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Classification of finite irreducible conformal modules over Lie conformal algebra W(a,b,r)

  • We study a family of non-simple Lie conformal algebras W(a,b,r) (a,b,rC) of rank three with free C[]-basis {L,W,Y} and relations [LλL]=(+2λ)L, [LλW]=(+aλ+b)W, [LλY]=(+λ)Y, [YλW]=rW and [YλY]=[WλW]=0. In this paper, we investigate the irreducibility of all free nontrivial W(a,b,r)-modules of rank one over C[] and classify all finite irreducible conformal modules over W(a,b,r).

    Citation: Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra W(a,b,r)[J]. Electronic Research Archive, 2021, 29(3): 2445-2456. doi: 10.3934/era.2020123

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  • We study a family of non-simple Lie conformal algebras W(a,b,r) (a,b,rC) of rank three with free C[]-basis {L,W,Y} and relations [LλL]=(+2λ)L, [LλW]=(+aλ+b)W, [LλY]=(+λ)Y, [YλW]=rW and [YλY]=[WλW]=0. In this paper, we investigate the irreducibility of all free nontrivial W(a,b,r)-modules of rank one over C[] and classify all finite irreducible conformal modules over W(a,b,r).



    Lie conformal algebras, introduced by Kac in [9,11], encode the singular part of the operator product expansion of chiral fields in conformal field theory [3]. The theory of Lie conformal algebras plays an important role in quantum field theory and vertex algebras. They are related to vertex algebras in a similar manner as Lie algebras are related to their universal enveloping algebras (see [4]). Furthermore, Lie conformal algebras have close connections to the Hamiltonian formalism in the theory of nonlinear evolution equations (see [2]). They have also proved to be an useful tool in the study of infinite dimensional Lie algebras satisfying the locality property in [10].

    The theory of finite simple Lie conformal algebras has been developed in recent years. Finite simple Lie conformal algebras were classified in [7], which shows that a finite simple Lie conformal algebra is isomorphic to either the Virasoro Lie conformal algebra Vir or a current Lie conformal algebra Curg, where g is a finite dimensional simple Lie algebra. All finite irreducible conformal modules of these simple Lie conformal algebras were further classified in [5] and of their extensions in [6]. Moreover, the cohomology theory of these Lie conformal algebras was developed in [1,8]. However, the structure theory, representation theory and cohomology theory of finite non-simple Lie conformal algebras is far from being well developed.

    The Virasoro algebra, that is the universal central extension of the Witt algebra, plays a fundamental role in two dimensional conformal quantum field theory and has important applications in many fields of mathematics and physics. From a semi-direct sum of the centerless Virasoro algebra and its intermediate series module A(a,b), a class of Lie conformal algebras W(b) of rank two was first obtained in [17], and their conformal modules of rank one were also constructed. Finite irreducible conformal modules over W(b) were classified in [16] and of the extensions in [12]. A more general class of Lie conformal algebras W(a,b), constructed as a semi-direct sum of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one, was introduced in [14]. A complete classification of finite irreducible conformal modules of W(a,b) was recently given in [15]. W(1,0) is just the Heisenberg-Virasoro Lie conformal algebra. The study of its structure, representation, and cohomology can be seen in [13,16,18]. W(2,0) is just the W(2,2) Lie conformal algebra. Its structure and representation were studied in [16,19].

    As for finite non-simple Lie conformal algebras, it is very natural to first study some important examples. In this paper, we focus on the Lie conformal algebra W(a,b,r) with three parameters a,b,rC, which is a free C[]-module generated by L, W and Y, satisfies the following λ-brackets,

    [LλL]=(+2λ)L,[LλW]=(+aλ+b)W,[LλY]=(+λ)Y, [YλW]=rW,  [YλY]=[WλW]=0. (1)

    The Lie conformal algebra W(a,b,r) is obviously finite and has a nontrivial abelian conformal ideal C[]W, thus it is not simple. Note also that W(a,b,r) contains the Virasoro Lie conformal algebra Vir, W(b) and W(a,b) as subalgebras. Therefore, the results of this paper generalize the related ones of the Lie conformal algebras W(b) and W(a,b) to some extent. We hope that our work here will shed some light on the representation theory of non-simple Lie conformal algebras. Moreover, it is known that a formal distribution Lie algebra can be regarded as a Lie conformal algebra. And the category of Lie conformal algebras is almost equivalent to that of formal distribution Lie algebras (see [9]). Therefore our results are also beneficial for understanding the formal distribution Lie algebras corresponding to W(a,b,r). This is our motivation to present this work.

