2-Local derivations on the Super Virasoro algebra and Super W(2,2) algebra
aa r X i v : . [ m a t h . R A ] A ug W(2 , ALGEBRA
MUNAYIM DILXAT, SHOULAN GAO, AND DONG LIU
Abstract:
The present paper is devoted to study 2-local superderivations onthe super Virasoro algebra and the super W(2 ,
2) algebra. We prove that all 2-local superderivations on the super Virasoro algebra as well as the super W(2 , Key Words:
Lie superalgebra, super W(2 ,
2) algebra, 2-Local superderivation, superderivation.
MSC: Introduction
In 1997, ˇSemrl [7] introduced the notion of 2-local derivations on algebras.Namely, a map ∆ :
L → L (not necessarily linear) on an algebra L is called a if, for every pair of elements x, y ∈ L , there exists a derivation D x,y : L → L such that D x,y ( x ) = ∆( x ) and D x,y ( y ) = ∆( y ). For a given algebra L , the main problem concerning these notions is to prove that they automaticallybecome derivations or to give examples of 2-local derivations of L , which are notderivations. Solutions of such problems for finite-dimensional Lie algebras overalgebraically closed field of zero characteristic were obtained in [1]. Namely, in[1] it is proved that every 2-local derivation on a semi-simple Lie algebra is aderivation and that each finite-dimensional nilpotent Lie algebra with dimensionlarger than two admits 2-local derivation which is not a derivation. Recently,there are some studies on 2-local derivations on the infinite-dimensional Lie al-gebras [2, 9, 10]. The authors prove that 2-local derivations on the Witt algebra[2], some class of generalized Witt algebra (or their Borel subalgebras) [10] andW-algebra W(2,2) [9] are derivations.However, 2-local derivations for many Lie superalgebras are not studied up tonow. In the present paper, we study 2-local superderivations on some infinite-dimensional Lie superalgebras.This paper is arranged as follows. In Section 2, we give some preliminariesconcerning the super Virasoro algebra. In Section 3, we prove that every 2-localsuperderivation on the super Virasoro algebra is automatically a superderivation.In Section 4, we prove that every 2-local superderivation on the super W(2 , C , Z to denote the sets of the complexnumbers and the integers, respectively. All algebras (vector spaces) are based onthe field C . Preliminaries
In this section we give some necessary definitions and preliminary results.
Definition 2.1. [5, 8]
Let L be a Lie superalgebra, we call a linear map D : L → L a superderivation of L if D ([ x, y ]) = [ D ( x ) , y ] + ( − | D || x | [ x, D ( y )] , ∀ x, y ∈ L. Denote by Der( L ) and Der τ ( L ) the set of all superderivations and all su-perderivations of degree τ of L ( τ ∈ Z ), respectively. Obviously, Der( L ) =Der ( L ) L Der ( L ). For all a ∈ L , the map ad( a ) on L defined as ad( a ) x =[ a, x ] , x ∈ L is a superderivation and superderivations of this form are called in-ner superderivations. Denote by IDer( L ) the set of all inner superderivations of L .Recall that a map ∆ : L → L (not linear in general) is called a 2-local su-perderivation if, for every pair of elements x, y ∈ L , there exists a superderivation D x,y : L → L (depending on x, y ) such that D x,y ( x ) = ∆( x ) and D x,y ( y ) = ∆( y ).For a 2-local superderivation on L and k ∈ C , x ∈ L , we have∆( kx ) = D x,kx ( kx ) = kD x,kx ( x ) = k ∆( x ) . Superconformal algebras have a long history in mathematical physics. Thesimplest examples, after the Virasoro algebra itself (corresponding to N = 0)are the N = 1 superconformal algebras: the Neveu-Schwarz algebra ( ǫ = ) andthe Ramond algebra ( ǫ = 0). These infinite dimensional Lie superalgebras arealso called the super-Virasoro algebras as they can be regarded as natural supergeneralizations of the Virasoro algebra. Definition 2.2.
For ǫ = 0 , , the super Virasoro algebra SVir[ ǫ ] is a Lie super-algebra spanned by { L m , G r , C | m ∈ Z , r ∈ Z + ǫ } , equipped with the followingrelations: [ L m , L n ] = ( m − n ) L m + n + 112 δ m + n, ( m − m ) C, [ L m , G r ] = ( m − r ) G m + r , [ G r , G s ] = 2 L r + s + 13 δ r + s, ( r −
14 ) C for all m, n ∈ Z , r, s ∈ Z + ǫ . Lemma 2.3. [3]
For the super Virasoro algebra, we have
Der(SVir[ ǫ ]) = IDer(SVir[ ǫ ]) . The even part of the super Virasoro algebra is the Virasoro algebra. In [2] and[10], 2-local derivations over the Virasoro algebra are determined.
