The derivations, central extensions and automorphism group of the Lie algebra $W$

0.16 MB·English

by  Shoulan Gao

THE DERIVATIONS, CENTRAL EXTENSIONS AND
AUTOMORPHISM GROUP OF THE LIE ALGEBRA W∗
8
0 SHOULAN GAO, CUIPO JIANG†, AND YUFENG PEI
0
2
Abstract. In this paper, we study the derivations, central extensions and the au-
n
tomorphisms of the infinite-dimensional Lie algebra W which appeared in [8] and
a
J Dong-Zhang’s recent work [22] on the classification of some simple vertex operator

5 algebras.
2

]
A
R 1. Introduction
.
h
It is well known that the Virasoro algebra Vir plays an important role in many areas
t
a
of mathematics and physics (see [13], for example). It can be regarded as the universal
m
central extension of the complexification of the Lie algebra Vect(S1) of (real) vector
[
fields on the circle S1:
1
v m3 −m
1 [L ,L ] = (m−n)L +δ c, (1.1)
m n m+n m+n,0
1 12
9
where c is a central element such that [L ,c] = 0. The Virasoro algebra admits many
3 n
. interesting extensions and generalizations, for example, the W -algebras [21], W
1 N 1+∞
0 [14], thehigher rankVirasoroalgebra [17], andthetwisted Heisenberg-Virasoro algebra
8
[1, 3, 11] etc. Recently M. Henke et al. [8, 9] investigated a Lie algebra W in their
0
: study of ageing phenomena which occur widely in physics [7]. The Lie algebra W is an
v
i abelian extension of centerless Virasoro algebra, and is isomorphic to the semi-direct
X
r product Lie algebraL⋉I, where L isthe centerless Virasoro algebra(Wittalgebra) and
a
I istheadjointL-module. Inparticular, W istheinfinite-dimensional extensions ofthe
Poincare algebra p . In their classification of the simple vertex operator algebras with
3
2 generators, Dong and Zhang [22] studied a similar infinite-dimensional Lie algebra
W(2,2) and its representation theory. Although the algebra W(2,2) is an extension of
the Virasoro algebra, as they remarked, the representation theory for W(2,2) is totally
different from that of the Virasoro algebra.
The purpose of this paper is to study the structure of the Lie algebra W and its
universal central extension W. We will see that there is a natural surjective homomor-
f
phism from the universal covering algebra W to W(2,2). We show that the second
f
cohomology group with trivial coefficients for the Lie algebra W is two dimensional.
Keywords: derivation, central extension, automorphism.
∗ Supported in part by NSFC grant 10571119and NSF grant 06ZR14049of Shanghai City.
† Corresponding author: [email protected].
1
2 GAO,JIANG, ANDPEI
Furthermore, by the Hochschild-Serre spectral sequence [20] and R. Farnsteiner’s the-
orem [5], we determine the derivation algebras of W and W, which both have only one
f
outerderivation. Finally, theautomorphism groupsofW andW arealsocharacterized.
Throughout the paper, we denote by Z the set of all intfegers and C the field of
complex numbers.
2. The Universal Central Extension of W
The Lie algebra W over the complex field C has a basis {L ,I | m ∈ Z} with the
m m
following bracket
[L ,L ] = (m−n)L , [L ,I ] = (m−n)I , [I ,I ] = 0,
m n m+n m n m+n m n
for all m,n ∈ Z. It is clear that W is isomorphic to the semi-direct product Lie algebra
W ≃ L ⋉ I, where L = CL is the classical Witt algebra and I = CI
Ln∈Z i Ln∈Z n
can be regarded as the adjoint L-module. Moreover, W = W is a Z-graded Lie
L m
m∈Z
algebra, where W = CL ⊕CI .
m m m
Let g be a Lie algebra. Recall that a bilinear function ψ : g× g −→ C is called a
2-cocycle on g if for all x,y,z ∈ g, the following two conditions are satisfied:
ψ(x,y) = −ψ(y,x),
ψ([x,y],z)+ψ([y,z],x)+ψ([z,x],y) = 0. (2.1)
For any linear function f : g −→ C, one can define a 2-cocycle ψ as follows
f
ψ (x,y) = f([x,y]), ∀ x,y ∈ g.
f
Such a 2-cocycle is called a 2-coboudary on g.