    The following theorem is the main result of this paper.

    Main Theorem. (1) If a1 or b0 or r0, then any finite nontrivial irreducible W(a,b,r)-module M is free of rank one over C[], and M is isomorphic to Mα,β,γ defined in (16) with α0 or γ0.

    (2) If a=1, b=0 and r=0, then any finite nontrivial irreducible W(1,0,0)-module M is free of rank one over C[], and M is isomorphic to Mα,β,γ,Δ defined in (17) with α0 or γ0 or Δ0.

    The organization of the present paper is as follows. In section 2, we recall some basic definitions and related known results about Lie conformal algebras. In section 3, we first investigate the structure of the extended annihilation algebra of W(a,b,r), then we study the irreducibility of all free nontrivial W(a,b,r)-modules of rank one over C[]. Based on this, we finally give a complete classification of all finite nontrivial irreducible conformal modules of W(a,b,r).

    Throughout this paper, we denote by C, Z, Z+ the sets of complex numbers, integers and nonnegative integers respectively. In addition, all vector spaces, linear maps and tensor products are assumed to be over C.

    In this section, we summarize some basic definitions and results concerning Lie conformal algebras. For more details, one can refer to [1,5,9].

    Definition 2.1. A Lie conformal algebra A is a C[]-module endowed with a C-linear map AAC[λ]A, ab[aλb], satisfying

    [(a)λb]=λ[aλb],[aλb]=(+λ)[aλb](conformal sesquilinearity), (2)
    [aλb]=[bλa]                                   (skew-symmetry), (3)
    [aλ[bμc]]=[[aλb]λ+μc]+[bμ[aλc]]                (Jacobi identity), (4)

    where a,b,cA.

    The ideals, subalgebras and homomorphisms of Lie conformal algebras can be defined naturally. A Lie conformal algebra is called simple if it is neither abelian nor does it contain a nontrivial ideal. A Lie conformal algebra is called finite if it is finitely generated as a C[]-module. The rank of a Lie conformal algebra A, denoted by rank(A), is its rank as a C[]-module.

    There are two important examples of Lie conformal algebras. One is the Virasoro Lie conformal algebra Vir. It is defined by

    Vir=C[]L,[LλL]=(+2λ)L.

    It is known that a torsion-free Lie conformal algebra of rank one is either trivial or isomorphic to Vir. The other is the current Lie conformal algebra associated to g, where g is a Lie algebra, which is defined by

    Curg=C[]g,[aλb]=[a,b],a,bg.

    The Lie conformal algebra W(a,b) with two parameters a,bC is a free C[]-module generated by L and W, satisfying

    [LλL]=(+2λ)L,[LλW]=(+aλ+b)W,[WλW]=0.

    Note that the Virasoro Lie conformal algebra Vir and Lie conformal algebra W(a,b) are both the subalgebras of W(a,b,r) defined in (1).

    For a given Lie conformal algebra A, from [9], we know that there is an important Lie algebra associated to it. For each jZ+, regarding [aλb]C[λ]A as a formal polynomial in λ, we can define the jth product a(j)b by the coefficient of λj in [aλb], i.e. a(j)b for all a,bA as follows:

    [aλb]=jZ+(a(j)b)λjj!. (5)

    Let ˆA=AC[x,x1]. We denote axn=axnˆA, (axn)=(a)xn and x(axn)=naxn1, then we extend and x linearly to the whole of ˆA. Define

    [axm,bxn]=jZ+(mj)(a(j)b)xm+nj (6)

    for all axm,bxnˆA. Then ˆA is a Lie algebra. Set ˆ=+x. It can be verified that ˆˆA is an ideal of ˆA. Then Lie(A)=ˆA/ˆˆA is also a Lie algebra. It is called the coefficient algebra of A. We denote a(n)=¯axnLie(A). The Lie subalgebra

    Lie(A)+=span{a(n)aA,nZ+}

    is called the annihilation algebra of A. The semi-direct sum of the 1-dimensional Lie algebra C and Lie(A)+ with the action (a(n))=na(n1) is called the extended annihilation algebra Lie(A)e.