Theorem 2.4. [2, 10]
Every 2-local derivation on the Virasoro algebra is a deriva-tion. -LOCAL DERIVATIONS ON INFINITE-DIMENSIONAL LIE ALGEBRAS 3
Based on the above methods, we do such researches for the super Virasoroalgebra and some related Lie superalgebras in this paper.3.
SVir[ ǫ ]Now we shall give the main result concerning 2-local superderivations on thesuper Virasoro algebra SVir[0]. Lemma 3.1.
Let ∆ be a 2-local superderivation on the super Virasoro algebra SVir[0] . Then for any x, y ∈ SVir[0] , there exists a superderivation D x,y of SVir[0] such that D x,y ( x ) = ∆( x ) , D x,y ( y ) = ∆( y ) and it can be written as D x,y = ad( X k ∈ Z ( a k ( x, y ) L k + b k ( x, y ) G k )) , (3.1) where a k , b k ( k ∈ Z ) are complex-valued functions on SVir[0] × SVir[0] .Proof.
By Lemma 2.3, the superderivation D x,y can obviously be written as theform of (3 . (cid:3) Theorem 3.2.
Every 2-local superderivation on the super Virasoro algebra
SVir[0] is a superderivation.
To prove Theorem 3.2, we need several lemmas.
Lemma 3.3.
Let ∆ be a 2-local superderivation on SVir[0] . For a given i ∈ Z ,if ∆( G i ) = 0 then D G i ,y = ad( a i ( G i , y ) L i ) , ∀ y ∈ SVir[0] , (3.2) where a i is a complex-valued function on SVir[0] × SVir[0] .Proof.
By Lemma 3.1, we can assume that D G i ,y = ad( X k ∈ Z ( a k ( G i , y ) L k + b k ( G i , y ) G k )) (3.3)for some complex-valued functions a k , b k ( k ∈ Z ) on SVir[0] × SVir[0].When ∆( G i ) = 0, in view of (3.3) we obtain0 = ∆( G i ) = D G i ,y ( G i )= [ X k ∈ Z ( a k ( G i , y ) L k + b k ( G i , y ) G k ) , G i ]= X k ∈ Z (( k − i ) a k ( G i , y ) G k + i + 2 b k ( G i , y ) L k + i + 13 δ k + i, ( k −
14 ) b k ( G i , y ) C ) . Then we have ( k − i ) a k ( G i , y ) = b k ( G i , y ) = 0 for all k ∈ Z , which deduces a k ( G i , y ) = 0 with k = 2 i and b k ( G i , y ) = 0 for all k ∈ Z . This with (3.3) impliesthat (3.2) holds. The proof is completed. (cid:3) MUNAYIM DILXAT, SHOULAN GAO, AND DONG LIU
Lemma 3.4.
Let ∆ be a -local superderivation on SVir[0] such that ∆( G ) =∆( G ) = 0 . Then ∆( G i ) = ∆( L i ) = 0 , ∀ i ∈ Z . Proof.
Since ∆( G ) = ∆( G ) = 0, by using Lemma 3.3, for any y ∈ SVir[0], wecan assume that D G ,y = ad( a ( G , y ) L ) , (3.4) D G ,y = ad( a ( G , y ) L ) , (3.5)where a , a are complex-valued functions on SVir[0] × SVir[0]. Let i ∈ Z be afixed index. Taking y = G i in (3.4) and (3.5) respectively, we get∆( G i ) = D G ,G i ( G i ) = [ a ( G , G i ) L , G i ] = − ia ( G , G i ) G i , ∆( G i ) = D G ,G i ( G i ) = [ a ( G , G i ) L , G i ] = (1 − i ) a ( G , G i ) G i +2 . By the above two equations, we have ia ( G , G i ) G i + (1 − i ) a ( G , G i ) G i +2 = 0 , which implies a ( G , G i ) = 0 with i = 0 and a ( G , G i ) = 0 with i = 1. Itconcludes that ∆( G i ) = 0.Similarly, setting y = L i in (3.4) and (3.5) respectively, we get∆( L i ) = D G ,L i ( L i ) = [ a ( G , L i ) L , L i ] = − ia ( G , L i ) L i , ∆( L i ) = D G ,L i ( L i ) = [ a ( G , L i ) L , L i ] = a ( G , L i )((2 − i ) L i +2 + 12 δ i +2 , C ) . By the above two equations, it follows that ia ( G , L i ) L i + a ( G , L i )((2 − i ) L i +2 + 12 δ i +2 , C ) = 0 , which implies a ( G , L i ) = 0 with i = 0 and a ( G , L i ) = 0 with i = 2. Itconcludes that ∆( L i ) = 0. The proof is finished. (cid:3) Lemma 3.5.