Let g be a perfect Lie algebra, i.e., [g,g] = g. Denote by C2(g,C) the vector space of
2-cocycles on g, B2(g,C) the vector space of 2-coboundaries on g. The quotient space:
H2(g,C) = C2(g,C)/B2(g,C)
is called the second cohomology group of g with trivial coefficients C. It is well-
known that H2(g,C) is one-to-one correspondence to the equivalence classes of one-
dimensional central extensions of the Lie algebra g.
We will determine the second cohomology group for the Lie algebra W.
Lemma 2.1 (See also [16]). Let (g,[, ] ) be a perfect Lie algebra over C and V a g-
0
module such that g·V = V. Consider the semi-direct product Lie algebra (g⋉V,[, ])
with the following bracket
[x,y] = [x,y] , [x,v] = x·v, [u,v] = 0, ∀ x,y ∈ g, u,v ∈ V.
0
Let V∗ be the dual g-module and
Bg(V) = {f ∈ Hom(V⊗V,C)|f(u,v) = −f(v,u), f(x·u,v)+f(u,x·v) = 0, ∀x ∈ g,u,v ∈ V}.
DERIVATIONS, CENTRAL EXTENSIONS AND AUTOMORPHISM GROUP OF W 3
Then we have
H2(g⋉V,C) = H2(g,C)⊕H1(g,V∗)⊕Bg(V).
Proof. Letαbea2-cocycleong⋉V. Obviously, α|g ∈ H2(g,C).DefineDα ∈ HomC(g,V∗)
by
D (x)(v) = α(x,v),
α
for x ∈ g,v ∈ V. Then D ∈ H1(g,V∗). In fact, for any x,y ∈ g,v ∈ V,
α
α([x,y],v) = α(x·v,y)−α(y ·v,x)
= −D (y)(x·v)+D (x)(y ·v)
α α
= (x·D (y))(v)−(y ·D (x))(v).
α α
Therefore
D ([x,y]) = x·D (y)−y ·D (x),
α α α
for all x,y ∈ g. Define f (u,v) = α(u,v) for any u,v ∈ V, then f ∈ Bg(V). It is
α α
straightforward to check α ,D and f are linearly independent. We get the desired
|g α α
(cid:3)
formula.
Corollary 2.2. Let V = g be the adjoint g-module. Then
H2(g⋉g,C) = H2(g,C)⊕H1(g,g∗)⊕Bg(g). (2.2)
Theorem 2.3. H2(W,C) = Cα⊕Cβ, where
m3 −m
α(L ,L ) = δ , α(L ,I ) = α(I ,I ) = 0,
m n m+n,0 m n m n
12
m3 −m
β(L ,I ) = δ , β(L ,L ) = β(I ,I ) = 0.
m n m+n,0 m n m n
12
for any m,n ∈ Z.
Proof. Since H2(L,C) = Cα and H1(L,L∗) ≃ Cβ (see [6, 10] for the details), we need
to prove that BL(I) = 0. Let f ∈ BL(I), then
(i−j)f(I ,I )+(k −i)f(I ,I ) = 0. (2.3)
i+j k k+i j
Letting i = 0 in (2.3), we get
(j +k)f(I ,I ) = 0.
j k
So f(I ,I ) = 0 for j +k 6= 0. Let k = −i−j in (2.3), then we obtain
j k
(i−j)f(I ,I )+(2i+j)f(I ,I ) = 0. (2.4)
i+j −i−j j −j
Let j = −i, then 2if(I ,I ) = if(I ,I ), which implies that f(I ,I ) = 0 for all i ∈ Z.