    Definition 2.2. A conformal module M over a Lie conformal algebra A is a C[]-module endowed with a C-linear map AMC[λ]M, avaλv, satisfying

    (a)λv=λaλv,aλ(v)=(+λ)aλv, (7)
    aλ(bμv)bμ(aλv)=[aλb]λ+μv, (8)

    where a,bA and vM.

    A conformal module M is called finite if M is finitely generated over C[]. The rank of a conformal module M is its rank as a C[]-module. A conformal module M is called irreducible if it has no nontrivial submodules.

    In the following, we only consider conformal modules, thus we simply abbreviate the term conformal modules to modules.

    Suppose that M is a module over a Lie conformal algebra A. An element mM is called invariant if Aλm=0. Obviously, the set of all invariant elements in M is a conformal submodule of M, denoted by M0. An A-module M is called trivial if M0=M, i.e. a module on which A acts trivially. For all aC, we obtain a natural trivial A-module Ca which is determined by a, such that Ca=C and m=am, Aλm=0 for all mCa. The modules Ca with aC exhaust all trivial irreducible A-modules. Therefore, we only need to consider nontrivial modules.

    Let M be an A-module. An element mM is called a torsion element if there exists a nonzero polynomial p()C[] such that p()m=0. From [5,16], we have the following two lemmas.

    Lemma 2.3. Let A be a Lie conformal algebra and M be an A-module.

    (1) If m=am for some aC and mM, then Aλm=0.

    (2) If M is a finite module without any nonzero invariant element, then M is a free C[]-module.

    Lemma 2.4. Let A be a Lie conformal algebra and M be a finite nontrivial irreducible A-module. Then M has no nonzero torsion elements and is free of a finite rank as a C[]-module.

    Similar to the definition of the jth product a(j)b of two elements a,bA, we can also define jth actions of A on M for each jZ+, i.e. a(j)v for all aA,vM by

    aλv=jZ+(a(j)v)λjj!. (9)

    If V is an A-module, by (9), it can be regarded as a module over Lie(A)e naturally. Conversely, if V is a module over Lie(A)e such that for all vV,aA

    a(n)v=0,nN,

    where N is a nonnegative integer depending on a and v. By (9), V can be viewed as an A-module.

    A close connection between the module of a Lie conformal algebra and that of its extended annihilation algebra was also given in [5].

    Lemma 2.5. Let A be a Lie conformal algebra and M be an A-module. Then M is precisely a module over Lie(A)e satisfying the property

    a(n)v=0,nN, (10)

    for aA,vM, where N is a nonnegative integer depending on a and v.

    In the following, by abuse of notations, we also call a Lie(A)e-module satisfying the above equation (10) a conformal Lie(A)e-module.

    Let L be a Lie algebra over C with a distinguished element and a descending sequence of subspaces LL0L1L2Ln, such that [,Lk]=Lk1 for all k>0. Let V be an L-module such that given any vV there exists a nonnegative integer N (depending on v) such that LNv=0. Let Vn={vVLnv=0}, and let N be the minimal nonnegative integer such that VN0 (it exists by definition). Then from [5], we have the following lemma.

    Lemma 2.6. Suppose that N1. Then C[]VN=C[]VN and therefore C[]VNVN=VN. In particular VN is a finite dimensional vector space if V is finite.

    In this section, we firstly consider the extended annihilation algebra Lie(W(a,b,r))e of W(a,b,r). Then we discuss the irreducibility of the free nontrivial W(a,b,r)-modules of rank one. Based on this, we further give the proof our main Theorem.

    Since the Virasoro Lie conformal algebra Vir and Lie conformal algebra W(a,b) are both the subalgebras of W(a,b,r), we recall some results about their modules. From [5,15], we have the following two propositions.

