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2010

Harish-Chandra Modules Over the Q Heisenberg-Virasoro Algebra

Xiangqian Guo Zhengzhou University

Xuewen Liu Zhengzhou University

Kaiming Zhao Wilfrid Laurier University, [email protected]

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Recommended Citation Guo, Xiangqian; Liu, Xuewen; and Zhao, Kaiming, "Harish-Chandra Modules Over the Q Heisenberg- Virasoro Algebra" (2010). Mathematics Faculty Publications. 51. https://scholars.wlu.ca/math_faculty/51

This Article is brought to you for free and open access by the Mathematics at Scholars Commons @ Laurier. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact [email protected]. J. Aust. Math. Soc. 89 (2010), 9–15 doi:10.1017/S1446788710001436

HARISH-CHANDRA MODULES OVER THE Q HEISENBERG–VIRASORO ALGEBRA

XIANGQIAN GUO ˛, XUEWEN LIU and KAIMING ZHAO

(Received 28 July 2009; accepted 6 June 2010)

Communicated by J. Du

Abstract

In this paper, it is proved that all irreducible Harish-Chandra modules over the Q Heisenberg–Virasoro algebra are of the intermediate series (all weight spaces are at most one-dimensional).

2000 Mathematics subject classiﬁcation: primary 17B10; secondary 17B65, 17B68, 17B70. Keywords and phrases: Virasoro algebra, Q Heisenberg–Virasoro algebra, modules of the intermediate series.

1. Introduction The Heisenberg–Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one. It contains the classical Heisenberg algebra and the Virasoro algebra as subalgebras. The structure of their irreducible highest weight modules was studied in [1, 3]. Irreducible Harish-Chandra modules over the Heisenberg–Virasoro algebra were classiﬁed in [13]. They are either highest weight modules, lowest weight modules, or modules of the intermediate series. The representation theory of the Heisenberg–Virasoro algebra is closely related to those of other Lie algebras, such as the Virasoro algebra and toroidal Lie algebras, see [4, 5, 8]. For other results on Heisenberg–Virasoro algebras, see [11, 16] and the references therein. Recently, some authors have introduced generalized Heisenberg–Virasoro algebras and started to study their representations (see [12, 17]). In this paper, we give the classiﬁcation of irreducible Harish-Chandra modules over the Q Heisenberg–Virasoro algebra. As in the Q Virasoro algebra case [15], only modules of the intermediate series appear. The main ideas in our proof (Lemma 3.1) are similar to those of [15] and [6].

c 2010 Australian Mathematical Publishing Association Inc. 1446-7887/2010 $16.00

9 10 X. Guo, X. Liu and K. Zhao [2]

In this paper we denote by Z, Q, and C the sets of integers, rational numbers and complex numbers, respectively. Now we give the deﬁnitions of the generalized Heisenberg–Virasoro algebras and the Q Heisenberg–Virasoro algebra. DEFINITION 1.1. Suppose that G is an additive subgroup of C. The generalized Heisenberg–Virasoro algebra HVir[G] is the Lie algebra over C with a basis

{dg, I (g), CD, CDI , CI | g ∈ G}, subject to the Lie brackets given by g3 − g [dg, dh] = (h − g)dg+h + δg −h CD, , 12 2 [dg, I (h)] = hI (g + h) + δg,−h(g + g)CDI , [I (g), I (h)] = gδg,−hCI , [HVir[G], CD] = [HVir[G], CDI ] = [HVir[G], CI ] = 0. It is easy to see that HVir[G]' HVir[G0] if and only if there exists a nonzero 0 a ∈ C such that G = aG . If G = Z, then HVir[Z] = HVir is the classical Heisenberg– Virasoro algebra. When G = Q, we call HVir[Q] the Q Heisenberg–Virasoro algebra; this is the main object of this paper. An HVir[G] module V is said to be trivial if HVir[G]V = 0, and we denote the one-dimensional trivial module by T . The modules of the intermediate series V (α, β; F) over HVir[G] are deﬁned as follows, for all α, β, F ∈ C. As a vector space over C, the space V (α, β; F) has a basis {vg | g ∈ G} and the actions are