Let ∆ be a 2-local superderivation on SVir[0] such that ∆( G i ) = 0 for all i ∈ Z . Then for any x = P t ∈ Z ( α t L t + β t G t ) + aC ∈ SVir[0] , where α t , β t , a ∈ C , we have ∆( x ) = 0 . Proof.
For any x = P t ∈ Z ( α t L t + β t G t ) + aC ∈ SVir[0], where α t , β t , a ∈ C , since∆( G i ) = 0 for any i ∈ Z , from Lemma 3.3 we have∆( x ) = D G i ,x ( x ) = [ a i ( G i , x ) L i , x ]= X t ∈ Z ( α t a i ( G i , x )((2 i − t ) L t +2 i + 112 δ i + t, (8 i − i ) C )+( i − t ) β t a i ( G i , x ) G t +2 i ) . By taking enough diffident i ∈ Z in the above equation and, if necessary, let these i , s be large enough, we obtain that ∆( x ) = 0. (cid:3) -LOCAL DERIVATIONS ON INFINITE-DIMENSIONAL LIE ALGEBRAS 5 Now we are to prove Theorem 3.2.
Proof of Theorem D G ,G such that∆( G ) = D G ,G ( G ) , ∆( G ) = D G ,G ( G ) . Set ∆ = ∆ − D G ,G . Then ∆ is a 2-local superderivation such that ∆ ( G ) =∆ ( G ) = 0. By Lemma 3.4, ∆ ( G i ) = 0 for all i ∈ Z . It follows that ∆ = 0.Thus ∆ = D G ,G is a superderivation. The proof is completed. (cid:3) Using the similar method of proving Theorem 3.2, we can obtain the sameresult for the super Virasoro algebra SVir[ ], and then we get the main result ofthis section. Theorem 3.6.
Every 2-local superderivation on the super Virasoro algebra
SVir[ ǫ ] is a superderivation. W(2 , algebra In this section we shall study 2-local superderivations on the super W(2 , , N = (1 ,
1) supersymmetric extension of Galileanconformal algebra (SGCA) in 2d (see [4, 6]).By definition, SW(2 ,
2) is a Lie superalgebra over C with a basis { L m , I m , G m , Q m , C , C | m ∈ Z } and the following non-vanishing relations:[ L m , L n ] = ( m − n ) L n + m + 112 δ m + n, ( m − m ) C , [ L m , I n ] = ( m − n ) I m + n + 112 δ m + n, ( m − m ) C , [ L m , G r ] = ( m − r ) G m + r , [ L m , Q r ] = ( m − r ) Q m + r , [ G r , G s ] = 2 L r + s + 13 δ r + s, ( r −
14 ) C , [ G r , Q s ] = 2 I r + s + 13 δ r + s, ( r −
14 ) C , [ I m , G r ] = ( m − r ) Q m + r for all m, n, r, s ∈ Z . Lemma 4.1. [4]
For the Lie superalgebra
SW(2 , , Der(SW(2 , , M C D, where D is an outer derivation defined by D ( I m ) = I m , D ( Q r ) = Q r , D ( C ) = C , D ( L m ) = D ( G r ) = D ( C ) = 0 (4.1) MUNAYIM DILXAT, SHOULAN GAO, AND DONG LIU for all m, r ∈ Z . Lemma 4.2.
Let ∆ be a 2-local superderivation on SW(2 , . Then for any x, y ∈ SW(2 , , there exists a superderivation D x,y of SW(2 , such that D x,y ( x ) =∆( x ) , D x,y ( y ) = ∆( y ) and it can be written as D x,y = ad X k ∈ Z ( a k ( x, y ) L k + b k ( x, y ) I k + c k ( x, y ) G k + d k ( x, y ) Q k ) ! + λ ( x, y ) D, (4.2) where a k , b k , c k , d k and λ are complex-valued functions on SW(2 , × SW(2 , and D is given by (4.1) .Proof. By Lemma 4.1, the superderivation D x,y can obviously be written as theform of (4.2). (cid:3) Now we shall give the main result concerning 2-local superderivations on SW(2 , Theorem 4.3.