    Proposition 1. All free nontrivial Vir-modules of rank one over C[] are as follows:

    Mα,β=C[]v,Lλv=(+αλ+β)v, (11)

    where α,βC. Moreover, the module Mα,β is irreducible if and only if α0. The module M0,β contains a unique nontrivial submodule (+β)M0,β which is isomorphic to M1,β. The modules Mα,β with α0 exhaust all finite irreducible nontrivial Vir-modules.

    Proposition 2. If a1 or b0, all free nontrivial W(a,b)-modules of rank one over C[] are as follows:

    Mα,β=C[]v,Lλv=(+αλ+β)v,Wλv=0, (12)

    where α,βC. Moreover, the module Mα,β is irreducible if and only if α0.

    If a=1 and b=0, all free nontrivial W(1,0)-modules of rank one over C[] are as follows:

    Mα,β,γ=C[]v,Lλv=(+αλ+β)v,Wλv=γv, (13)

    where α,β,γC. Moreover, the module Mα,β,γ is irreducible if and only if α0 or γ0.

    Now, we investigate the extended annihilation algebra of W(a,b,r) and obtain the following result.

    Lemma 3.1. The annihilation algebra of W(a,b,r) is

    Lie(W(a,b,r))+=m1CLmn0CWns0CYs,

    with the following Lie brackets:

    [Lm,Ln]=(mn)Lm+n,[Lm,Wn]=bWm+n+1+((a1)(m+1)n)Wm+n,[Lm,Ys]=sYm+s,[Ys,Wn]=rWs+n,[Wm,Wn]=[Ym,Yn]=0. (14)

    And the extended annihilation algebra is Lie(W(a,b,r))e=CLie(W(a,b,r))+, which satisfies (14) and

    [,Lm]=(m+1)Lm1,[,Wn]=nWn1,[,Ys]=sYs1. (15)

    Proof. By (1) and (5), we get

    L(0)L=L,L(1)L=2L,L(0)Y=Y,L(1)Y=Y,L(0)W=(+b)W,L(1)W=aW,Y(0)W=rW,L(j)L=L(j)W=L(j)Y=Y(j1)W=W(j2)W=Y(j2)Y=0,j2.

    Then, for all m,nZ+, it follows from (6) that

    [L(m),L(n)]=jZ+(mj)(L(j)L)(m+nj)=(mn)L(m+n1),[L(m),W(n)]=jZ+(mj)(L(j)W)(m+nj)=bW(m+n)+((a1)mn)W(m+n1),[L(m),Y(s)]=jZ+(mj)(L(j)Y)(m+sj)=sY(m+s1),[Y(s),W(n)]=jZ+(sj)(Y(j)W)(s+nj)=rW(s+n),[W(m),W(n)]=[Y(m),Y(n)]=0.

    Taking Lm=L(m+1) for m1 and Wn=W(n), Yn=Y(n) for n0, we can prove that the Lie brackets in (14) hold. By definition, the extended annihilation algebra Lie(W(a,b,r))e is the semi-direct sum of the 1-dimensional Lie algebra C and Lie(W(a,b,r))+ with the action =x, we obtain (15). Hence, we complete the proof.

    From (14) and (15), the following lemma follows.

    Lemma 3.2. L1 is in the center of Lie(W(a,b,r))e.

    From now on, we denote LW(a,b,r)=Lie(W(a,b,r))e. Define W1=Y1=0 and LW(a,b,r)n=in(CLiCWiCYi) for 1nZ. Then, by Lemma 3.2, we have LW(a,b,r)1=Lie(W(a,b,r))+ and we obtain a filtration of subalgebras of LW(a,b,r):

    LW(a,b,r)LW(a,b,r)1LW(a,b,r)0LW(a,b,r)n.

    For this filtration, by Lemma 3.2, we can obtain the following lemmas step by step.

    Lemma 3.3. (1) [LW(a,b,r)m,LW(a,b,r)n]LW(a,b,r)m+n for m,nZ+. In particular, LW(a,b,r)n is an ideal of LW(a,b,r)0 for all nZ+.

    (2) For all nZ+, [,LW(a,b,r)n]=LW(a,b,r)n1.

    Lemma 3.4. (1) If a=0 and r=0, then [LW(a,b,r)0,LW(a,b,r)0]CW0+LW(a,b,r)1.

