The groups of automorphisms of the Witt W n and Virasoro Lie algebras
aa r X i v : . [ m a t h . R A ] A p r The groups of automorphisms of the Witt W n and VirasoroLie algebras V. V. Bavula
Abstract
Let L n = K [ x ± , . . . , x ± n ] be a Laurent polynomial algebra over a field K of characteristiczero, W n := Der K ( L n ), the Witt
Lie algebra, and Vir be the
Virasoro
Lie algebra. Weprove that Aut
Lie ( W n ) ≃ Aut K − alg ( L n ) ≃ GL n ( Z ) ⋉ K ∗ n and Aut Lie (Vir) ≃ Aut
Lie ( W ) ≃{± } ⋉ K ∗ . Key Words: Group of automorphisms, monomorphism, Lie algebra, the Witt algebra, theVirasoro algebra, automorphism, locally nilpotent derivation.Mathematics subject classification 2010: 17B40, 17B20, 17B66, 17B65, 17B30.
In this paper, module means a left module, K is a field of characteristic zero and K ∗ is its groupof units, and the following notation is fixed: • P n := K [ x , . . . , x n ] = L α ∈ N n Kx α is a polynomial algebra over K where x α := x α · · · x α n n , • G n := Aut K − alg ( P n ) is the group of automorphisms of the polynomial algebra P n , • L n := K [ x ± , . . . , x ± n ] = L α ∈ Z n Kx α is a Laurent polynomial algebra, • L n := Aut K − alg ( L n ) is the group of K -algebra automorphisms of L n , • ∂ := ∂∂x , . . . , ∂ n := ∂∂x n are the partial derivatives ( K -linear derivations) of P n , • D n := Der K ( P n ) = L ni =1 P n ∂ i is the Lie algebra of K -derivations of P n where [ ∂, δ ] := ∂δ − δ∂ , • G n := Aut Lie ( D n ) is the group of automorphisms of the Lie algebra D n , • W n := Der K ( L n ) = L ni =1 L n ∂ i is the Witt
Lie algebra where [ ∂, δ ] := ∂δ − δ∂ , • W n := Aut Lie ( W n ) is the group of automorphisms of the Witt Lie algebra W n , • δ := ad( ∂ ) , . . . , δ n := ad( ∂ n ) are the inner derivations of the Lie algebras D n and W n determined by ∂ , . . . , ∂ n where ad( a )( b ) := [ a, b ], • D n := L ni =1 K∂ i , • H n := L ni =1 KH i where H := x ∂ , . . . , H n := x n ∂ n , The group of automorphisms of the Witt Lie algebra W n . The aim of the paper is tofind the groups of automorphisms of the Witt algebra W n (Theorem 1.1) and the Virasoro algebraVir (Theorem 1.2). The following lemma is an easy exercise.1 (Lemma 2.8) L n ≃ GL n ( Z ) ⋉ T n where GL n ( Z ) is identified with a subgroup of L n via thegroup monomorphism GL n ( Z ) → L n , A = ( a ij ) σ a : x i Q nj =1 x a ji j and T n := { t λ ∈ L n | t λ ( x ) = λ x , . . . , t λ ( x n ) = λ n x n ; λ ∈ K ∗ n } ≃ K ∗ n is the algebraic n -dimensional torus . Theorem 1.1 W n = L n .Structure of the proof . (i) L n is a subgroup W n (Lemma 2.2) via the group monomorphism L n → W n , σ σ : ∂ σ ( ∂ ) := σ∂σ − . Let σ ∈ W n . We have to show that σ ∈ L n .(ii)(crux) σ ( H n ) = H n (Lemma 2.5), i.e. σ ( H ) = A σ H for some A σ ∈ GL n ( K )where H := ( H , . . . , H n ) T .(iii) A σ ∈ GL n ( Z ) (Corollary 2.7).(iv) There exists an automorphism τ ∈ L n such that τ σ ∈ Fix W n ( H , . . . , H n ) (Lemma 2.10).(v) Fix W n ( H , . . . , H n ) = T n ⊆ L n (Lemma 2.12) and so σ ∈ L n . (cid:3) The group of automorphisms of the Virasoro Lie algebra . The
Virasoro
Lie algebraVir = W ⊕ Kc is a 1-dimensional central extension of the Witt Lie algebra W where Z (Vir) = Kc is the centre of Vir and for all i, j ∈ Z ,[ x i H, x j H ] = ( j − i ) x i + j H + δ i, − j i − i c (1)where x = x and H = H . Theorem 1.2
Aut