dgvh = (α + h + gβ)vg+h, I (g)vh = Fvg+h, (1.1) CDvg = 0, CI vg = 0, CDI vg = 0 ∀g, h ∈ G. (1.2) ∼ It is well known that V (α, β; F) = V (α + g, β; F) for all α, β, F ∈ C and g ∈ G. So we always assume that α = 0 when α ∈ G. It is also easy to see that V (α, β; F) is reducible if and only if F = 0, α = 0, and β ∈ {0, 1}. The module V (0, 0; 0) has a one-dimensional submodule T and V (0, 0; 0)/T is irreducible; V (0, 1; 0) has a codimension one irreducible submodule. We denote the unique nontrivial irreducible subquotient of V (α, β; F) by V 0(α, β; F). It is easy to see that V 0(0, 0; 0) =∼ V 0(0, 1; 0). 0 0 0 REMARK 1.2. When F 6= 0, it is not hard to verify that V (α, β; F) =∼ V (α , β ; F ) if and only if α − α0 ∈ G and (β, F) = (β0, F0). Here is our main theorem.

THEOREM 1.3. Suppose that V is an irreducible nontrivial Harish-Chandra module 0 over HVir[Q]. Then V is isomorphic to V (α, β; F) for suitable α, β, F ∈ C. This paper is organized as follows. In Section 2, we collect some known results on the Virasoro algebra and on the Heisenberg–Virasoro algebra for later use. In Section 3, we give the proof of the main theorem. [3] Harsh-Chandra modules over HVir[Q] 11

2. Preliminaries Since generalized Heisenberg–Virasoro algebras are closely related to generalized Virasoro algebras, we ﬁrst recall some results on generalized Virasoro algebras.

DEFINITION 2.1. Let G be a nonzero additive subgroup of C. The generalized Virasoro algebra Vir[G] is a Lie algebra over C with a basis {dg, CD | g ∈ G} and is subject to the Lie brackets given by g3 − g [dg, dh] = (h − g)dg+h + δg −h CD, [Vir[G], CD] = 0. , 12 Notice that Vir[G] is a subalgebra of HVir[G]. If G = Z, then Vir[Z] = Vir is the classical Virasoro algebra and if G = Q, then Vir[Q] is called the Q Virasoro algebra. A module of the intermediate series V (α, β) has a C-basis {vg | g ∈ G} and the Vir-actions are

dgvh = (α + h + gβ)vg+h, CDvg = 0 ∀g, h ∈ G. It is well known that V (α, β) is reducible if and only if α ∈ G and β ∈ {0, 1}. The unique nontrivial irreducible subquotient of V (α, β) is denoted by V 0(α, β). It is also ∼ known that V (α, β) = V (α + g, β) for all α, β ∈ C and all g ∈ G. So we always assume that α = 0 if α ∈ G. Mazorchuk [15] classiﬁed irreducible Harish-Chandra modules over Vir[Q]. THEOREM 2.2. Every irreducible Harish-Chandra module over Vir[Q] is isomorphic 0 to V (α, β) for suitable α, β ∈ C. Now we consider the Q Heisenberg–Virasoro algebra HVir[Q], which contains the classical Heisenberg–Virasoro algebra HVir[Z] = HVir. The classiﬁcation of irreducible Harish-Chandra modules over HVir was given in [13].

THEOREM 2.3. Any irreducible Harish-Chandra module over HVir is isomorphic to either a highest weight module, a lowest weight module, or V 0(α, β; F) for suitable α, β, F ∈ C. 3. Proof of Theorem 1.3

We decompose Q as the union of a series of rings Qk = {n/k! | n ∈ Z}, where k ∈ N. Then we can view each HVir[Qk] as a subalgebra of HVir[Q] naturally, and thus HVir[ ] = S HVir[ ]. Clearly, HVir[ ]' HVir for all k ∈ . For Q k∈N Qk Qk N convenience, we write U(Q) = U(HVir[Q]) and U(Qk) = U(HVir[Qk]). Denote U(Q)q = {u ∈ U(Q) | [d0, u] = qu} and U(Qk)q = {u ∈ U(Qk) | [d0, u] = qu}, for all q ∈ Q.

LEMMA 3.1. Suppose that N is a ﬁnite-dimensional irreducible U(Q)0-module; then there exists k ∈ N such that N is an irreducible U(Qk)0-module.