    (2) If a=1 and r=0, then [LW(a,b,r)0,LW(a,b,r)0]=LW(a,b,r)1.

    (3) If r0 or a0,1 and r=0, then [LW(a,b,r)0,LW(a,b,r)0]=CW0+LW(a,b,r)1.

    Proof. It is obvious that (1) and (2) hold.

    Now we need to prove (3). If r0, from (14), by a straightforward calculation, one can show that (3) holds. If a0,1 and r=0, by (14), one can immediately get [LW(a,b,r)0,LW(a,b,r)0]CW0+LW(a,b,r)1. It also follows from (14) that LW(a,b,r)1[LW(a,b,r)0,LW(a,b,r)0]. Note that [L0,W1]=bW2+(a2)W1 and [L1,W0]=bW2+(2a2)W1. Comparing these two equations, then we obtain that aW1[LW(a,b,r)0,LW(a,b,r)0]. Since a0, we have W1[LW(a,b,r)0,LW(a,b,r)0]. From [L0,W0]=bW1+(a1)W0 and a1, we deduce that W0[LW(a,b,r)0,LW(a,b,r)0]. Thus CW0+LW(a,b,r)1[LW(a,b,r)0,LW(a,b,r)0] and (3) holds.

    Lemma 3.5. For each fixed NZ+, LW(a,b,r)0/LW(a,b,r)N is a finite dimensional solvable Lie algebra.

    Proof. It is obvious that LW(a,b,r)0/LW(a,b,r)N is a finite dimensional Lie algebra. Therefore, it only leaves us to prove its solvability. Denote the derived subalgebras of LW(a,b,r)0 by LW(a,b,r)(0)0=LW(a,b,r)0 and LW(a,b,r)(n+1)0=[LW(a,b,r)(n)0,LW(a,b,r)(n)0] for nZ+. By Lemma 3.4, we obtain

    LW(a,b,r)(1)0=[LW(a,b,r)0,LW(a,b,r)0]CW0+LW(a,b,r)1.

    Since that [W0,LW(a,b,r)1]LW(a,b,r)1, we have LW(a,b,r)(2)0LW(a,b,r)1. Suppose LW(a,b,r)(n)0LW(a,b,r)n1 for n2, using induction on n and Lemma 3.3, we can deduce that LW(a,b,r)(n+1)0=[LW(a,b,r)(n)0,LW(a,b,r)(n)0][LW(a,b,r)n1,LW(a,b,r)1]=LW(a,b,r)n. Thus LW(a,b,r)(n+1)0LW(a,b,r)n for all nZ+. This completes the proof of this lemma.

    Firstly, we give a characterization of free W(a,b,r)-modules of rank one.

    Proposition 3. If a1 or b0 or r0, all free nontrivial W(a,b,r)-modules of rank one over C[] are as follows:

    Mα,β,γ=C[]v,Lλv=(+αλ+β)v,Wλv=0,Yλv=γv, (16)

    where α,β,γC. If a=1, b=0 and r=0, all free nontrivial W(1,0,0)-modules of rank one over C[] are as follows:

    Mα,β,γ,Δ=C[]v,Lλv=(+αλ+β)v,Wλv=Δv,Yλv=γv, (17)

    where α,β,γ,ΔC.

    Proof. Suppose

    Lλv=φ(,λ)v,Wλv=ψ(,λ)v,Yλv=ω(,λ)v, (18)

    where φ(,λ),ψ(,λ),ω(,λ)C[,λ] are polynomials of λ and . By Proposition\, 1, we get φ(,λ)=0 or φ(,λ)=+αλ+β for some α,βC. From [WλW]=0, we have Wλ(Wμv)=Wμ(Wλv). Thus, we can obtain

    ψ(+λ,μ)ψ(,λ)=ψ(+μ,λ)ψ(,μ),

    which implies degψ(,λ)+degλψ(,λ)=degλψ(,λ), where degλψ(,λ) is the highest degree of λ in ψ(,λ). Therefore degψ(,λ)=0, i.e. ψ(,λ)=ψ(λ) for some ψ(λ)C[λ]. Similarly, ω(,λ)=ω(λ) for some ω(λ)C[λ]. Since [LλW]λ+μv=((+aλ+b)W)λ+μv and [LλY]λ+μv=((+λ)Y)λ+μv, we can deduce that