Lie (Vir) ≃ W ≃ L ≃ GL ( Z ) ⋉ T . The key point in the proof of Theorem 1.2 is to use Theorem 1.3 of which Theorem 1.2 is aspecial case (where G = Vir, W = W and Z = Kc , see Section 3). Theorem 1.3
Let G be a Lie algebra, Z be a subspace of the centre of G and W = G /Z . Supposethat1. every automorphism σ of the Lie algebra W can be extended to an automorphism b σ of theLie algebra G ,2. Z ⊆ [ G, G ] , and3. W = [ W, W ] .Then, for each σ , the extension b σ is unique and the map Aut
Lie ( W ) → Aut
Lie ( G ) , σ b σ , is agroup isomorphism. The groups Aut
Lie ( u n ) and Aut Lie ( D n ) where found in [3] and [4] respectively. The Lie algebras u n have been studied in great detail in [1] and [2]. In particular, in [1] it was proved that everymonomorphism of the Lie algebra u n is an automorphism but this is not true for epimorphisms.2 Proof of Theorem 1.1
This section can be seen as a proof of Theorem 1.1. The proof is split into several statements thatreflect ‘Structure of the proof of Theorem 1.1’ given in the Introduction.By the very definition, H n = L ni =1 KH i is an abelian Lie subalgebra of W n of dimension n .Each element H of H n is a unique sum H = P ni =1 λ i H i where λ i ∈ K . Let us define the bilinearmap H n × Z n → K, ( H, α ) ( H, α ) := n X i =1 λ i α i . The Witt algebra W n is a Z n -graded Lie algebra . The Witt algebra W n = M α ∈ Z n n M i =1 Kx α ∂ i = M α ∈ Z n x α H n (2)is a Z n -graded Lie algebra, that is [ x α H n , x β H n ] ⊆ x α + β H n for all α, β ∈ Z n . This follows fromthe identity [ x α H, x β H ′ ] = x α + β (( H, β ) H ′ − ( H ′ , α ) H ) . (3)In particular, [ H, x α H ′ ] = ( H, α ) x α H ′ . (4)So, x α H n is the weight subspace W n,α := { w ∈ W n | [ H, w ] = (
H, α ) w } of W n with respect to theadjoint action of the abelian Lie algebra H n on W n . The direct sum (2) is the weight decompositionof W n and Z n is the set of weights of H n .Let G be a Lie algebra and H be its Lie subalgebra. The centralizer C G ( H ) := { x ∈ G | [ x, H ] =0 } of H in G is a Lie subalgebra of G . In particular, Z ( G ) := C G ( G ) is the centre of the Liealgebra G . The normalizer N G ( H ) := { x ∈ G | [ x, H ] ⊆ H} of H in G is a Lie subalgebra of G , it isthe largest Lie subalgebra of G that contains H as an ideal. Each element a ∈ G determines thederivation of the Lie algebra G by the rule ad( a ) : G → G , b [ a, b ], which is called the innerderivation associated with a . An element a ∈ G is called a locally finite element if so is the innerderivation ad( a ) of the Lie algebra G , that is dim K ( P i ∈ N K ad( a ) i ( b )) < ∞ for all b ∈ G . LetLF( G ) be the set of locally finite elements of G . The Cartan subalgebra H n of W n . A nilpotent Lie subalgebra C of a Lie algebra G suchthat C = N G ( C ) is called a Cartan subalgebra of G . We use often the following obvious observation: An abelian Lie subalgebra that coincides with its centralizer is a maximal abelian Lie subalgebra . Lemma 2.1 H n = C W n ( H n ) is a maximal abelian Lie subalgebra of W n .2. H n is a Cartan subalgebra of W n .Proof . Both statements follow from (2) and (4) . (cid:3) The next lemma is very useful and can be applied in many different situations. It allows oneto see the group of automorphisms of a ring as a subgroup of the group of automorphisms of itsLie algebra of derivations.
