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. 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