Every 2-local superderivation on
SW(2 , is a superderivation. To prove Theorem 4.3, we need several lemmas.
Lemma 4.4.
Let ∆ be a 2-local superderivation on SW(2 , . ( i ) For a given r ∈ Z , if ∆( G r ) = 0 , then for any y ∈ SW(2 , , D G r ,y = ad( a r ( G r , y ) L r + b r ( G r , y ) I r ) + λ ( G r , y ) D ; (4.3)( ii ) If ∆( I + Q ) = 0 , then for any y ∈ SW(2 , we have D I + Q ,y = ad( a ( I + Q , y ) L + a ( I + Q , y ) L + X k ∈ Z b k ( I + Q , y ) I k + c ( I + Q , y ) G + X k ∈ Z d k ( I + Q , y ) Q k ) , (4.4) where a r , b r , a , a , c , b k , d k ( k ∈ Z ) and λ are complex-valued functions on SW(2 , × SW(2 , .Proof. By Lemma 4.2, we can assume that D G r ,y = ad( X k ∈ Z ( a k ( G r , y ) L k + b k ( G r , y ) I k + c k ( G r , y ) G k + d k ( G r , y ) Q k ))+ λ ( G r , y ) D, (4.5) D I + Q ,y = ad( X k ∈ Z ( a k ( I + Q , y ) L k + b k ( I + Q , y ) I k + c k ( I + Q , y ) G k + d k ( I + Q , y ) Q k )) + λ ( I + Q , y ) D, (4.6)where a k , b k , c k , d k and λ are complex-valued functions on SW(2 , × SW(2 , -LOCAL DERIVATIONS ON INFINITE-DIMENSIONAL LIE ALGEBRAS 7 (i) When ∆( G r ) = 0, in view of (4.5) we obtain0 = ∆( G r ) = D G r ,y ( G r )= [ X k ∈ Z ( a k ( G r , y ) L k + b k ( G r , y ) I k + c k ( G r , y ) G k + d k ( G r , y ) Q k ) , G r ]+ λ ( G r , y ) D ( G r )= X k ∈ Z (( k − r ) a k ( G r , y ) G k + r + ( k − r ) b k ( G r , y ) Q k + r + c k ( G r , y )(2 L k + r + 13 δ k + r, ( k −
14 ) C ) + d k ( G r , y )(2 I k + r + 13 δ k + r, ( k −
14 ) C ))From the above equation, we have ( k − r ) a k ( G r , y ) = ( k − r ) b k ( G r , y ) = 0 forall k ∈ Z , which deduces a k ( G r , y ) = b k ( G r , y ) = 0 with k = 2 r . We also get c k ( G r , y ) = d k ( G r , y ) = 0 for all k ∈ Z . Then Equation (4.3) holds.(ii) When ∆( I + Q ) = 0, in view of (4.6) we obtain0 = ∆( I + Q ) = D I + Q ,y ( I + Q )= [ X k ∈ Z ( a k ( I + Q , y ) L k + b k ( I + Q , y ) I k + c k ( I + Q , y ) G k + d k ( I + Q , y ) Q k ) , I + Q ] + λ ( I + Q , y ) D ( I + Q )= X k ∈ Z ( ka k ( I + Q , y ) + 2 c k ( I + Q , y )) I k + X k ∈ Z ( k a k ( I + Q , y ) + kc k ( I + Q , y )) Q k + 112 δ k, ( k − k ) a k ( I + Q , y ) C + 13 δ k, ( k −
14 ) c k ( I + Q , y ) C + λ ( I + Q , y )( I + Q )If k = 0, then 2 c ( I + Q , y ) I + λ ( I + Q , y ) I + λ ( I + Q , y ) Q = 0, weget λ ( I + Q , y ) = c ( I + Q , y ) = 0 and a ( I + Q , y ) = 0. If k = 0, then ka k ( I + Q , y ) + 2 c k ( I + Q , y ) = 0 and kc k ( I + Q , y ) + k a k ( I + Q , y ) = 0,we get a k ( I + Q , y ) = c k ( I + Q , y ) = 0 with k = 1. Then (4.4) holds. Theproof is completed. (cid:3) Lemma 4.5.