0 0 i −i i −i
Therefore, f(I ,I ) = 0 for all m,n ∈ Z.
m n
(cid:3)
4 GAO,JIANG, ANDPEI
Let g be a Lie algebra, (g,π) is called a central extension of g if π : g −→ g is a
surjective homomorphism wehose kernel lies in the center of the Lie algebraeg. The pair
(g,π)iscalledacoveringofgifgisperfect. Acovering(g,π)iscalledauniveersalcentral
eextension of g if for every centrael extension (g′,ϕ) of g theere is a unique homomorphism
ψ : g −→ g′ for which ϕψ = π. It follows froem [7] that every perfect Lie algebra has a
univeersal ceentral extension.
Let W = W C C C C be a vector space over the complex field C with a basis
L 1L 2
{L ,If,C ,C | n ∈ Z} satisfying the following relations
n n 1 2
m3 −m
[L ,L ] = (m−n)L +δ C ,
m n m+n m+n,0 1
12
m3 −m
[L ,I ] = (m−n)I +δ C ,
m n m+n m+n,0 2
12
[I ,I ] = 0, [C ,W] = 0, [C ,W] = 0,
m n 1 2
f f
for all m,n ∈ Z. By Theorem 2.3, W is a universal covering algebra of W.
There is a Z-grading on W: f
f
W = MWn,
f f
n∈Z
where W = span{L ,I ,δ C ,δ C }. Set W = W and W = W , then
n n n n,0 1 n,0 2 + L n − L n
f f n>0f f n<0f
there is a triangular decomposition on W:
f
W = W−MW0MW+.
f f f f
Remark 2.4. Let C = C = C, then we get the Lie algebra W(2,2) appeared in [22].
1 2
3. The Derivation Algebra of W
In this section, we consider the derivation algebra of W.
Let G be a commutative group, g = g a G-graded Lie algebra. A g-module V
L g
g∈G
is called G-graded, if
V = MVg∈G, ggVh ⊆ Vg+h, ∀ g,h ∈ G.
g∈G
Let g be a Lie algebra and V a g-module. A linear map D : g −→ V is called a
derivation, if for any x,y ∈ g,
D[x,y] = x.D(y)−y.D(x).
If there exists some v ∈ V such that D : x 7→ x.v, then D is called an inner derivation.
Denote by Der(g,V) the vector space of all derivations, Inn(g,V) the vector space of
all inner derivations. Set
H1(g,V) = Der(g,V)/Inn(g,V).
DERIVATIONS, CENTRAL EXTENSIONS AND AUTOMORPHISM GROUP OF W 5
Denote by Der(g) the derivation algebra of g, Inn(g) the vector space of all inner
derivations of g.
Firstly, we have the short exact sequence of Lie algebras
0 → I → W → L → 0,
which induces an exact sequence
H1(W,I) → H1(W,W) → H1(W,L). (3.1)
The right-hand side can be computed from the initial terms
0 → H1(L,L) → H1(W,L) → H1(I,L)W (3.2)
ofthefour-termsequenceassociatedtotheHochschild-Serrespectralsequence[20]. Itis
known that H1(L,L) = 0, while the term H1(I,L)W is equal to Hom (I/[I,I],L).
U(W)
Therefore, it suffices to compute the terms H1(W,I) and Hom (I/[I,I],L).
U(W)
By Proposition 1.1 in [5], we have the following lemma.
Lemma 3.1.
Der(W,I) = MDer(W,I)n,
n∈Z
where Der(W,I) (I ) ⊆ I for all m,n ∈ Z.
n m m+n
(cid:3)
Lemma 3.2. For any nonzero integer m, we have
H1(W ,I ) = 0,
0 m
where I = CI .
m m
Proof. Let m 6= 0, ϕ : W −→ CI a derivation. Assume that
0 m
ϕ(L ) = aI , ϕ(I ) = bI ,
0 m 0 m
where a,b ∈ C. Since ϕ[L ,I ] = [ϕ(L ),I ]+ [L ,ϕ(I )], it is easy to deduce b = 0.
