZ]gavethepositiveanswerstotheabovequestionandconjecture
to the Virasoro algebra, Shen and Su [SS] also also gave a similar result for the twisted
Heisenberg-Virasoro algebra.
In this paper, we also give the positive answers to the above question and conjecture
for the W-algebra W(2,2). Due to many differences between the W(2,2) and the twisted
Heisenberg-Virasoro algebra, some new methods are given in our paper.
Our main result is the following:
Theorem 1.4 Let M be an irreducible weight L -module. Assume that there exists λ ∈ C
such that dimM = ∞. Then Supp(M) = λ+Z, and for every k ∈ Z, we have dimM =
λ λ+k
∞.
The paper is organized as follow: Some lemma for the proof of Theorem 1.4 are given
in Section 2. The Proof of the main Theorem is given in Section 3 where some corollaries
from this theorem are also discussed.
2 Point modules over the W-algebra
We first recall a main result about the weight Virasoro-module in [MZ]:
Theorem 2.1 Let V be an irreducible weight Virasoro-module. Assume that there exists
λ ∈ C, such that dimV = ∞. Then Supp(V) = λ + Z, and for every k ∈ Z, we have
λ
dimV = ∞.
λ+k
Lemma 2.2 Assume that there exists µ ∈ C and a non-zero element v ∈ M , such that
µ
I v = L v = L I v = L v = 0 or I v = L v = L I v = L v = 0.
1 1 −1 2 2 −1 −1 1 −2 −2
Then M is a Harish-Chandra module.
Proof. Suppose that I v = L v = L v = 0 for v ∈ V , it is clear that L v = 0 and
1 1 2 µ >0
I v = 0 for m ≥ 3. Moreover L I v = 0 and I I v = 0 for m ≥ 3 or m = 1.
m >0 2 m 2
4
But L I v = 0, then L L I v = [L ,L ]I v +L L I v = −1L I v = 0. So I v = 0
−1 2 1 −1 2 1 −1 2 −1 1 2 2 0 2 2
if µ 6= −2. Then L v = 0. Hence v is a highest weight vector, and hence, M is a
>0
Harish-Chandra module.
If µ = −2 and w = I v 6= 0, then L w = L I v = [L ,I ]v +I L v = 0 for any n ∈ N.
2 n n 2 n 2 2 n
Moreover I w = 0 and L I w = [L ,I ]I v+I L I v = 0. Then I w = 0 since I w ∈/ V .
1 −1 2 −1 2 2 2 −1 2 2 2 0
So L w = 0. Hence w is either a highest weight vector, and hence, M is a Harish-Chandra
>0
(cid:3)
module. Similar for the lowest weight case.
Assume now that M is an irreducible weight L -module such that there exists λ ∈ C
satisfying dimM = ∞.
λ
Lemma 2.3 There exists at most one i ∈ Z such that dimM < ∞.
λ+i
Proof. Assume that
dimM < ∞ and dimM < ∞ for some different i,j ∈ Z.
λ+i λ+j
Without loss of generality, we may assume i = 1 and j > 1. Set
V := Ker(I : M → M )∩Ker(L ,L I : M → M )∩Ker(I : M → M )
1 λ λ+1 1 −1 2 λ λ+1 j λ λ+j
∩Ker(L : M → M ),
j λ λ+j
which is a subspace of M . Since
λ
dimM = ∞, dimM < ∞ and dimM < ∞,
λ λ+1 λ+j
we have, dimV = ∞. Since
[L ,L ] = (k −1)L 6= 0 and [I ,L ] = (l−1)I 6= 0 for k,l ∈ Z, k,l ≥ 2,
1 k k+1 1 l l+1
we get
L V = 0, k = 1,j,j +1,j +2,··· , and
k
(2.1)
I V = 0, l = 1,j,j +1,j +2··· .
l
If there would exist 0 6= v ∈ V such that L v = 0, then I v = L v = L I v = L v = 0 and
2 1 1 −1 2 2
M would be a Harish-Chandra module by Lemma 2.2. It is a contradiction. Hence L v 6= 0
2
for all v ∈ V. In particular,
dimL V = ∞.