    (λμ+aλ+b)ψ(λ+μ)=ψ(μ)φ(,λ)φ(+μ,λ)ψ(μ), (19)
    μω(λ+μ)=φ(+μ,λ)ω(μ)ω(μ)φ(,λ). (20)

    Obviously, if φ(,λ)=0, then (19) and (20) give that ψ(λ)=ω(λ)=0. That is to say this module action is trivial. Therefore, we must have φ(,λ)=+αλ+β for some α,βC. Taking this into (19), we obtain

    ((a1)λμ+b)ψ(λ+μ)=μψ(μ). (21)

    It is not hard to check that ψ(μ)=Δ for some ΔC. Then by (21), we obtain ψ(,λ)=0 if a1 or b0 and ψ(,λ)=Δ for some ΔC if a=1 and b=0. Similarly, φ(,λ)=+αλ+β together with (20) shows that

    μω(λ+μ)=μω(μ). (22)

    It follows that ω(λ)=γ for some γC. In addition, if a=1 and b=0, from [YλW]λ+μv=rWλ+μv, we obtain

    ψ(+λ,μ)ω(λ)ω(+μ,λ)ψ(μ)=rψ(λ+μ).

    If r=0, then the above equation obviously holds. If r0, we have rΔ=0, which implies Δ=0. Then the proposition follows.

    Now we discuss the irreducibility of W(a,b,r)-modules of rank one in Proposition 3.

    Proposition 4. (1) If a1 or b0 or r0, then Mα,β,γ is an irreducible W(a,b,r)-module if and only if α0 or γ0.

    (2) If a=1, b=0 and r=0, then Mα,β,γ,Δ is an irreducible W(a,b,r)-module if and only if α0 or γ0 or Δ0.

    Proof. (1) At first, we prove the necessary condition. Suppose α=γ=0, then M0,β,0 as a W(a,b,r)-module is actually equivalent to M0,β as a Vir-module. Therefore, by Proposition 1, we have (+β)M0,β,0M1,β,0. In addition, (+β)M0,β,0 is the unique nontrivial submodule of M0,β,0, which contradicts the irreducibility of M0,β,0.

    Now we deal with the sufficient condition. If α=0 and γ0, assume that U is a nonzero submodule of M0,β,γ. Then, there exists an element u=φ()vU for some nonzero φ()C[]. If degφ()=0, then vU. It follows that U=M0,β,γ. If degφ()=k>0, then Wλu=φ(+λ)(Wλv)=0 and Yλu=φ(+λ)(Yλv)=γφ(+λ)vU[λ]. The coefficient of λk in γφ(+λ)v is a nonzero multiple of v, which implies vU and U=M0,β,γ. Thus, M0,β,γ is an irreducible W(a,b,r)-module. If α0, then Mα,β,γ is an irreducible Vir-module. Therefore, it is also an irreducible W(a,b,r)-module.

    (2) To prove the necessary condition, we suppose α=γ=Δ=0, then M0,β,0,0 as a W(a,b,r)-module is actually equivalent to M0,β as a Vir-module. Therefore, by Proposition 1, we have (+β)M0,β,0,0M1,β,0,0. In addition, (+β)M0,β,0,0 is the unique nontrivial submodule of M0,β,0,0, which contradicts the irreducibility of M0,β,0,0.

    Now, we prove the sufficient condition. If α=0,γ=0 and Δ0, assume that U is a nonzero submodule of M0,β,0,Δ. Then, there exists an element t=ψ()vU for some nonzero ψ()C[]. If degψ()=0, then vU. It follows that U=M0,β,0,Δ. If degψ()=k>0, then Yλw=ψ(+λ)(Yλv)=0, Wλw=ψ(+λ)(Wλv)=Δψ(+λ)vU[λ]. The coefficient of λk in Δψ(+λ)v is a nonzero multiple of v, which implies vU and U=M0,β,0,Δ. Thus, M0,β,0,Δ is an irreducible W(1,0,0)-module.

