aa r X i v : . [ m a t h . R A ] S e p On cohomology of filiform Lie superalgebras
Yong Yang a and Wende Liu b ∗ a School of Mathematics, Jilin University, Changchun 130012, China b School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China
Abstract : Suppose the ground field F is an algebraically closed field of characteristicdifferent from 2, 3. We determine the Betti numbers and make a decomposition ofthe associative superalgebra of the cohomology for the model filiform Lie superalge-bra. We also describe the associative superalgebra structures of the (divided power)cohomology for some low-dimensional filiform Lie superalgebras. Keywords : filiform Lie superalgebra; Betti number; associative superalgebra
Mathematics Subject Classification 2010 : 17B30, 17B56
In 1970, in the study of the reducibility of the varieties of nilpotent Lie algebras, Vergneintroduced the concept of filiform Lie algebras and showed that every filiform Lie algebracan be obtained by an infinitesimal deformation of the model filiform Lie algebra L n (see[1]). Since then, the study of the filiform Lie algebras, especially the model filiform Liealgebra, has become an important subject. Many conclusions on cohomology of the modelfiliform Lie algebra with coefficients in the trivial module have been obtained. For example,the Betti numbers for L n with coefficients in the trivial module over a field of characteristiczero have been calculated in [2–4]. A result, in [5], states that the filiform Lie algebras L n and m ( n ) have the same Betti numbers over a field of characteristic two, which isdifferent from the case of characteristic zero. Moreover, the first three Betti numbers of L n and m ( n ) over Z have been calculated in [6]. As what happens in the Lie case, everyfiliform Lie superalgebra can be obtained by an infinitesimal deformation of the modelfiliform Lie superalgebra L n,m . Many conclusions on cohomology of the model filiform Liesuperalgebra with coefficients in the adjoint module have been obtained. For example,Khakimdjanov and Navarro gave a complete description of the second cohomology of L n,m with coefficients in the adjoint module in [7–11]. The first cohomology of L n,m withcoefficients in the adjoint module has been described in [12] by calculating the derivations.However, in the trivial module case, less of work is done for L n,m .Throughout this paper, the ground field F is an algebraically closed field of charac-teristic different from 2, 3 and all vector spaces, algebras are over F . In the character-istic zero case, for any non-negative integer k , we make a decomposition of H k ( L n,m )by the Hochschild-Serre spectral sequences, moreover, we can describe completely theBetti number of H • ( L n,m ) and the superalgebra structures of the cohomology for some ∗ Correspondence: [email protected] n cohomology of filiform Lie superalgebras A Lie superalgebra is a Z -graded algebra whose multiplication satisfies the skew-supersymmetryand the super Jacobi identity (see [13]). For a Lie superalgebra L , we inductively definetwo sequences: L = L ¯0 , L i +1¯0 = [ L ¯0 , L i ¯0 ]and L = L ¯1 , L i +1¯1 = [ L ¯0 , L i ¯1 ] . If there exists ( m, n ) ∈ N such that L m ¯0 = 0, L m − = 0 and L n ¯1 = 0, L n − = 0, thepair ( m, n ) is called the super-nilindex of L . In particular, a Lie superalgebra L is called filiform if its super-nilindex is (dim L ¯0 − , dim L ¯1 ) (see [7]).Denote by F n,m the set of filiform Lie superalgebras of super-nilindex ( n, m ). Let F be a filiform Lie superalgebra over C . If F ∈ F n,m , there exists a basis { X , X , . . . , X n | Y , . . . , Y m } of F such that:[ X , X i ] = X i +1 , ≤ i ≤ n − , [ X , X n ] = 0;[ X , X ] ∈ C X + · · · + C X n ;[ X , Y i ] = Y i +1 , ≤ i ≤ m − , [ X , Y m ] = 0 . Recall the following classifications up to isomorphism (see [14]):The classification of F , :(1) F , : [ X , Y ] = Y ;(2) F , : [ X , Y ] = [ X , Y ] = Y ;(3) F , : [ X , Y ] = Y , [ Y , Y ] = X . The classification of F , :(1) F , : [ X , X ] = X , [ X , Y ] = Y ;(2) F , : [ X , X ] = 2[ Y , Y ] = X , [ X , Y ] = Y , [ Y , Y ] = X ;(3) F , : [ X , X ] = [ Y , Y ] = X , [ X , Y ] = Y ;(4) F , : [ X , X ] = X , [ X , Y ] = [ X , Y ] = Y ;(5) F , : [ X , X ] = [ Y , Y ] = X , [ X , Y ] = [ X , Y ] = Y . Let L n,m be the filiform Lie superalgebra with a homogeneous basis { X , X , . . . , X n | Y , . . . , Y m } and Lie super-brackets are given by[ X , X i ] = X i +1 , ≤ i ≤ n −
1; [ X , Y j ] = Y j +1 , ≤ j ≤ m − . We call L n,m the model filiform Lie superalgebra and { X , X , . . . , X n | Y , . . . , Y m } the standard basis of L n,m .Obviously, we have: F , = L , , F , = L , . n cohomology of filiform Lie superalgebras In this section, we introduce the definitions of cohomology and divided power cohomologyof g with coefficients in the trivial module. For more details, the reader is referred to[13, 15, 16].Let g be a finite-dimensional Lie superalgebra, denote by g ∗ the dual superspace of g .Fix an ordered basis of g { x , . . . , x m | x m +1 , . . . , x m + n } , (2.0.1)where | x | = · · · = | x m | = 0, | x m +1 | = · · · = | x m + n | = 1, and write { x ∗ , . . . , x ∗ m | x ∗ m +1 , . . . , x ∗ m + n } , for the dual basis.For k ∈ Z , we let V k g ∗ be the k -th super-exterior product of g ∗ . Let V • g ∗ = L k ∈ N V k g ∗ .Then V • g ∗ can be viewed as a g -module in a natural manner. Note that V • g ∗ also has a Z -grading structure given by setting k x k = · · · = k x m + n k = 1. Hereafter k x k denotes the Z -degree of a Z -homogeneous element x in a Z -graded superspace. Let d : V • g ∗ −→ V • g ∗ bethe linear operator induced by the dual of the Lie superalgebra bracket map g ∗ −→ V g ∗ .Then d (1) = 0 ,d ( x ∗ i ) = X ≤ k
2, we introduce the definition of divided powercohomology of g . For a multi-index r = ( r , . . . , r m + n ), where r , . . . , r m are 0 or 1, and r m +1 , . . . , r m + n are non-negative integers, we set u ( r i ) i = x r i i r i ! and u ( r ) = m + n Y i =1 u ( r i ) i . Clearly, their multiplication relations are u ( r ) u ( s ) = (cid:18) m Y i =1 min(1 , − r i − s i ) (cid:19) ( − m P j =1 m + n P k = m +1 r k s j + P ≤ j Let D : • ^ I ∗ −→ • ^ I ∗ ,x X · x. Let