We study a family of non-simple Lie conformal algebras
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
[1] |
Wenjun Liu, Yukun Xiao, Xiaoqing Yue .
Classification of finite irreducible conformal modules over Lie conformal algebra |
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We study a family of non-simple Lie conformal algebras
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 Cur
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
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
[LλL]=(∂+2λ)L,[LλW]=(∂+aλ+b)W,[LλY]=(∂+λ)Y, [YλW]=rW, [YλY]=[WλW]=0. | (1) |
The Lie conformal algebra
The following theorem is the main result of this paper.
Main Theorem. (1) If
(2) If
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
Throughout this paper, we denote by
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)λb]=−λ[aλb],[aλ∂b]=(∂+λ)[aλb](conformal sesquilinearity), | (2) |
[aλb]=−[b−λ−∂a] (skew-symmetry), | (3) |
[aλ[bμc]]=[[aλb]λ+μc]+[bμ[aλc]] (Jacobi identity), | (4) |
where
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
There are two important examples of Lie conformal algebras. One is the Virasoro Lie conformal algebra
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
Curg=C[∂]⊗g,[aλb]=[a,b],a,b∈g. |
The Lie conformal algebra
[LλL]=(∂+2λ)L,[LλW]=(∂+aλ+b)W,[WλW]=0. |
Note that the Virasoro Lie conformal algebra Vir and Lie conformal algebra
For a given Lie conformal algebra
[aλb]=∑j∈Z+(a(j)b)λjj!. | (5) |
Let
[axm,bxn]=∑j∈Z+(mj)(a(j)b)xm+n−j | (6) |
for all
Lie(A)+=span{a(n)∣a∈A,n∈Z+} |
is called the annihilation algebra of
Definition 2.2. A conformal module
(∂a)λv=−λaλv,aλ(∂v)=(∂+λ)aλv, | (7) |
aλ(bμv)−bμ(aλv)=[aλb]λ+μv, | (8) |
where
A conformal module
In the following, we only consider conformal modules, thus we simply abbreviate the term conformal modules to modules.
Suppose that
Let
Lemma 2.3. Let
Lemma 2.4. Let
Similar to the definition of the
aλv=∑j∈Z+(a(j)v)λjj!. | (9) |
If
a(n)⋅v=0,n≥N, |
where
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(n)⋅v=0,n≥N, | (10) |
for
In the following, by abuse of notations, we also call a
Let
Lemma 2.6. Suppose that
In this section, we firstly consider the extended annihilation algebra
Since the Virasoro Lie conformal algebra Vir and Lie conformal algebra
Proposition 1. All free nontrivial
Mα,β=C[∂]v,Lλv=(∂+αλ+β)v, | (11) |
where
Proposition 2. If
M′α,β=C[∂]v,Lλv=(∂+αλ+β)v,Wλv=0, | (12) |
where
If
M′α,β,γ=C[∂]v,Lλv=(∂+αλ+β)v,Wλv=γv, | (13) |
where
Now, we investigate the extended annihilation algebra of
Lemma 3.1. The annihilation algebra of
Lie(W(a,b,r))+=∑m≥−1CLm⊕∑n≥0CWn⊕∑s≥0CYs, |
with the following Lie brackets:
[Lm,Ln]=(m−n)Lm+n,[Lm,Wn]=bWm+n+1+((a−1)(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
[∂,Lm]=−(m+1)Lm−1,[∂,Wn]=−nWn−1,[∂,Ys]=−sYs−1. | (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(j−1)W=W(j−2)W=Y(j−2)Y=0,j≥2. |
Then, for all
[L(m),L(n)]=∑j∈Z+(mj)(L(j)L)(m+n−j)=(m−n)L(m+n−1),[L(m),W(n)]=∑j∈Z+(mj)(L(j)W)(m+n−j)=bW(m+n)+((a−1)m−n)W(m+n−1),[L(m),Y(s)]=∑j∈Z+(mj)(L(j)Y)(m+s−j)=−sY(m+s−1),[Y(s),W(n)]=∑j∈Z+(sj)(Y(j)W)(s+n−j)=rW(s+n),[W(m),W(n)]=[Y(m),Y(n)]=0. |
Taking
From (14) and (15), the following lemma follows.
Lemma 3.2.
From now on, we denote
LW(a,b,r)⊃LW(a,b,r)−1⊃LW(a,b,r)0⊃⋯⊃⋯⊃LW(a,b,r)n⊃⋯. |
For this filtration, by Lemma
Lemma 3.3. (1)
Lemma 3.4.
Proof. It is obvious that (1) and (2) hold.
Now we need to prove (3). If
Lemma 3.5. For each fixed
Proof. It is obvious that
LW(a,b,r)(1)0=[LW(a,b,r)0,LW(a,b,r)0]⊂CW0+LW(a,b,r)1. |
Since that
Firstly, we give a characterization of free
Proposition 3. If
Mα,β,γ=C[∂]v,Lλv=(∂+αλ+β)v,Wλv=0,Yλv=γv, | (16) |
where
Mα,β,γ,Δ=C[∂]v,Lλv=(∂+αλ+β)v,Wλv=Δv,Yλv=γv, | (17) |
where
Proof. Suppose
Lλv=φ(∂,λ)v,Wλv=ψ(∂,λ)v,Yλv=ω(∂,λ)v, | (18) |
where
ψ(∂+λ,μ)ψ(∂,λ)=ψ(∂+μ,λ)ψ(∂,μ), |
which implies
(−λ−μ+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
((a−1)λ−μ+b)ψ(λ+μ)=−μψ(μ). | (21) |
It is not hard to check that
μω(λ+μ)=μω(μ). | (22) |
It follows that
ψ(∂+λ,μ)ω(λ)−ω(∂+μ,λ)ψ(μ)=rψ(λ+μ). |
If
Now we discuss the irreducibility of
Proposition 4.
Proof. (1) At first, we prove the necessary condition. Suppose
Now we deal with the sufficient condition. If
(2) To prove the necessary condition, we suppose
Now, we prove the sufficient condition. If
If
If
This completes the proof.
Note that
Lemma 3.6. For a conformal module
Proof. By Lemma
Lm⋅v=0,m≥N1,Wn⋅v=0,n≥N2,Ys⋅v=0, s≥N3. |
Choosing
Finally, we can prove the main theorem of this paper.
Proof of Main Theorem. Assume that
Let
(23) |
By the Poincar
(24) |
where
Note that
(25) |
Denote
(26) |
for
Note that
By Lemma
Case 1.
In this case,
for some
Now,
Case 2.
In this case,
Since
(27) |
where
(28) |
If
(29) |
Comparing the coefficients of
(30) |
Since
Case 3.
In this case,
Therefore, the main theorem holds.
The authors would like to thank the referees for their careful reading and useful suggestions.
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