    If α=0 and γ0, assume that U is a nonzero submodule of M0,β,γ,Δ. Then, there exists an element u=ω()vU for some nonzero ω()C[]. If degω()=0, then vU. It follows that U=M0,β,γ,Δ. If degω()=k>0, then Yλu=ω(+λ)(Yλv)=γω(+λ)vU[λ]. The coefficient of λk in γω(+λ)v is a nonzero multiple of v, which implies vU and U=M0,β,γ,Δ. Thus, M0,β,γ,Δ is an irreducible W(1,0,0)-module.

    If α0, then Mα,β,γ,Δ is an irreducible Vir-module. Therefore, it is also an irreducible W(1,0,0)-module.

    This completes the proof.

    Note that W(a,b,r) is of finite rank as C[]-module, we can obtain the following result.

    Lemma 3.6. For a conformal module V over W(a,b,r) and an element vV, there exists an integer mZ+ such that LW(a,b,r)mv=0.

    Proof. By Lemma 2.5, V as an LW(a,b,r)-module has the following property: for each vV, there exists N11, N20 and N30 such that

    Lmv=0,mN1,Wnv=0,nN2,Ysv=0, sN3.

    Choosing m=max, we have for all . Therefore, .

    Finally, we can prove the main theorem of this paper.

    Proof of Main Theorem. Assume that is a finite nontrivial irreducible -module. Let . By Lemma , there exists some such that . Let be the minimal integer such that and set . Now, we discuss the possible values of . If , then . Hence is a trivial -module, which is a contradiction. Thus, we can deduce that . Note that we take for here. Thus by Lemma 2.6, is finite dimensional. By Lemma 3.2, is central in . Using the definition of , it stabilizes . By Schur's Lemma, there exists some such that for all . Thus for all .

    Let . Then has a decomposition of vector spaces

    (23)

    By the Poincar-Birkhoff-Witt Theorem, we have the universal enveloping algebra of :

    (24)

    where , as a vector space over .

    Note that th actions of the elements in on can be regarded as the formal polynomials of with their coefficients still belonging to . Considering that is a -module at the same time, we obtain that as a -submodule. Obviously, . Otherwise, we can deduce that is free of rank one, which contradicts to for some in Proposition . Since is irreducible and , we must have . Furthermore,

    (25)

    Denote ) to be the right (left) multiplication by in the universal enveloping algebra of . Using and the binomial formula, we obtain

    (26)

    for .

    Note that is an ideal of and is an -module. From , we can conclude that is an -module. By Lemma , is a finite dimensional solvable Lie algebra. Therefore, by Lie's Theorem, there exists a nonzero common eigenvector under the action of and there exists a linear function on such that for all .

    By Lemma , depends on the value of and . Thus we consider the following three cases.

    Case 1. and .

    In this case, . From for , we claim that . In fact, if we have , which leads to ; if by Lemma , there exists an satisfies . For each , is a linear combination of . We have

    for some . Thus, we can cancel by appropriate iteration and obtain an equation which only has and the terms of the form . That is to say, the action of on is equal to the action of those linear combinations of the elements in on . Then the claim is proved. Furthermore, one can obtain for . Combining this with for and for , we can conclude that .

    Now, and . Therefore and is determined by and . Suppose that for some and for some . Since and is an ideal of , considering the commutative condition of with and , one can get by and . Thus, the irreducibility of as a -module is equivalent to that of as a -module. It follows that is free of rank one over . Then the conclusion can be obtained by Proposition .

    Case 2. and .

    In this case, , then , , and . Suppose that for some , for some and for some .

    Since is a free -module of finite rank, there exists and such that

    (27)

    where is -linearly independent and . Note that , , , applying to both sides of (27), we obtain that

    (28)

    If , then we can deduce that , which is similar to Case 1. Thus with or . Otherwise, by , we obtain that

    (29)

    Comparing the coefficients of on both sides of , we can obtain that

    (30)

    Since and , we have . Let . is free of rank one over . Obviously, . By , we have for all . Therefore, is a submodule of . By the irreducibility of , is free of rank one. Thus, with or or , which is irreducible by Proposition and Proposition .

    Case 3. or and .

    In this case, . Then, using the similar discussion in Case 1, we can conclude that with or .

    Therefore, the main theorem holds.

    The authors would like to thank the referees for their careful reading and useful suggestions.



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