On the Study of Some Twisted Deformative Schr?dinger Virasoro Algebra

TANG Jia，GAO Shoulan，GU Haixia

（School of Science，Huzhou University，Huzhou 313000，China）

On the Study of Some Twisted Deformative Schr?dinger Virasoro Algebra

TANG Jia，GAO Shoulan，GU Haixia

（School of Science，Huzhou University，Huzhou 313000，China）

In this paper，we study a kind of twisted deformative Schr?dinger-Virasoro Lie algebra with two parameters.The calculation of all the derivations of certain 1-dimensional center extension of the Lie algebra proves that the Lie algebra has 7 outer derivations.The result will be helpful to further study the representation theory of this Lie algebra.

Schr?dinger-Virasoro Lie algebra；central extension；derivation

MSC 2000：17B40

0 Introduction

The infinite-dimensional Schr?dinger Lie algebra and Virasoro algebra are of great implications in many fields of mathematics and physics.In 1994，Henkel introduced the Schr?dinger-Virasoro Lie algebra［1］.Then many generations and extensions of the Schr?dinger-Virasoro Lie algebra appear and they are studied extensively.The twisted deformative Schr?dinger-Virasoro Lie algebra Lλ，μover the complex field was introduced in［2］as follows：for complex numbersλ，μ，the vector space Lλ，μhas a basis｛Ln，Mn，Yn|n∈Z｝with the following Lie brackets：

and others are zero.2-cocycles of all the Lie algebras Lλ，μwere determined in［3］.According to Theorem 2.1 in［3］，we have the one-dimensional central extension of L，forμ?Z，λ∈C.For simpliciλμty，denote the Lie algebra by S.That is，the Lie algebra S has a basis｛ Ln，Mn，Yn，C1n∈Z｝equipped with the Lie brackets：

and others are zero，where m，n∈Z andμ?1Z. 3

Throught the paper，denote the set of integers，the complex field and the set of nonzero complex numbers by Z，C and C*，respectively.All the vector spaces are assumed over the complex field.

1 The derivations of S

Definition 1.1［4］Let g be a Lie algebra，V a g-module.A linear map D：g→V is called a derivation，if for any x，y∈g，we have D［ x，y］=x.D（ y）-y.D（x）.If there exists some v∈V such that D：x?xv.，then D is called an inner derivation.

Let g be a Lie algebra，V a module of g.Denote by Der（ g，V）the vector space of all derivations，Inn（ g，V）the vector space of all inner derivations［4］.Set

Denote by Der（g）the derivation algebra of g，Inn（ g）the vector space of all inner derivations of g.

Definition 1.2［4］Let G be a commutative group，a G-graded Lie algebra.A g module V is called G-graded，if

In this section，we will determine the derivation algebra of S.

It is easy to see that S is finitely generated.Define a Z-grading on S by

By Proposition 1.1 in［4］，we have the following lemma.

Theorem 1.4

and others are zero.

Theorem 1.5 H1（S，S）.That is，the derivation algebra of S is

2 Proof of Theorem 1.4

Proof For any m∈Z，D∈（Der S）m，by Lemma 1.3，we can assume

where a1（n），a2（n），a3（n），x11，b1（n），b2（n），b3（n），x12，c1（n），c2（n），c3（n），x13，y∈C.

By D［Li，Mj］=［D（Li），Mj］+［Li，D（Mj）］，we can get

From D［Li，Yj］=［D（Li），Yj］+［Li，D（Yj）］，we can obtain

By D［Yi，Yj］=［D（Yi），Yj］+［Yi，D（Yj）］，we have

Case 1 m=0.Letting i=0 in（1）～（13），we can obtain

for all j∈Z.