Lemma 2.2
Let R be a commutative ring such that there exists a derivation ∂ ∈ Der( R ) suchthat r∂ = 0 for all nonzero elements r ∈ R (eg, R = P n , L n and δ = ∂ ). Then the grouphomomorphism Aut( R ) → Aut
Lie (Der( R )) , σ σ : δ σ ( δ ) := σδσ − , is a monomorphism. roof . If an automorphism σ ∈ Aut( R ) belongs to the kernel of the group homomorphism σ σ then, for all r ∈ R , r∂ = σ ( r∂ ) σ − = σ ( r ) σ∂σ − = σ ( r ) ∂ , i.e. σ ( r ) = r for all r ∈ R .This means that σ is the identity automorphism. Therefore, the homomorphism σ σ is amonomorphism. (cid:3) The ( Z , λ ) -grading and the filtration F λ on W n . Each vector λ = ( λ , . . . , λ n ) ∈ Z n determines the Z -grading on the Lie algebra W n by the rule W n = M i ∈ Z W n,i ( λ ) , W n,i ( λ ) := M ( λ,α )= i x α H n , ( λ, α ) := n X i =1 λ i α i , [ W n,i ( λ ) , W n,j ( λ )] ⊆ W n,i + j ( λ ) for all i, j ∈ Z as follows from (3) and (4). The Z -grading aboveis called the ( Z , λ )- grading on W n . Every element a ∈ W n is the unique sum of homogeneouselements with respect to the ( Z , λ )-grading on W n , a = a i + a i + · · · + a i s , a i ν ∈ W n,i ν ( λ ) , and i < i < · · · < i s . The elements l + λ ( a ) := a i s and l − λ ( a ) := a i are called the leading term andthe least term of a respectively. So, a = l + λ ( a ) + · · · ,a = l − λ ( a ) + · · · , where the three dots denote smaller and larger terms respectively. For all a, b ∈ W n ,[ a, b ] = [ l + λ ( a ) , l + λ ( b )] + · · · , (5)[ a, b ] = [ l − λ ( a ) , l − λ ( b )] + · · · , (6)where the three dots denote smaller and larger terms respectively (the brackets on the RHS canbe zero). The Newton polygon of an element of W n . Each element a ∈ W n is the unique finite sum a = P α ∈ Z n λ α x α H α were λ α ∈ K and H α ∈ H n . The set Supp( a ) := { α ∈ Z n | λ α = 0 } is calledthe support of a and its convex hull in R n is called the Newton polygon of a , denoted by NP( a ). Lemma 2.3
Let a be a locally finite element of W n . Then the elements l + λ ( a ) and l − λ ( a ) are locallyfinite for all λ ∈ Z n .Proof . The statement follows from (5) and (6). (cid:3) Let LF( W n ) h be the set of homogeneous (with respect to the Z n -grading on W n ) locally finiteelements of the Lie algebra W n . Lemma 2.4
LF( W n ) h = H n .Proof . H n ⊆ LF( W n ) h since every element of H n is a semi-simple element of W n : for all H = P ni =1 λ i H i where λ i ∈ K ,[ H, x α H ′ ] = ( λ, α ) x α H ′ for all α ∈ Z n , H ′ ∈ H n . (7)It suffices to show that every homogeneous element x α H ′ that does not belong to H n , i.e. α = 0,is not locally finite. Fix i such that α i = 0. Let δ = ad( x α H ′ ).Suppose that ( H ′ , α ) = 0. This is the case for n = 1. Then δ m ( x α H ′ ) = ( m − m − ( H ′ , α ) m x (1+2 m ) α H ′ for m ≥ . x α H ′ is not locally finite.Suppose that ( H ′ , α ) = 0. Then necessarily n ≥