Let ∆ be a 2-local superderivation on SW(2 , such that ∆( G ) =∆( G ) = 0 . Then ∆( G i ) = ∆( L i ) = 0 , ∀ i ∈ Z . Proof.
It is essentially same as that of Lemma 3.4. Since ∆( G ) = ∆( G ) = 0,by using Lemma 4.4, for any y ∈ SW(2 , D G ,y = ad( a ( G , y ) L + b ( G , y ) I ) + λ ( G , y ) D, (4.7) D G ,y = ad( a ( G , y ) L + b ( G , y ) I ) + λ ( G , y ) D, (4.8) MUNAYIM DILXAT, SHOULAN GAO, AND DONG LIU where a , a , b , b , λ are complex-valued functions on SW(2 , × SW(2 , i ∈ Z be a fixed index. Taking y = G i in(4.7) and (4.8) respectively, weget ∆( G i ) = D G ,G i ( G i )= [ a ( G , G i ) L + b ( G , G i ) I , G i ] + λ ( G , G i ) D ( G i )= − ia ( G , G i ) G i − ib ( G , G i ) Q i and ∆( G i ) = D G ,G i ( G i )= [ a ( G , G i ) L + b ( G , G i ) I ) , G i ] + λ ( G , G i ) D ( G i )= (1 − i ) a ( G , G i ) G i +2 + (1 − i ) b ( G , G i ) Q i +2 . By the above two equations, it follows that ia ( G , G i ) G i + ib ( G , G i ) Q i + (1 − i ) a ( G , G i ) G i +2 + (1 − i ) b ( G , G i ) Q i +2 = 0 , which implies a ( G , G i ) = b ( G , G i ) = 0 with i = 0 and a ( G , G i ) = b ( G , G i ) =0 with i = 1. It concludes that ∆( G i ) = 0.Similarly, setting y = L i we can also get ∆( L i ) = 0. Taking y = L i in(4.7) and(4.8) respectively, we get∆( L i ) = D G ,L i ( L i )= [ a ( G , L i ) L + b ( G , L i ) I , L i ] + λ ( G , L i ) D ( L i )= − ia ( G , L i ) L i − ib ( G , L i ) I i and ∆( L i ) = D G ,L i ( L i )= [ a ( G , L i ) L + b ( G , L i ) I ) , L i ] + λ ( G , L i ) D ( L i )= a ( G , L i )((2 − i ) L i +2 + δ i +2 , C ) + (2 − i ) b ( G , L i ) I i +2 . By the above two equations, it follows that ia ( G , L i ) L i + ib ( G , L i ) I i + a ( G , L i )((2 − i ) L i +2 + 12 δ i +2 , C )+(2 − i ) b ( G , L i ) I i +2 = 0 , which implies a ( G , L i ) = b ( G , L i ) = 0 with i = 0 and a ( G , L i ) = b ( G , L i ) =0 with i = 2. It concludes that ∆( L i ) = 0. The proof is finished. (cid:3) Lemma 4.6.
Let ∆ be a 2-local superderivation on SW(2 , such that ∆( G i ) = 0 for all i ∈ Z . Then for any x = P t ∈ Z ( α t L t + β t I t + γ t G t + δ t Q t ) + aC + bC ∈ SW(2 , , we have ∆( x ) = µ x X t ∈ Z ( β t I t + δ t Q t ) + bC , -LOCAL DERIVATIONS ON INFINITE-DIMENSIONAL LIE ALGEBRAS 9 where α t , β t , γ t , δ t , a, b, µ x ∈ C .Proof. For x = P t ∈ Z ( α t L t + β t I t + γ t G t + δ t Q t ) + aC + bC ∈ SW(2 , G i ) = 0 for any i ∈ Z , from Lemma 4.4 we have∆( x ) = D G i ,x ( x ) = [ a i ( G i , x ) L i + b i ( G i , x ) I i , x ] + λ ( G i , x ) D ( x )= P t ∈ Z α t a i ( G i , x )((2 i − t ) L i + t + δ i + t, (4 i − i ) C )+ P t ∈ Z α t b i ( G i , x )((2 i − t ) I i + t + δ i + t, (4 i − i ) C )+ P t ∈ Z β t a i ( G i , x )((2 i − t ) I i + t + δ i + t, (4 i − i ) C )+ P t ∈ Z (( i − t ) γ t a i ( G i , x ) G i + t + (( i − t ) δ t a i ( G i , x ) + γ t b i ( G i , x )) Q i + t )+ P t ∈ Z λ ( G i , x )( β t I t + δ t Q t ) + bC . By taking enough diffident i ∈ Z in the above equation and, if necessary, letthese i , s be large enough, we obtain that ∆( x ) = µ x P t ∈ Z ( β t I t + δ t Q t ) + bC .Furthermore, µ x = λ ( G i , x ) is a constant since it is independent on i . (cid:3) Lemma 4.7.