PROOF. There is an associative algebra homomorphism 8: U(Q)0 → gl(N), where gl(N) is the general linear associative algebra of N. 12 X. Guo, X. Liu and K. Zhao [4]

Since N and hence gl(N) are both ﬁnite-dimensional, U(Q)0/ker(8) is ﬁnite- dimensional. Take y1, y2,..., ym ∈ U(Q)0 such that y1, y2,..., ym form a basis of U(Q)0/ker(8). Then there exists k ∈ N such that y1, y2,..., ym are all in U(Qk)0. We claim that N is an irreducible U(Qk)0-module. Let M be a proper U(Qk)0- submodule of N. For all y ∈ U(Q)0, there exist y0 ∈ ker(8) and ai ∈ C such that Pm y = y0 + i=1 ai yi . Thus m X X yM = y0 + ai yi M ⊆ ai yi M ⊆ M, i=1 that is, M is a U(Q)0-submodule of N, which forces M = 0. Thus N is irreducible over U(Qk)0. 2 Now we ﬁx a nontrivial irreducible Harish-Chandra module V over HVir[Q]. Then there exists α ∈ such that V = L V , where V = {v ∈ V | d v = (α + q)v}. C q∈Q q q 0 We deﬁne the support of V as supp V = {q ∈ Q | Vq 6= 0}.

LEMMA 3.2. supp V = Q or Q \ {−α} and dim Vq = 1 for all q ∈ supp V.

PROOF. First view V as a Vir[Q] module; by Theorem 2.2, dim Vp = dim Vq for all p, q ∈ Q \ {−α}. In particular, V is a uniformly bounded module, that is, the dimensions of all weight spaces are bounded by a positive integer. Suppose that dim Vq > 1 for some q ∈ supp V . Then it is easy to see that Vq is an irreducible U[Q]0-module. By Lemma 3.1, there is some k ∈ N such that Vq is an irreducible U(Qk)0-module. We consider the HVir[Qk]-module W = U(Qk)Vq , which is uniformly bounded. Then, by Theorem 2.3, there is a composition series of HVir[Qk]-modules, namely, 0 = W (0) ⊂ W (1) ⊂ W (2) ⊂ · · · ⊂ W (m) = W. Each factor W (i)/W (i−1) is either trivial or of the intermediate series, and in both cases all weight spaces are one-dimensional. (i) (i) (i) (i−1) Take the ﬁrst i such that dim Wq 6= 0. Then dim Wq = dim(W /W )q = 1. (i) But, on the other hand, Wq is a U(Qk)0-module, that is, a nontrivial proper U(Qk)0- submodule of Vq . This is a contradiction, so dim Vq = 1 for all q ∈ supp V . Since V is nontrivial, there exists q ∈ supp V \ {−α}. Our result follows since dim Vp = dim Vq = 1 for all p ∈ Q \ {−α}. 2 Now we can give the proof of our main theorem.

PROOFOF THEOREM 1.3. By Lemma 3.2, we know that supp V = Q for some α ∈ C or supp V = \ {−α}. Denote V (0) = 0 and V (k) = L V for all k ≥ 1. Then Q q∈Qk q (k) (k) supp V = {q ∈ Q | Vq 6= 0} = supp V ∩ Qk for all k ≥ 1. There is a vector space ﬁltration of V , namely, ∞ [ 0 = V (0) ⊂ V (1) ⊂ V (2) ⊂ · · · ⊂ V (k) ⊂ · · · ⊂ V and V = V (k). k=0 [5] Harsh-Chandra modules over HVir[Q] 13