0 0 0 0 0 0
a
Let E = − I , then we have ϕ(L ) = [L ,E ] and ϕ(I ) = [I ,E ]. Therefore,
m m 0 0 m 0 0 m
m
ϕ ∈ Inn(W ,I ), that is,
0 m
H1(W ,I ) = 0, ∀ m ∈ Z\{0}.
0 m
(cid:3)
Lemma 3.3. Hom (W ,I ) = 0 for all m,n ∈ Z, m 6= n.
W0 m n
Proof. Let f ∈ Hom (W ,I ), where m 6= n. Then for any E ∈ W , we have
W0 m n m m
f([L ,E ]) = [L ,f(E )].
0 m 0 m
Note [L ,E ] = −kE for all k ∈ Z. Then we get
0 k k
−mf(E ) = [L ,f(E )] = −nf(E ).
m 0 m m
So f(E ) = 0 for all m 6= n. Therefore, we have f = 0. (cid:3)
m
6 GAO,JIANG, ANDPEI
By Lemma 4.3-4.4 and Proposition 1.2 in [5], we have the following Lemma.
Lemma 3.4. Der(W,I) = Der(W,I) +Inn(W,I).
0
Lemma 3.5. H1(W,I) = CD , where
1
D (L ) = 0, D (I ) = I ,
1 m 1 m m
for all m ∈ Z .
Proof. For any D ∈ Der(W,I) , assume
0
D(L ) = a I , D(I ) = b I ,
m m m m m m
where a ,b ∈ C. By the definition of derivation and the Lie bracket in W, we have
m m
a = a +a , b = b , m 6= n.
m+n m n m+n n
Obviously, b = b for all m ∈ Z and a = 0,a = −a . By induction on m > 0, we
m 0 0 −m m
deduce that a = (m− 2)a + a for m > 2. Then it is easy to infer that a = 2a .
m 1 2 2 1
Consequently, we get a = ma for all m ∈ Z. So for all m ∈ Z, we have
m 1
D(L ) = ma I , D(I ) = b I .
m 1 m m 1 m
Set D = −ad(a I ) ∈ Inn(W), then D(L ) = D (L ), D(I ) = D (I ) + b I .
0 1 0 m 0 m m 0 m 1 m
Therefore,
D¯(L ) = 0,D¯(I ) = b I ,
m m 1 m
for all m ∈ Z, where D¯ = D −D is an outer derivation. The lemma holds.
0
(cid:3)
Lemma 3.6. Hom (I/[I,I],L) = 0.
U(W)
Proof. As a matter of fact,
Hom (I/[I,I],L) = Hom (I,L).
U(W) U(W)
For any f ∈ Hom (I,L), we have [L ,f(I )] = f([L ,I ]), i.e.,
U(W) 0 m 0 m
ad(−L )(f(I )) = mf(I ),
0 m m
for all m ∈ Z. This suggests f(I ) ∈ CL . Assume f(I ) = x L for all m ∈ Z,
m m m m m
where x ∈ C. By the relation that [L ,f(I )] = f([L ,I ]) for all m,n ∈ Z, we have
m n m n m
x = x , m 6= n.
m+n m
Obviously, x = x for all m ∈ Z. So there exists some constant a ∈ C such that
m 0
f(I ) = aL ,
m m
for all m ∈ Z. Since f([I ,I ]) = 0 = [I ,f(I )] = −aL , we have a = 0. Hence f = 0.
0 1 0 1 1
(cid:3)
DERIVATIONS, CENTRAL EXTENSIONS AND AUTOMORPHISM GROUP OF W 7
Theorem 3.7. H1(W,W) = CD, where
D(L ) = 0, D(I ) = I ,
m m m
for all m ∈ Z .