2. Fix β ∈ Z n such that ( H ′ , β ) = 1. Then δ m ( x β H ′ ) = x β + mα H ′ for m ≥ . Therefore, the element x α H ′ is not locally finite. (cid:3) Lemma 2.5 σ ( H n ) = H n for all σ ∈ W n .Proof . Let σ ∈ W n and H ∈ H n . We have to show that H ′ := σ ( H ) ∈ H n . The element H is a locally finite element, hence so is H ′ . By Lemma 2.3 and Lemma 2.4, the Newton polygonNP( H ′ ) has the single vertex 0, i.e. H ′ ∈ H n . (cid:3) Let H = ( H , . . . , H n ) T where T stands for the transposition. By Lemma 2.5, σ ( H ) = A σ H for all σ ∈ W n (8)where A σ = ( a ij ) ∈ GL n ( K ) and σ ( H i ) = P nj =1 a ij H j . Let σ W n be the W n -module W n twistedby the automorphism σ ∈ W n . As a vector space, σ W n = W n , but the adjoint action is twistedby σ : w · x α H ′′ = [ σ ( w ) , x α H ′′ ]for all w ∈ W n and α ∈ Z n . The map σ : W n → σ W n , w σ ( w ), is a W n -module isomorphism.By Lemma 2.5, every weight subspace x α H n of the H n -module W n = L α ∈ Z n x α H n is also aweight subspace for the H n -module σ W n , and vice versa. Moreover, W n,α = x α H n = ( σ W n ) A σ α for all α ∈ Z n (9)where α = ( α , . . . , α n ) T ∈ Z n is a column: for all H ′ = P ni =1 λ i H i ∈ H n ,[ σ ( H ′ ) , x α H ′′ ] = n X i,j =1 λ i a ij α j x α H ′′ = ( H ′ , A σ α ) x α H ′′ . (10)Since σ ( H n ) = H n and σ : W n → σ W n is a W n -module isomorphism, the automorphism σ permutes the weight components { W n,α = x α H n } α ∈ Z n . There is a bijection σ ′ : Z n → Z n , α σ ′ ( α ), such that σ ( W n,α ) = W n,σ ′ ( α ) for all α ∈ Z n . Lemma 2.6
For all σ ∈ W n and α ∈ Z n , σ ′ ( α ) = A σ − α .Proof . By (10),( H ′ , σ ′ ( α )) σ ( x α H ′′ ) = [ H ′ , σ ( x α H ′′ )] = σ ([ σ − ( H ′ ) , x α H ′′ ]) = σ (( H ′ , A σ − α ) x α H ′′ )= ( H ′′ , A σ − α ) σ ( x α H ′′ ) . Therefore, σ ′ ( α ) = A σ − α . (cid:3) Corollary 2.7
For all σ ∈ W n , A σ ∈ GL n ( Z ) .Proof . This follows from Lemma 2.6. (cid:3) The group of automorphisms L n = Aut Lie ( L n ). The group L n contains two obvioussubgroups: the algebraic n -dimensional torus T n = { t λ | λ ∈ K ∗ n } ≃ K ∗ n where t λ ( x i ) = λ i x i for i = 1 , . . . , n and GL n ( Z ) which can be seen as a subgroup of L n via the group monomorphismGL n ( Z ) → L n , A σ A : x i n Y j =1 x a ji j . (11)For all α ∈ Z n , σ A ( x α ) = x Aα . Hence σ AB = σ A σ B and σ − A = σ A − .5 emma 2.8 L n = GL n ( Z ) ⋉ T n .Proof . The group of units L ∗ n of the algebra L n is equal to the direct product of its twosubgroups K ∗ × X where X = { x α | α ∈ Z n } ≃ Z n via x α α . Since σ ( K ∗ ) = K ∗ for all σ ∈ L n ,there is a group homomorphism (where Aut gr ( G ) is the group of automorphisms of a group G ) θ : L n → Aut gr ( L n /K ∗ ) , σ σ : K ∗ x α K ∗ σ ( x α ) . Notice that Aut gr ( L n /K ∗ ) ≃ Aut gr ( Z n ) ≃ GL n ( Z ) and θ | GL n ( Z ) : GL n ( Z ) → Aut gr ( L n /K ∗ ), A A . Then L n ≃ GL n ( Z ) ⋉ ker( θ ) but ker( θ ) = T n . Clearly, L n = GL n ( Z ) ⋉ T n . (cid:3) Lemma 2.9