Let j=-i in（1）and use（17），and then we haveLet j=1，i=2 and j=3，i=2 in（1）respectively.Then we get a1（3）=a1（1）+a1（2）and a1（5）=a1（3）+a1（2）.So a1（2）=2a1（1）.Leting j=0 in（1）and using induction on i，we have

Letting j=-i in（4）and（17），we have y=0.Letting j=0 in（30），we get

Subcase 1.1 If there exists some n0∈Z such that 2μ-n0λ=0.Sinceμ≠0，we have n0≠0.Let j=0 in（6），and then we have（2μ-λi）［b2（i）-a1（i）-b2（0）］=0.Hence

Letting i=j=n0in（6），we get b2（2n0）=a1（n0）+b2（0）.According to（19），we can obtain b2（n0）= n0a1（1）+b2（0）.So b2（i）=a1（i）+b2（0）=ia1（1）+b2（0）for all i∈Z.By（18），we have

Letting j=-i≠0 in（14）and using（20），

Subcase 1.2 2μ-nλ≠0 for all n∈Z.Letting j=0 in（6），we have b2（i）=ia1（1）+b2（0）for all i∈Z.By（18），we have

Letting j=-i≠0 in（14）and using（21），we can obtainTherefore，

Therefore，by Subcase 1.1 and Subcase1.2，we always have Hence

Thus we obtain

So Der S（）0=Inn S（）0⊕CD-1⊕CD-2⊕CD-3.

Case 2 m≠0.Let i=0 in（1）～（16）.Then we have

①λ≠0，-1，-2.Let j=0 in（31）.Then we get a2（0）=0.Let j=-i in（31）.Then we have

Let j=1，i=-2 in（31）.and then we obtain a22（）=2a21（）.Let i=1 in（31）and use induction on j＞1，and then we can get that a2j（）=ja21（）for all j∈Z.Hence，we have D（Mn）=D（C1）=0 and

②λ=0.By（32）and（24），we have

Then（31）becomes

Let i=1，and then we have（j-1）a2（1+j）=-a2（1）+ja2j（）.Hence we have

Let i=-j in（34），and then we get

Let j=-2 in（35），ang then we obtain a20（）=2a21（）-a2（0）.So

Thus we have D（Mn）=D（C1）=0 and

Set a1=a2（1）-a2（0），a2=a2（0）.Then we can check.So

③λ=-1.By（31）～（33），we have

Let i=1 in（36）.Then we have（j-1）a2（1+j）=-j+1（）a2（1）+j+1（）a2j（）.So we can deduce

Hence D（Mn）=D（C1）=0 and

④λ=-2.By（31）～（33），we have

Let i=1 in（37），and then we have

Use induction on j＞1，and then we can deduce

Let j=0 in（38），and then we get a2（0）=0.Let j=-i in（38），and then we get a2（-i）=-a2（i）for all i∈Z.Then we canall j∈Z.Hence

［1］HENKEL M.Schr?dinger invariance and strongly anisotropic critical systems［J］.Journal of Statistical Physics，1994，75（5/6）：1 023-1 061.

［2］ROGER C，UNTERBERGER J.The Schr?dinger-Virasoro Lie group and algebra：representation theory and cohomological study［J］.Ann Henri Poincare，2006，7（7-8）：1 477-1 529.

［3］LI J.2-cocycles of twisted deformative Schr?dinger-Virasoro algebras［J］.Comm Algebra，2012，40（6）：1 933-1 950.

［4］FARNSTEINER R.Derivations and extensions of finitely generated graded Lie algebras［J］.J Algebra，1988，118（1）：34-35.

［5］JIANG C，MENGD.The derivations，algebra of the associative algebra Cq［X，Y，X-1，Y-1］［J］.Comm Algebra，1998，6（2）：1 723-1 736.

［6］BENKART G，MOODY R.Derivations，central extensions and affine Lie algebras［J］.Algebras Groups Geom，1986，3（4）：456-492.

一類扭形變Schr?dinger-Virasoro代數的研究

唐 佳，高壽蘭，顧海霞
（湖州師范學院理學院，浙江湖州313000）

研究了一類含有兩個參數的扭形變Schr?dinger-Virasoro李代數，計算了這類李代數的一維中心擴張的所有導子，證明它有7個外導子.此結果為繼續研究這個李代數的表示理論提供了依據.

Schr?dinger-Virasoro李代數；中心擴張；導子

O152.5

O152.5 Document code：A Article ID：1009-1734（2016）04-0007-07

［責任編輯 高俊娥］

Received date：2016-03-05

s：Supported by National Nature Science Foundation（11201141，11371134）and Natural Science Foundation of Zhejiang Province（LQ12A01005，LZ14A010001）.

Biography：Gao Shoulan，Doctor，Research Interests：Lie algebra.E-mail：gaoshoulan@hutc.zj.cn

MSC 2000：17B40