(k) It is clear that each V can be viewed as an HVir[Qk]-module, and each HVir[Qk] is isomorphic to HVir. (k) If −α∈ / supp V , then each V is irreducible over HVir[Qk] when k ∈ Z, by Lemma 3.2 and Theorem 2.3. If −α ∈ supp V , then α = 0 by assumption. Since V is an irreducible HVir[Q]- module, there exists p ∈ Q \{0} such that u p ∈ U(HVir[Q])p and u p V0 = Vp, as well as q ∈ Q \{0} such that uq ∈ U(HVir[Q])q and uq V−q = V0. Then there exists k0 such ∈ [ ] (k0) [ ] that u p, uq U(HVir Qk0 ), and thus V is irreducible as an HVir Qk0 -module. (k0) [ ] (k) [ ] Now V is irreducible over HVir Qk0 , and so V is irreducible over HVir Qk whenever k ≥ k0. By Theorem 2.3, we see that CD V = CI V = CDI V = 0, that I (0) (k) ∼ 0 acts as a scalar F ∈ C, and that there exist αk, βk ∈ C such that V = V (αk, βk; F) as modules over HVir[Qk] (deﬁned by (1.5) and (1.6) with G = Qk) when k ≥ k0. = = Write α αk0 and β βk0 for short. If F = 0, V is an irreducible Vir(Q)-module and I (p)V = 0 for all p ∈ Q. Theorem 1.3 follows from Theorem 2.2. Next we assume that F 6= 0. ∼ 0 Now we need to prove that V = V (α, β; F) as modules over HVir[Q], that is, we can choose a basis of V such that (1.1) and (1.2) hold. We proceed by choosing a basis of each V (k) inductively, such that (1.1) and (1.2) hold when one only considers the (k+1) actions of HVir[Qk], and that the basis of each V is the extension of the basis of (k) V for all k ≥ k0. (k ) Naturally, there is such a basis for V 0 . Now suppose that m > k0 and that we have (k) found such bases for V whenever k0 ≤ k ≤ m − 1. In particular, there is a basis (m−1) {vq ∈ Vq | q ∈ supp V } with the property that dpvq = (α + q + pβ)vp+q and (m−1) (m) I (p)vq = Fvp+q for all p, q ∈ supp V . Now we consider V . Let 8 be the canonical isomorphism HVir → HVir[Qm] given by 8(dn) = m! dn/m! (m) and 8(I (n)) = m! I (n/m!) for all n ∈ Z. Then V can be viewed as an HVir module via 8 and is isomorphic to V (α0, β0; F0) as modules over HVir for suitable 0 0 0 0 (m) (m) α , β , F ∈ C, by Theorem 2.3. Then there is a basis {vq | q ∈ supp V } of V such that, for all i/m!, k/m! ∈ supp V (m),

0 0 0 0 0 0 0 (m! di/m!)vk/m! = (α + k + iβ )v(k+i)/m!,(m! I (i/m!))vk/m! = F v(k+i)/m!. That is, 0 0 0 α k i 0 0 0 F 0 d !v = + + β v , I (i/m!)v = v , i/m k/m! m! m! m! (k+i)/m! k/m! m! (k+i)/m! or equivalently,

α0 F0 d v0 = + q + pβ0 v0 , I (p)v0 = v0 ∀p, q ∈ supp V (m). p q m! p+q q m! p+q

(m−1) (m) 0 0 0 We take p = 0 and q ∈ supp V ⊂ supp V , and then d0vq = (α /m! + q)vq 0 0 0 0 0 and I (0)vq = (F /m!)vq , which shows that α = m! α and F = m! F. 14 X. Guo, X. Liu and K. Zhao [6]

0 (m−1) 0 0 Assume that vq = cq vq for all q ∈ supp V . Compare I (p)vq = Fvp+q and 0 I (p)vq = Fvp+q . We see that cp is independent of p. Then we may choose vp = vp for all p ∈ supp V (m−1). By the remark in Section 1 we see that β0 = β. Denote 0 (m) (m−1) (m) vq = vq for all q ∈ supp V \ supp V . Thus the basis {vq | q ∈ supp V } (m−1) is an extension of {vp | p ∈ supp V } such that dpvq = (α + q + pβ)vp+q and (m) I (p)vq = Fvp+q for all p, q ∈ supp V . By induction, we can produce a basis {vq | q ∈ supp V } for V such that dpvq = (α + q + pβ)vp+q and I (p)vq = Fvp+q for all p, q ∈ supp V and further ∼ 0 CD V = CI V = CDI V = 0. That is, V = V (α, β; F). This completes the proof. 2 Recall from [10] that the rank of an additive subgroup G of C, denoted by rank(G), is the maximal number r for which we can ﬁnd g1,..., gr ∈ G \{0} such that Zg1 + · · · + Zgr is a direct sum. If such an r does not exist, then we deﬁne rank(G) = ∞. Now we can see that the proof of Theorem 1.3 is also valid when G is an inﬁnitely generated additive subgroup of C of rank one. Thus we have the following result. THEOREM 3.3. Let G be an inﬁnitely generated additive subgroup of C of rank one. Then any nontrivial irreducible Harish-Chandra module over HVir[G] is isomorphic 0 to V (α, β; F) for suitable α, β, F ∈ C.

Acknowledgements The authors are grateful to the referee and the editor for valuable suggestions to make the paper more readable.

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XIANGQIAN GUO, Department of Mathematics, Zhengzhou University, Zhengzhou 450001, Henan, PR China e-mail: [email protected]

XUEWEN LIU, Department of Mathematics, Zhengzhou University, Zhengzhou 450001, Henan, PR China e-mail: [email protected]

KAIMING ZHAO, Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5 and Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, PR China e-mail: [email protected]