2
Since dimM < ∞, and the actions of I and L on L V map L V (which is an
λ+1 −1 −1 2 2
infinite dimensional subspace of M ) to M (which is finite dimensional), there exists
λ+2 λ+1
5
0 6= w ∈ L V such that I w = L w = 0. Let w = L v for some v ∈ V. For all k ≥ j,
2 −1 −1 2
using (2.1), we have
L w = L L v = L L v +(2−k)L v = 0+0 = 0.
k k 2 2 k k+2
Hence L w = 0 for all k = 1,j,j +1,j +2,··· . Since
k
[L ,L ] = (l+1)L 6= 0 and [I ,I ] = (l+1)I 6= 0 for all l > 1,
−1 l l−1 −1 l l−1
we get inductively L w = I w = 0 for all k = 1,2,··· . Hence M is a Harish-Chandra
k k
(cid:3)
module by Lemma 2.2. A contradiction. The lemma follows.
Because of Lemma 2.3, we can now fix the following notation: M is an irreducible weight
L -module, µ ∈ C is such that dimM < ∞ and dimM = ∞ for every i ∈ Z\{0}.
µ µ+i
Lemma 2.4 Let 0 6= v ∈ M and µ 6= −1 such that I v = L v = L I v = 0. Then
µ−1 1 1 −1 2
(1) There exists a nonzero u ∈ M such that L v = I v = 0 for all m ≥ 1.
1 m
(2) I L v = 0 for all m ≥ 1.
m 2
Proof. Since L I v = 0, then L L I v = [L ,L ]I v +L L I v = −1L I v = 0. So
−1 2 1 −1 2 1 −1 2 −1 1 2 2 0 2
I v = 0 since µ 6= −1. By [L ,I ] = (k −1)I we have I v = 0 for all k ≥ 2. Moreover
2 1 k k+1 k
I L v = [I ,L ]v +L I v = (2−m)I v +L I v = 0. (cid:3)
m 2 m 2 2 m m+2 2 m
Lemma 2.5 Let 0 6= w ∈ M and µ 6= 1 such that I w = L w = L I w = 0. Then
µ+1 −1 −1 1 −2
(1)L w = I w = 0 for all m ≥ 1.
−1 −m
(2) I L w = 0 for all m ≥ 1.
−m −2
(cid:3)
Proof. It is similar to that in Lemma 2.4.
3 Proof of Theorem 1.4
Proof of Theorem 1.4. Due to Lemma 2.3, we can suppose that dimM < +∞ and
µ
dimM = +∞ for all i ∈ Z,i 6= 0.
µ+i
Set
V := Ker{L : M → M }∩Ker{I : M → M }
1 µ−1 µ 1 µ−1 µ
∩Ker{L I : M → M }∩Ker{L L : M → M } ⊂ M .
−1 2 µ−1 µ −1 2 µ−1 µ µ−1
6
For any v ∈ V, L v = I v = L I v = 0
1 1 −1 2
Since dimM = ∞ and dimM < ∞, we have dimV = ∞. For any v ∈ V, consider
µ−1 µ
the element L v. By Lemma 2.2, L v = 0 would imply that M is a Harish-Chandra module,
2 2
a contradiction. Hence L v 6= 0, in particular, dimL V = ∞.
2 2
Since the actions of I , L ,L L and L I on L V map L V (which is an infinite
−1 −1 1 −2 1 −2 2 2
dimensional subspace of M ) to M (which is finite dimensional), there exists w = L v ∈
µ+1 µ 2
L V for some v ∈ V, such that 0 6= w ∈ M and I w = L w = L I w = L L w = 0.
2 µ+1 −1 −1 1 −2 1 −2
(1) If µ 6= ±1, then
I w = 0, k = 1,2,··· (3.1)
k
from Lemma 2.4 and
I w = 0, k = 1,2,··· (3.2)
−k
from Lemma 2.5.
This means that I act trivially on M for all k ∈ Z, and so M is simply an irreducible
k
module over the Virasoro algebra. Thus, Theorem 1.3 follows from Theorem 2.1 in the case
µ 6= ±1.
(2) If µ = ±1, we only show that µ = 1 is not possible and for µ = −1 the statement
will follow by applying the canonical involution on L.