Let ∆ be a 2-local superderivation on SW(2 , such that ∆( G i ) = 0 and ∆( I + Q ) = 0 for all i ∈ Z . Then for any odd integer p and y ∈ SW(2 , ,there exist some ζ yp , η yp ∈ C such that D L p + I p + Q p ,y = ad( ζ yp L p + η yp I p + ζ yp I p + ζ yp Q p ) . Proof.
By ∆( G i ) = 0 for all i ∈ Z and Lemma 4.6, we have∆( L p + I p + Q p ) = µ L p + I p + Q p ( I p + Q p ) , (4.9)where µ L p + I p + Q p ∈ C . In view of ∆( I + Q ) = 0 and Lemma 4.4, we know that( x = L p + I p + Q p )∆( L p + I p + Q p )= D I + Q ,L p + I p + Q p ( L p + I p + Q p )= [ a ( I + Q , x ) L + a ( I + Q , x ) L + P k ∈ Z b k ( I + Q , x ) I k + c ( I + Q , x ) G + P k ∈ Z d k ( I + Q , x ) Q k , L p + I p + Q p ]= − pa ( I + Q , x )( L p + 2 I p + 2 Q p )+ a ( I + Q , x )((1 − p ) L p +1 + (1 − p ) I p +1 + ( − p ) Q p +1 )+ P k ∈ Z ( k − p ) b k ( I + Q , x ) I p + k + c ( I + Q , x )((1 − p ) G p +1 +(1 − p ) Q p +1 + 2 I p +1 ) + P k ∈ Z ( k − p ) d k ( I + Q , x ) Q k + p . Together with (4.9), we get µ L p + I p + Q p ( I p + Q p )= − pa ( I + Q , x )( L p + 2 I p + 2 Q p )+ a ( I + Q , x )((1 − p ) L p +1 + (1 − p ) I p +1 + ( − p ) Q p +1 )+ P k ∈ Z ( k − p ) b k ( I + Q , x ) I p + k + c ( I + Q , x )((1 − p ) G p +1 +(1 − p ) Q p +1 + 2 I p +1 ) + P k ∈ Z ( k − p ) d k ( I + Q , x ) Q k + p . From the above equation we can easily see that a ( I + Q , x ) = 0. Next, observethe coefficient of I p , and then we get µ L p + I p + Q p = 0. So it follows that∆( L p + I p + Q p ) = 0 . (4.10)Next, for any y ∈ SW(2 , D L p + I p + Q p ,y = ad ( X k ∈ Z ( a k ( L p + I p + Q p , y ) L k + b k ( L p + I p + Q p , y ) I k + c k ( L p + I p + Q p , y ) G k + d k ( L p + I p + Q p , y ) Q k ))+ λ ( L p + I p + Q p , y ) D. (4.11)From (4.10) and (4.11), we have ( x = L p + I p + Q p )∆( x ) = D x,y ( x )= [ P k ∈ Z ( a k ( x, y ) L k + b k ( x, y ) I k + c k ( x, y ) G k + d k ( x, y ) Q k ) , x ]+ λ ( x, y ) D ( x )= P k ∈ Z a k ( x, y )(( k − p ) L p + k + δ p + k, ( k − k ) C + ( k − p ) I k +2 p + δ p + k, ( k − k ) C + ( k − p ) Q k +2 p ) + P k ∈ Z b k ( x, y )(( k − p ) I p + k + δ p + k, ( k − k ) C ) + P k ∈ Z c k ( x, y )(2 I p + k + δ p + k, ( k − ) C +( k − p ) G p + k + ( k − p ) Q k +2 p ) + P k ∈ Z ( k − p ) d k ( x, y ) Q k + p + λ ( x, y )( I p + Q p ) = 0 . From this, it is easy to see that ( k − p ) a k ( x, y ) L k + p = 0 for all k ∈ Z . So a k ( x, y ) = 0 for all k = p . Similarly, ( k − p ) c k ( x, y ) G p + k = 0 for all k ∈ Z . Hence c k ( x, y ) = 0. Using this conclusion and observing the coefficients of I p , Q p inthe above equation, we get − pa p ( x, y ) + (2 p − p ) b p ( x, y ) = 0 , ( p − p ) a p ( x, y ) + (2 p − p d p ( x, y ) = 0 , which implies a p ( x, y ) = b p ( x, y ) = d p ( x, y ). Furthermore, by observing thecoefficient of I k , k = 3 p in the above equation we get λ ( x, y ) = 0 and b k ( x, y ) = 0 -LOCAL DERIVATIONS ON INFINITE-DIMENSIONAL LIE ALGEBRAS 11 for all k = p, p . Observing the coefficient of Q k , k = 3 p in the above equation,we get d k ( x, y ) = 0 for all k = 2 p . Finally, set ζ yp = a p ( x, y ), η yp = b p ( x, y ), andthen we finish the proof. (cid:3) Lemma 4.8.