Furthermore, it follows from Theorem 2.2 in [2] that
Corollary 3.8. H1(W,W) ≃ H1(W,W).
f f
4. The Automorphism Group of W
Denote by Aut(W) and Z the automorphism group and the inner automorphism
group of W respectively. Obviously, Z is generated by exp(kadI ), m ∈ Z, k ∈ C, and
m
Z is an abelian subgroup. Note that I is the maximal proper ideal of W, so we have
the following lemma.
Lemma 4.1. For any σ ∈ Aut(W), σ(I ) ∈ I for all n ∈ Z.
n
(cid:3)
t
For any exp(k adI ) ∈ Z, we have
Q ij ij
j=s
t t t
Yexp(kijadIij)(In) = In, Yexp(kijadIij)(Ln) = Ln +Xkij(ij −n)Iij+n.
j=s j=s j=s
Lemma 4.2. For any σ ∈ Aut(W), there exist some τ ∈ Z and ǫ ∈ {±1} such that
σ¯(L ) = anǫL +anλnI , (4.1)
n ǫn ǫn
σ¯(I ) = anµI , (4.2)
n ǫn
where σ¯ = τ−1σ, a,µ ∈ C∗ and λ ∈ C. Conversely, if σ¯ is a linear operator on sv
satisfying (4.1)-(4.2) for some ǫ ∈ {±1}, a,µ ∈ C∗ and λ ∈ C, then σ¯ ∈ Aut(W). e
Proof. For any σ ∈ Aut(W), denote σ| = σ′. Then σ′ is an automorphism of the
L
classical Witt algebra, so σ′(L ) = ǫamL for all m ∈ Z, where a ∈ C∗ and ǫ ∈ {±1}.
m ǫm
Assume that
q
σ(L0) = ǫL0 +XλiIi +λ0I0,
i=p
q
where i 6= 0. Let τ = exp(λiadI ), then
Q ǫi i
i=p
q
τ(ǫL0) = ǫL0 +XλiIi.
i=p
Therefore, σ(L ) = τ(ǫL )+λ I . Set σ¯ = τ−1σ, then σ¯(L ) = ǫL +λ I . Assume
0 0 0 0 0 0 0 0
σ¯(Ln) = anǫLǫn +anXλ(ni)Ini, n 6= 0,
8 GAO,JIANG, ANDPEI
where each formula is of finite terms and λ(n ),µ(n ) ∈ C. For m 6= 0, through the
i j
relation [σ¯(L ),σ¯(L )] = −mσ¯(L ), we get
0 m m
λ0mIǫm = Xλ(mi)[m−ǫmi]Imi.
This forces that λ = 0. Then [σ¯(L ),σ¯(L )] = [ǫL ,σ¯(L )] = −mσ¯(L ), that is,
0 0 m 0 m m
−adL (σ¯(L )) = ǫmσ¯(L ).
0 m m
So for all m ∈ Z, we obtain
σ¯(L ) = amǫL +amλ(ǫm)I .
m ǫm ǫm
ComparingthecoefficientsofI onthebothsideof[σ¯(L ),σ¯(L )] = (m−n)σ¯(L ),
ǫ(m+n) m n m+n
we get
λ(ǫm)+λ(ǫn) = λ(ǫ(m+n)).
Note λ = 0, we deduce that λ(ǫm) = mλ(ǫ) for all m ∈ Z. Therefore,
0
σ¯(L ) = amǫL +ammλ(ǫ)I .
m ǫm ǫm
Finally, by [σ(L ),σ(I )] = −mσ(I ), we have
0 m m
−adL (σ(I )) = ǫmσ(I ),
0 m m
for all m ∈ Z. Then by Lemma 4.1, we may assume
σ(I ) = anµ(ǫn)I ,
n ǫn
for all n ∈ Z. By [σ(L ),σ(I )] = (m−n)σ(I ), we get
m n m+n
µ(ǫn) = µ(ǫ(m+n)), m 6= n.