In fact, if µ = 1, then for v ∈ V, L v = I v = L I v = L L v = L v = 0. By Lemma
1 1 −1 2 −1 2 0
2.4, we have I v = 0,k = 1,2,···.
k
For any v ∈ V, consider the element L v. By Lemma 2.2, L v = 0 would imply that M
2 2
is a Harish-Chandra module, a contradiction. Hence L v 6= 0, in particular, dimL V = ∞.
2 2
SincetheactionsofI ,L andL I onL V mapL V (whichisaninfinitedimensional
−1 −1 1 −2 2 2
subspace of M ) to M (which is finite dimensional), there exists w = L v ∈ L V for some
2 1 2 2
v ∈ V, such that w 6= 0 and I w = L w = L I w = 0. Moreover we have
−1 −1 1 −2
I w = 0,k = 1,2,··· (3.3)
k
from Lemma 2.4. So
I w = 0. (3.4)
0
If L w = L L v = 0, then from L L v = 0 we have L L L v = [L ,L ]L v +
1 1 2 −1 2 1 −1 2 1 −1 2
L L L v = 0. So L L v = 2L V = 0 since L v ∈ M . Hence L v = 0 and then M is a
−1 1 2 0 2 2 2 2 2
highest weight module.
Then we can suppose that L w 6= 0 for any w ∈ L V.
1 2
For any w ∈ L V, consider the element L w. If L w = 0, then L w = I w =
2 −2 −2 −k −k
0, k = 1,2,···. Then M is a Harish-Chandra module. Hence L L V 6= 0, in particular,
−2 2
7
dimL L V = ∞. Let W = L V, then L maps L W to M has infinite dimensional
−2 2 2 1 −2 1
Kernel K. Let 0 6= L w ∈ K, then L L w = 0. But L L = [L ,L ] + L L and
−2 1 −2 1 −2 1 −2 −2 1
[L ,L ]w = (−3)L w = 0, hence L L w = 0. Setting u = L w 6= 0, we have L u = 0,
1 −2 −1 −2 1 1 −2
I u = I L w = [I ,L ]w +L I w = 0. Moreover by induction we have I u = 0 for
−1 −1 1 −1 1 1 −1 −m
all m ≥ 3.
I u = [I ,L ]w+L I w = (1−m)I w+L I w = 0 for all m ≥ 0 by (3.3) and (3.4).
m m 1 1 m m+1 1 m
So I u = 1[L ,I ]u = 0. Then I u = 0 for all k ∈ Z.
2 6 −2 4 k
By L u = 0, we have I L u = 0. Therefore c = 0.
−2 2 −2 1
This means that I ,k ∈ Z,C act trivially on the irreducible M for all k ∈ Z, and so
k 1
M is simply a module over the Virasoro algebra. Thus, Theorem 1.3 follows from Theorem
(cid:3)
2.1.
Theorem 1.4 also implies the following classification of all irreducible weight L -modules
which admit a nontrivial finite dimensional weight space:
Corollary 3.1 Let M be an irreducible weight L -module. Assume that there exists λ ∈ C
such that 0 < dimM < ∞. Then M is a Harish-Chandra module. Consequently, M
λ
is either an irreducible highest or lowest weight module or an irreducible module from the
intermidiate series.
Proof. Assume that M is not a Harish-Chandra module. Then there should exists i ∈ Z
such that dimM = ∞. In this case, Theorem 1.3 implies dimM = ∞, a contradiction.
λ+i λ
Hence M is a Harish-Chandra module, and the rest of the statement follows from Theorem
(cid:3)
1.3.
ACKNOWLEDGMENTS
ProjectissupportedbytheNNSF(Grant10671027,10701019,10571119),theZJZSF(Grant
Y607136), and Qianjiang Excellence Project of Zhejiang Province (No. 2007R10031). Au-
thors give their thanks to Prof. Congying Dong for his useful comments.
REFERENCES
[ADKP] Arbarello, E.; DeConcini, C.; Kac, V.G.; Procesi, C., Moduli spaces of curves and
representation theory Commun. Math. Phys., 117 (1988), 1-36.
[B] Billig, Y., Representations of the twisted Heisenberg-Virasoro algebra at level zero,
Canad. Math. Bull. 46 (2003), no. 4, 529–537.