Let ∆ be a 2-local superderivation on SW(2 , such that ∆( G ) =∆( G ) = ∆( I + Q ) = 0 . Then ∆( x ) = 0 for all x ∈ SW(2 , .Proof. Take any but fixed x = P t ∈ Z ( α t L t + β t I t + γ t G t + δ t Q t ) + aC + bC ∈ SW(2 , α t , β t , γ t , δ t , a, b ∈ C for any t ∈ Z .Since ∆( G ) = ∆( G ) = 0, it follows by Lemma 4.5 that∆( G i ) = 0 , ∀ i ∈ Z . (4.12)This, together with Lemma 4.6, gives∆( x ) = ∆( X t ∈ Z ( α t L t + β t I t + γ t G t + δ t Q t ) + aC + bC )= µ x X t ∈ Z ( β t I t + δ t Q t ) + bC (4.13)for some µ x ∈ C . By (4.12) and ∆( I + Q ) = 0, we obtain by Lemma 4.7 that∆( x ) = D L p + I p + Q p ,x ( x )= [ ζ xp L p + η xp I p + ζ xp I p + ζ xp Q p , X t ∈ Z ( α t L t + β t I t + γ t G t + δ t Q t ) + aC + bC ]= X t ∈ Z ζ xp α t (( p − t ) L t + p + 112 δ p + t, ( p − p ) C )+ X t ∈ Z ζ xp β t (( p − t ) I t + p + 112 δ p + t, ( p − p ) C )+ X t ∈ Z η xp α t (( p − t ) I t + p + 112 δ p + t, ( p − p ) C )+ X t ∈ Z ζ xp α t ((2 p − t ) I t +2 p + 112 δ p + t, (8 p − p ) C )+ X t ∈ Z ζ xp γ t (2 I t +2 p + 13 δ p + t, (4 p −
14 ) C )+ X t ∈ Z ( p − t ) ζ xp γ t G t + p + X t ∈ Z (( p − t ) ζ xp δ t + ( p − t ) η xp γ t ) Q t + p + X t ∈ Z (( p − t ) ζ xp γ t + (2 p − t ζ xp α t ) Q t +2 p . (4.14) Next the proof is divided into two cases according to the cases of ( α t ) t ∈ Z and( γ t ) t ∈ Z . Case 1. ( α t ) t ∈ Z is not a zero sequence, i.e., x = P t ∈ Z ( α t L t + β t I t + γ t G t + δ t Q t ) + aC + bC . Hence there is a nonzero term α t L t in x = P t ∈ Z ( α t L t + β t I t + γ t G t + δ t Q t ) + aC + bC for some t ∈ Z . Take two integers p = p and p = p in (4.14) such that p i − t = 0 , i = 1 ,
2, then by ( p i − t ) ζ xp i α t L p i + t = 0in (4.14) we have ζ xp i = 0. By (4.13) and (4.14), we have∆( x ) = µ x P t ∈ Z ( β t I t + δ t Q t ) + bC = P t ∈ Z η xp i α t (( p i − t ) I t + p i + δ p i + t, ( p i − p i ) C )+( p i − t ) η xp i γ t Q p i + t ) , i = 1 , . Hence µ x X t ∈ Z β t I t = X t ∈ Z η xp i α t (( p i − t ) I t + p i + 112 δ p i + t, ( p i − p i ) C ) ,µ x X t ∈ Z δ t Q t = X t ∈ Z ( p i − t ) η xp i γ t Q p i + t , i = 1 , . By taking p and p in the above equation such that p , p , p − p are largeenough, we see that ∆( x ) = 0. Case 2. ( α t ) t ∈ Z is a zero sequence, i.e., x = P t ∈ Z ( β t I t + γ t G t + δ t Q t )+ aC + bC . Subcase 2.1.