Let σ A ∈ L n be as in (11) where A ∈ GL n ( Z ) , ∂ = ( ∂ , . . . , ∂ n ) T , H = ( H , . . . , H n ) T and diag( λ , . . . , λ nn ) be the diagonal matrix with the diagonal elements λ , . . . , λ nn . Then1. σ ( ∂ ) = C σ ∂ where C σ = diag( σ ( x ) − , . . . , σ ( x n ) − ) A − diag( x , . . . , x n ) .2. σ ( H ) = A − H .Proof . 1. Let ∂ ′ i = σ ( ∂ i ) and x ′ j = σ ( x j ). Clearly, σ ( ∂ ) = C σ ∂ for some matrix C σ = ( c ij ) ∈ M n ( L n ). Applying the automorphism σ to the equalities δ ij = ∂ i ∗ x j where i, j = 1 , . . . , n , weobtain the equalities δ ij = σ∂ i σ − σ ( x j ) = ∂ ′ i ∗ x ′ j = ( n X k =1 c ik ∂ k ) ∗ n Y l =1 x a lj l = ( n X k,l =1 c ik x − k a kj ) x ′ j where i, j = 1 , . . . , n . Equivalently, C σ diag( x − , . . . , x − n ) A = diag( x ′− , . . . , x ′− n ), and statement1 follows.2. Statement 2 follows from statement 1: σ ( H ) = σ (diag( x , . . . , x n ) ∂ ) = σ (diag( x , . . . , x n )) σ ( ∂ ) = diag( σ ( x ) , . . . , σ ( x n )) C σ ∂ = diag( σ ( x ) , . . . , σ ( x n ))(diag( σ ( x ) − , . . . , σ ( x n ) − ) A − diag( x , . . . , x n ) = A − H. (cid:3) Let group G acts on a set S and T ⊆ S . Then Fix G ( T ) := { g ∈ G | gt = t for all t ∈ T } is the fixator of the the set T . Fix G ( T ) is a subgroup of G . Lemma 2.10
Let σ ∈ W n . Then σ ( H ) = A σ − H for some A σ − ∈ GL n ( Z ) (see (8) and Lemma2.7) and σ A σ − σ ∈ Fix W n ( H , . . . , H n ) where σ A σ − ∈ GL n ( Z ) ⊆ L n , see (11).Proof . The statement follows from Lemma 2.9.(2). (cid:3) Sh n := { s λ ∈ G n | s λ ( x ) = x + λ , . . . , s λ ( x n ) = x n + λ n } is the shift group of automorphisms of the polynomial algebra P n where λ = ( λ , . . . , λ n ) ∈ K n ;Sh n ⊂ Aut K − alg ( P n ) ⊆ Aut
Lie ( D n ). Proposition 2.11
Fix W n ( ∂ , . . . , ∂ n ) = { e } .Proof . Let σ ∈ F := Fix W n ( ∂ , . . . , ∂ n ). We have to show that σ = e . Let N := Nil W n ( ∂ , . . . , ∂ n ) := { w ∈ W n | δ si ( w ) = 0 for some s = s ( w ) and all i = 1 , . . . , n } . Clearly, N = D n . The automor-phisms σ and σ − preserve the space N = D n , that is σ ± ( D n ) ⊆ D n . Hence σ ( D n ) = D n and σ | D n ∈ Fix G n ( ∂ , . . . , ∂ n ) = Sh n , [4]. The only element s λ of Sh n that can be extended to anautomorphism of W n is e (since s λ ( x − i ∂ i ) = ( x i + λ ) − ∂ i ). Therefore, σ = e . In more detail,suppose that s λ can be extended to an automorphism of the Witt algebra W n and λ i = 0, weseek a contradiction. Then applying s λ to the relation [ x − i ∂ i , x i ∂ i ] = 3 ∂ i we obtain the rela-tion [ s λ ( x − i ∂ i ) , ( x i + λ i ) ∂ i ] = 3 ∂ i . On the other hand, [( x i + λ i ) − ∂ i , ( x i + λ i ) ∂ i ] = 3 ∂ i inthe Lie algebra K ( x i ) ∂ i . Hence, s λ ( x − i ∂ i ) − ( x i + λ i ) − ∂ i ∈ C := C K ( x i ) ∂ i (( x i + λ i ) ∂ i ). Since C = K · ( x + λ i ) ∂ i , we see that s λ ( x − i ∂ i ) W n , a contradiction. In more detail, let α = ( x i + λ i ) .Then β∂ i ∈ C where β ∈ K ( x i ) iff (where α ′ := dαdx i , etc) 0 = [ α∂ i , β∂ i ] = ( αβ ′ − α ′ β ) ∂ i = α ( βα ) ′ ∂ i iff ( βα ) ′ = 0 iff βα ∈ ker K ( x i ) ( ∂ i ) = K . Hence, β ∈ Kα , as required. (cid:3) emma 2.12 Fix W n ( H , . . . , H n ) = T n .Proof . The inclusion T n ⊆ F := Fix W n ( H , . . . , H n ) is obvious. Let σ ∈ F . We have to showthat σ ∈ T n . In view of Proposition 2.11, it suffices to show that σ ( ∂ ) = λ ∂ , . . . , σ ( ∂ n ) = λ n ∂ n for some λ = ( λ , . . . , λ n ) ∈ K ∗ n since then t λ σ ∈ Fix W n ( ∂ , . . . , ∂ n ) = { e } (Proposition 2.11),and so σ = t − λ ∈ T n . Since σ ∈ F , the automorphism respects the weight components of the Liealgebra W n , that is σ ( x α H n ) = x α H n for all α ∈ Z n . In particular, for i = 1 , . . . , n , ∂ ′ i = σ ( ∂ i ) = σ ( x − i H i ) = x − i n X j =1 λ ij H j = − x − i n X j =1 λ ij x j ∂ j , (12) ∂ ′ = D − Λ D∂ where D = diag( x , . . . , x n ) and D − Λ D ∈ GL( L n ), and so Λ = ( λ ij ) ∈ GL n ( K ).In view of (12), e have to show that Λ is a diagonal matrix. The elements ∂ , . . . , ∂ n commute, sodo ∂ ′ , . . . , ∂ ′ n : for all i, j = 1 , . . . , n ,0 = [ ∂ ′ i , ∂ ′ j ] = [ x − i n X k =1 λ ik H k , x − j n X l =1 λ jl H l ] . Therefore, foro all i, j, l = 1 , . . . , n , λ ij λ jl = λ ji λ il . For each i = 1 , . . . , n , let c i := P nj =1 λ ji . Theabove equalities yield the equalities n X j =1 λ ij λ jl = c i λ il for i, l = 1 , . . . , n. Equivalently, Λ = diag( c , . . . , c n )Λ. Therefore, Λ = diag( c , . . . , c n ) since Λ ∈ GL n ( K ), asrequired. (cid:3) Proof of Theorem 1.1 . Let σ ∈ W n . We have to show that σ ∈ L n . By Lemma 2.10 andLemma 2.12, τ σ ∈ Fix W n ( H , . . . , H n ) = T n for some τ ∈ L n , hence σ ∈ L n . (cid:3) The aim of this section is to find the group of automorphisms of the Virasoro algebra (Theorem1.3). The key idea is to use Theorem 1.3.
Proof of Theorem 1.3 . Let a K -linear map s : W → G be a section to the surjection π : G → W , a a + Z , i.e. πs = id W . The map σ is an injection and we identify the vector space W with its image in G via s . Then, G = W ⊕ Z , a direct sum of vector spaces.(i) b σ is unique : Suppose we have another extension, say b σ . Then τ := b σ − b σ ∈ G := Aut Lie ( G )and φ ( w ) := τ ( w ) − w ∈ Z for all w ∈ W, where φ ∈ Hom K ( W, Z ). By condition 2, the inclusion Z ⊆ [ G , G ] = [ W + Z, W + Z ] = [ W, W ]implies that τ ( z ) = z for all z ∈ Z . For all w , w ∈ W ,[ w , w ] = [ w , w ] W + z ( w , w ) (13)where [ · , · ] and [ · , · ] W are the Lie brackets in G and W respectively and z ( w , w ) ∈ Z . Moreover,[ w , w ] W means s ([ w , w ] W ). Applying the automorphism τ to the above equality we have[ w , w ] = [ τ ( w ) , τ ( w )] = τ ([ w , w ]) = τ ([ w , w ] W + z ( w , w ))= [ w , w ] W + φ ([ w , w ] W ) + z ( w , w ) = [ w , w ] + φ ([ w , w ] W ) . Hence, φ ([ w , w ] W ) = 0 for all w , w ∈ W . By condition 3, φ = 0, that is τ ( w ) = w for all w ∈ W . Together with the condition τ ( z ) = z for all z ∈ Z , this gives τ = e . So, b σ = b σ .7ii) The map σ b σ is a monomorphism : Let b σ and b τ be the extensions of σ and τ respectively.By the uniqueness, b σ b τ is the extension of στ , that is c στ = b σ b τ , and so the map σ b σ is ahomomorphism. Again, by the uniqueness, σ b σ is a monomorphism.(iii) The map σ b σ is an isomorphism : By condition 1, the map σ b σ is a surjection, hencean isomorphism, by (ii). (cid:3) Proof of Theorem 1.2 . The conditions of Theorem 1.3 are satisfied for the Virasoro algebra: Z = Z (Vir) = Kc , Vir /Z ≃ W , [ W , W ] = W (since W is a simple Lie algebra), Z ⊆ [Vir , Vir]and each automorphism σ ∈ W = Aut Lie ( W ) = Aut K − alg ( L ) = GL ( Z ) ⋉ T is extended to anautomorphism b σ ∈ Aut