8
[GJP] S. Gao; C. Jiang; Y. Pei, Structure of extensions of the Virasoro algebra and its
modules of tensor densities, preprint.
[GLZ] R. Lu; K. Zhao, Classification of irreducible weight modules over the loop-Virasoro
algebra, preprint.
[HL] Hu, N.; Liu, D., Decompositions of bosonic modules of Lie algebras W and
1+∞
W (gl ), Chinese Math. Ann. Vol.26 (Ser. B)(2005), 1-10.
1+∞ N
[JJ] Jiang, Q.F., Jiang, C., Representations of the twisted Heisenberg-Virasoro algebra and
the full Toroidal Lie Algebras (to appear in Algebra Colloquium).
[K] Kawamoto, N., Generalizations of Witt algebras over a field of characteristic zero,
Hiroshinma. Math. J. 16 (1986) 417-426.
[Kac] Kac, V. G., Some problems of infinite-dimensional Lie algebras and their representa-
tions, Lecture Notes in Mathematics 933, 117-126, Springer, 1982.
[Kas] Kassel, C., Cyclic homology of differential operators, the Virasoro algebra and a q-
analogue, Comm. Math. Phys. 164 (1992), 343–356.
[KP] Kac, V.G.; Peterson, D.H., Spin and wedge representations of infinite-dimensional Lie
algebras and groups, Proc. Natl. Acad. Sci. USA 78 (1981), 3308–3312.
[KPS] Kirkman, E.; Procesi, C.; Small, L., A q-analog for the Virasoro algebra, Comm. in
Algebra. 22 (10) (1994), 3755–3774.
[LLZ] Liu, D.; Zhu, L., Classification of Harish-Chandra modules over the W-algebra
W(2,2), Preprint.
[LJ] Liu, D.; Jiang, C.P., Harish-Chandra modules over the twisted Heisenberg-Virasoro
algebra, J. Math. Phys., In press (2008).
[LZ] Lu, R.; Zhao, K., Classification of irreducible weight modules over the twisted
Heisenberg-Virasoro algebra, math-ST/0510194.
[M] V. Mazorchuk, On simple mixed modules over the Virasoro algebra, Mat. Stud.
22(2004), 121-128.
[MZ] V. Mazorchuk, K. Zhao, Classification of simple weight Virasoro modules with a finite
dimensional weight space, J. Algebra 307 (2007), no. 1, 209–214..
9
[OR] V.Yu. Ovsienko, ,C. Roger, Extension of Virasoro group and Virasoro algebra by
modules of tensor densities on S1, Funct.Anal.Appl. 31, 4(1996).
[P] Pianzola, A., Automorphisms of toroidal Lie algebras and their central quotients, J.
Algebra Appl. 1 (2002), no. 1, 113–121.
[SS] Shen, R.; Su, Y.C., Classification of irreducible weight modules with a finite dimen-
sional weight space over twisted Heisenberg-Virasoro algebra, math.RT/0512209
[S1] Su, Y., Classification of Harish-Chandra modules over the higher rank Virasoro alge-
bras, Comm. Math. Phys. 240 (2003), no. 3, 539–551.
[S2] Su, Y., 2-cocycles on the Lie algebras of generalized differential operators, Comm. Alg.
(30) (2002), 763-782.
[SZ1] Su, Y.; Zhao, K., Generalized Virasoro and super-Virasoro algebras and modules of
the intermediate series, J. Alg. 252 (2002) 1–19.
[SZ2] Su,Y.; Zhao,K.,SimplealgebrasofWeyltype, ScienceinChinaA,44(2001)419-426.
[SZ3] Su, Y.; Zhao, K., Structure of Lie algebras of Weyl type, Comm. Alg. 32 (2004)
1052–1059.
[X] X. Xu, Pointed representations of Virasoro algebra, A Chinese summary appears in
Acta Math. Sinica 40 (1997) no. 3, 479. Acta Math. Sinica (N.S.) 13(1997), 161-168.
[Z] Zhao, K., Lie algebras of derivations of algebras of differential operators, Chin. Sci.
Bull. 38 (10) (1993), 793-798.
[ZD] W. Zhang; C. Dong, W-algebra W(2,2) and the Vertex operator algebra L(1,0) ⊗
2
L(1,0) , math.QA/, 071146.
2
10