If ( γ t ) t ∈ Z is not a zero sequence, there is a nonzero term γ t G t in x = P t ∈ Z ( β t I t + γ t G t + δ t Q t )+ aC + bC for some t ∈ Z . Take two integers p = p and p = p in (4.14) such that p i − t = 0 , i = 1 ,
2, then by ( p i − t ) ζ xp i γ t G p i + t = 0in (4.14) we have ζ xp i = 0. By (4.13) and (4.14), we have∆( x ) = µ x P t ∈ Z ( β t I t + δ t Q t ) + bC = P t ∈ Z ( p i − t ) η xp i γ t Q p i + t , i = 1 , . Hence ∆( x ) = 0. Subcase 2.2.
If ( γ t ) t ∈ Z is a zero sequence, i.e., x = P t ∈ Z ( β t I t + δ t Q t ) + aC + bC . Then by (4.13) and (4.14) we have∆( x ) = µ x P t ∈ Z ( β t I t + δ t Q t ) + bC = P t ∈ Z ( ζ xp β t (( p − t ) I t + p + δ p + t, ( p − p ) C ) + ( p − t ) ζ xp δ t Q p + t ) . By taking enough diffident p in the above equation and, if necessary, let these p ′ sbe large enough, we obtain that ∆( x ) = 0. The proof is completed. (cid:3) Now we are to prove Theorem 4.3.
Proof of Theorem , D G ,G such that∆( G ) = D G ,G ( G ) , ∆( G ) = D G ,G ( G ) . Set ∆ = ∆ − D G ,G . Then ∆ is a 2-local superderivation such that ∆ ( G ) =∆ ( G ) = 0. By Lemma 4.5, ∆ ( G i ) = 0 for all i ∈ Z . Combining with Lemma -LOCAL DERIVATIONS ON INFINITE-DIMENSIONAL LIE ALGEBRAS 13 ( I + Q ) = µ I + Q ( I + Q ) for some µ I + Q ∈ C . Now, set∆ = ∆ − µ I + Q D , then ∆ is a 2-local derivation such that∆ ( G ) = ∆ ( G ) − µ I + Q D ( G ) = 0 , ∆ ( G ) = ∆ ( G ) − µ I + Q D ( G ) = 0 , ∆ ( I + Q ) = ∆ ( I + Q ) − µ I + Q D ( I + Q )= µ I + Q ( I + Q ) − µ I + Q ( I + Q ) = 0 . By Lemma 4.8, it follows that ∆ = ∆ − D G ,G − µ I + Q D ≡
0. Thus ∆ = D G ,G + µ I + Q D is a superderivation. The proof is completed. (cid:3) References [1] Sh.A Ayupov, K.K. Kudaybergenov, I.S. Rakhimov, 2-Local derivations on finite-dimensional Lie algebras, Linear Algebra and its Applications, 474(2015) 1-11.[2] Sh.A Ayupov, B. Yusupov, 2-Local derivations of infinite-dimentional Lie algebras, J.Algebra Appl.,ID:2050100(2020).[3] S. Gao, Q. Meng, Y. Pei, The first cohomology of N=1 super Virasoro algebras, Comm.Algebra, 47(10)(2019)4230-4246.[4] S. Gao, Y. Pei, C. Bai, Some algebraic properties of the supersymmetric extension of GCAin 2d, J. Phys. A: Math. Theor. 47 (2014) 225202 (19pp).[5] V. Kac, Lie superalgebras, Adv. Math. 26(1)(1977)8-96.[6] I. Mandal, Supersymmetric extension of GCA in 2d, J. High Energy Phys. 11(2010)1-28.[7] P. ˇSemrl, Local automorphisms and derivations on B ( H ) , Proc. Amer. Math. Soc., 125(9)(1997)2677-2680.[8] M. Scheunert, Theory of Lie Superalgebras (Lecture Notes in Mathematics, 716). Springer-Verlag (1976).[9] X. Tang, 2-Local derivations on the W-algebra W(2,2), arXiv:2003.05627.[10] Y. Zhao, Y. Chen, K. Zhao, 2-local Derivations on Witt Algebras, J. Algebra Appl.,inpress,doi: 10.1142/S0219498821500687(2020).
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