Consequently, µ(ǫm) = µ(0) for all m ∈ Z. Set µ(0) = µ, then for all n ∈ Z, we have
σ¯(I ) = anµI .
n ǫn
(cid:3)
Denote by σ¯(ǫ,λ,a,µ) the automorphism of W satisfying (4.1)-(4.2), then
σ¯(ǫ ,λ ,a ,µ )σ¯(ǫ ,λ ,a ,µ ) = σ¯(ǫ ǫ ,λ +µ λ ,aǫ2a ,µ µ ), (4.3)
1 1 1 1 2 2 2 2 1 2 1 1 2 1 2 1 2
and σ¯(ǫ ,λ ,a ,µ ) = σ¯(ǫ ,λ ,a ,µ ) if and only if ǫ = ǫ ,λ = λ ,a = a ,µ = µ .
1 1 1 1 2 2 2 2 1 2 1 2 1 2 1 2
Let
π¯ = σ¯(ǫ,0,1,1), σ¯ = σ¯(1,λ,1,1), σ¯ = σ¯(1,0,a,µ)
ǫ λ a,µ
and
a = {π¯ | ǫ = ±1}, t = {σ¯ | λ ∈ C}, b = {σ¯ | a,µ ∈ C∗}.
ǫ λ a,µ
By (4.3), we have the following relations:
σ¯(ǫ,λ,a,µ) = σ¯(ǫ,0,1,1)σ¯(1,λ,1,1)σ¯(1,0,a,µ) ∈ atb,
σ¯(ǫ,λ,a,µ)−1 = σ¯(ǫ,−λµ−1,a−ǫ,µ−1),
π¯ π¯ = π¯ , σ¯ σ¯ = σ¯ , σ¯ σ¯ = σ¯ ,
ǫ1 ǫ2 ǫ1ǫ2 λ1 λ2 λ1+λ2 a1,µ1 a2,µ2 a1a2,µ1µ2
DERIVATIONS, CENTRAL EXTENSIONS AND AUTOMORPHISM GROUP OF W 9
σ¯ π¯ = π¯ σ¯ , π¯−1σ¯ π¯ = σ¯ , σ¯ σ¯ σ¯−1 = σ¯ .
λ ǫ ǫ λ ǫ a,µ ǫ aǫ,µ a,µ λ a,µ µλ
Hence, the following lemma holds.
Lemma 4.3. a,t and b are all subgroups of Aut(W). Furthermore, t is an abelian
normal subgroup commutative with Z.
Aut(W) = (Zt)⋊(a⋉b),
where a ∼= Z = {±1},t ∼= C, b ∼= C∗ ×C∗. (cid:3)
2
Let C∞ = {(ai)i∈Z | ai ∈ C, all but a finite number of the ai are zero }. Then C∞
is an abelian group.
Lemma 4.4. Zt is isomorphic to C∞.
Proof. Define f : Zt −→ C∞ by
s
f(σ¯λYexp(αkiadIki)) = (ap)p∈Z,
i=1
where a = α for k < 0, a = λ, a = α for k ≥ 0, and the others are zero,
ki ki i 0 ki+1 ki i
k ∈ Z and k < k < ··· < k . Since every element of Zt has the unique form of
i 1 2 s
s
σ¯ exp(α adI ), it is easy to check that f is an isomorphism of group.
λ Q ki ki
i=1
(cid:3)
Theorem 4.5. Aut(W) ∼= C∞ ⋊(Z ⋉(C∗ ×C∗)).
2
(cid:3)
Since W is centerless, it follows from Corollary 6 in [18] that Aut(W) = Aut(W),
f
that is,
Aut(W) ∼= C∞ ⋊(Z ⋉(C∗ ×C∗)).
2
f
References
[1] E.Arbarello,C. De Concini, V.G. Kac,C.Procesi,Moduli spaces of curves and representation
theory, Comm. Math. Phys. 117 (1988), 1-36.