Lie (Vir) by the rule b σ ( c ) = c . The last condition is obvious for σ ∈ T butfor e = σ ∈ GL ( Z ) = {± } , i.e. σ : L → L , x x − , i.e. σ : W → W , x i H
7→ − x − H for all i ∈ Z , it follows from the relations (1). (cid:3) Corollary 3.1
1. Each automorphism σ of the Witt algebra W is uniquely extended to anautomorphism b σ of the Virasoro algebra Vir . Moreover, b σ ( c ) = c .2. All the automorphisms of the Virasoro algebra Vir act trivially on its centre.
When we drop condition 3 of Theorem 1.3, we obtain a more general result.
Corollary 3.2
Let G be a Lie algebra, Z be a subspace of the centre of G and W = G /Z . Fixa K -linear map s : W → G which is a section to the surjection π : G → W , a a + Z , andidentify W with im( s ) , and so G = W ⊕ Z (a direct sum of vector spaces). Let K := { τ = τ φ ∈ End K ( G ) | τ ( w ) = w + φ ( w ) and τ ( z ) = z for all w ∈ W and z ∈ Z , φ ∈ Hom K ( W, Z ) is such that φ ([ W, W ]) = 0 } . Suppose that1. every automorphism σ of the Lie algebra W can be extended to an automorphism b σ of theLie algebra G , and2. Z ⊆ [ G, G ] .Then the short exact sequence of groups → K → Aut
Lie ( G ) ψ → Aut
Lie ( W ) → is exact where ψ ( σ ) : a + Z σ ( a ) + Z for all a ∈ G .Proof . By condition 1, ψ is a group epimorphism. It remains to show that ker( ψ ) = K . Let τ ∈ ker( ψ ). Each element g ∈ G = W ⊕ Z is a unique sum g = w + z where w ∈ W and z ∈ Z .Then τ ( w ) = w + φ ( w ) for some φ ∈ Hom K ( W, Z ). We keep the notation of the proof of Theorem1.3. By condition 2, Z ⊆ [ G , G ] = [ W, W ], hence τ ( z ) = z for all elements z ∈ Z . Applying theautomorphism τ to the equality (13) yields φ ([ w , w ]) = 0 (see the proof of Theorem 1.3). Itfollows that ker( ψ ) = K . (cid:3) Acknowledgements
The work is partly supported by the Royal Society and EPSRC.
References [1] V. V. Bavula, Every monomorphism of the Lie algebra of triangular polynomial derivations is an auto-morphism,
C. R. Acad. Sci. Paris, Ser. I , (2012) no. 11-12, 553–556. (Arxiv:math.AG:1205.0797).[2] V. V. Bavula, Lie algebras of triangular polynomial derivations and an isomorphism criterion fortheir Lie factor algebras, Izvestiya: Mathematics , (2013), in print. (Arxiv:math.RA:1204.4908).[3] V. V. Bavula, The groups of automorphisms of the Lie algebras of triangular polynomial derivations,Arxiv:math.AG/1204.4910.
4] V. V. Bavula, The group of automorphisms of the Lie algebra of derivations of a polynomial algebra.Arxiv:math.RA:0695900.Department of Pure MathematicsUniversity of SheffieldHicks BuildingSheffield S3 7RHUKemail: v.bavula@sheffield.ac.uk4] V. V. Bavula, The group of automorphisms of the Lie algebra of derivations of a polynomial algebra.Arxiv:math.RA:0695900.Department of Pure MathematicsUniversity of SheffieldHicks BuildingSheffield S3 7RHUKemail: v.bavula@sheffield.ac.uk