[2] G.M.Benkart,R. V.Moody, Derivations, central extensions and affine Lie algebras, Algebras
Groups Geom. 3 (1986), 456-492.
[3] Y. Billig, Representations of the twisted Heisenberg-Virasoro algebra at level zero, Canadian
Mathematical Bulletin, 46 (2003), 529-537.
[4] D.Zˇ.Dokovi´c,K.Zhao,Derivations, isomorphisms, andsecondcohomology ofgeneralized Witt
algebras, Tran. Ams. Math. Soc. 350(2) (1998), 643-664.
[5] R.Farnsteiner,Derivations and extensions of finitely generated graded Lie algebras, J.Algebra
118 (1) (1988), 34-45.
[6] J.L.Loday,T.Pirashvili,Universal enveloping algebras of Leibniz algebras and (co)-homology,
Math. Ann. 296 (1993), 138-158.
[7] M. Henkel, Ageing, dynamical scaling and its extensions in many-particle systems without
detailed balance, J. Phys. Cond. Matt. 19 (2007).
10 GAO,JIANG, ANDPEI
[8] M. Henkel, R. Schott, S. Stoimenov, J. Unterberger, The Poincare algebra in the con-
text of ageing systems: Lie structure, representations, Appell systems and coherent states,
arXiv:math-ph/0601028(2006).
[9] M. Henkel, R. Schott, S. Stoimenov, J. Unterberger, On the dynamical symmetric algebra of
ageing: Lie structure, representations and Appell systems, QP-PQ: Quantum probability and
white-noise analysis 20 (2007), 233-240.
[10] N. Hu, Y. Pei, D. Liu, A cohomological characterization of Leibniz central extensions of Lie
algebras, Proc. Amer. Math. Soc. 136 (2008), 437-447.
[11] Q. Jiang, C. Jiang, Representations of the twisted Heisenberg-Virasoro algebra and the full
toroidal Lie algebras, Algebra Colloq. 14(1) (2007), 117-134.
[12] C.Jiang,D.Meng,ThederivationalgebraoftheassociativealgebraC [X,Y,X−1,Y−1],Comm.
q
Algebra 6 (1998), 1723-1736.
[13] V. Kac, A. Raina, Bombay lectures on highest weight representations of infinite-dimensional
Lie algebras, Advanced Series in Mathematics 2 (1987).
[14] V. Kac and A. Radul, Quasi-finite highest weight modules over the Lie algebra of differential
operators on the circle, Comm. Math. Phys. 157 (1993), 429-457.
[15] D. Liu, C. Jiang, The generalized Heisenberg-Virasoro algebra, arXiv:math/0510543v3
[math.RT]
[16] V. Ovsienko, C. Roger, Extensions of the Virasoro group and the Virasoro algebra by modules
of tensor densities on S1, Func. Anal. Appl. 30 (1996), 1573-8485.
[17] J. Patera, and H. Zassenhaus, The higher rank Virasoro algebras, Comm. Math. Phys. 136
(1991), 1-14.
[18] A. Pianzola, Automorphisms of toroidal Lie algebras and their central quotients, J. Algebra
Appl. 1 (2002), 113-121.
[19] R. Shen, C. Jiang, The derivation algebra and automorphism group of the twisted Heisenberg-
Virasoro algebra, Commun. Alg. 34 (7) (2006), 2547-2558.
[20] C.Weibel,Anintroduction to homological algebra, CambridgeStudiesinAdvancedMathemat-
ics, Cambridge University Press, 1994.
[21] A. B. Zamolodchikov, Infinite additional symmetries in two dimensional conformal quantum
field theory, Theor. Math. Phys. 65 (1985), 1205-1213.
1 1
[22] W. Zhang, C. Dong, W algebra W(2,2) and the vertex operator algebra L( ,0)⊗L( ,0),
2 2
arXiv:0711.4624v1 (2007).
Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China
E-mail address: [email protected]
Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China
E-mail address: [email protected]
Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China
E-mail address: [email protected]