D k = D | V k I ∗ . Then H k ( L n,m ) = Ker D k L (cid:16) F X ∗ V (cid:16)V k − I ∗ / Im D k − (cid:17)(cid:17) . Proof. From Eq. (2.0.3), we haveE p,k − p ∞ ∼ = H (cid:16) F X , V k I ∗ (cid:17) , p = 0;H (cid:16) F X , V k − I ∗ (cid:17) , p = 1;0 , p = 0 , . Then we haveH k ( L n,m ) = M p + q = k E p,q ∞ = H F X , k ^ I ∗ ! M H F X , k − ^ I ∗ ! . By the definitions of the low cohomology (see [13]), we haveH F X , k ^ I ∗ ! = Ker D k , Der F X , k − ^ I ∗ ! = F X ∗ ^ k − ^ I ∗ , Inder F X , k − ^ I ∗ ! = F X ∗ ^ Im D k − . Moreover, we obtain thatH k ( L n,m ) = Ker D k M F X ∗ ^ k − ^ I ∗ / Im D k − !! . Theorem 1. As a Z -graded superalgebra, we haveH • ( L n,m ) = Ker D ⋉ F X ∗ ^ • ^ I ∗ / Im D !! . n cohomology of filiform Lie superalgebras Proof. For any x, y ∈ Ker D , z ∈ V • I ∗ / Im D , we have D ( x ∧ y ) = − X · ( x ∧ y ) = − [( X · x ) ∧ y + x ∧ ( X · y )] , = D ( x ) ∧ y + x ∧ D ( y ) , = 0 , and ( X ∗ ∧ z ) ∧ x = X ∗ ∧ ( z ∧ x ) , = X ∗ ∧ ( z ∧ x ) ∈ F X ∗ ^ • ^ I ∗ / Im D ! . Thus, Ker D is a subalgebra of H • ( L n,m ) and F X ∗ V ( V • I ∗ / Im D ) is an ideal of H • ( L n,m )with trivial multiplication. From Lemma 3.1, the proof is complete.In order to calculate the dimension of H • ( L n,m ), it suffices to calculate the dimensionof Ker D .Let f i = 1( n − i )! X ∗ i , ≤ i ≤ n, g j = 1( m − j )! Y ∗ j , ≤ j ≤ m. Let x, y, h ∈ End F ( I ∗ ), such that x = D | I ∗ ,y ( f i ) = if i +1 , ≤ i ≤ n − , y ( f n ) = 0 ,h ( f i ) = ( n + 1 − i ) f i , ≤ i ≤ n. Let x ′ , y ′ , h ′ ∈ End F ( I ∗ ), such that x ′ = D | I ∗ ,y ′ ( g j ) = jg j +1 , ≤ j ≤ m − , y ′ ( g m ) = 0 ,h ′ ( g j ) = ( m + 1 − j ) g j , ≤ j ≤ m. Obviously, span F { x, y, h } , span F { x ′ , y ′ , h ′ } are subalgebras of gl ( I ∗ ) and gl ( I ∗ ), respec-tively. Moreover, the following Lie algebra isomorphisms hold:span F { x, y, h } ∼ = span F { x ′ , y ′ , h ′ } ∼ = sl (2) . By Weyl’s Theorem and representation theory of sl (2) (see [17]), I ∗ and I ∗ are simplemodules of sl (2) and, for k ≥ V k I ∗ is a completely reducible module of sl (2). Moreover,we can obtain the following theorem. Lemma 3.2. Suppose k ≥ 0. Thendim Ker D k = k X k =0 f n,m ( k , k − k ) , where f n,m ( k , k − k ) = card (cid:26) ( i , . . . , i k , j , . . . , j k − k ) ∈ Z k | ≤ i < . . . < i k ≤ n, ≤ j < . . . < j k − k ≤ m + k − k − , k P a =1 i a + k − k P b =1 j b = (cid:22) k ( n + 1) + ( k − k )( m + 1)2 (cid:23) +( k − k )( k − k − (cid:27) . n cohomology of filiform Lie superalgebras Proof. For k ≥ 0, set k ^ I ∗ = r M i =1 V i , where V , . . . , V r are simple sl (2)-modules. Moreover,Ker D k = r M i =1 Ker D k \ V i . So we obtain that for any v ∈ V i , D k ( v ) = 0 if and only if v is a maximal vector of V i .Moreover, the following conclusions hold:dim Ker D k = r = dim k ^ I ∗ ! + dim k ^ I ∗ ! , (3.1.1)where (cid:16)V k I ∗ (cid:17) , (cid:16)V k I ∗ (cid:17) are weight spaces of weight 0, 1, respectively.Let sl (2) = span F { X + , H, X − } , where F H is a Cartan subalgebra of sl (2), and f i ∧ . . . ∧ f i k ∧ g j ∧ . . . ∧ g j k − k is a standard basis of V k I ∗ , where 0 ≤ k ≤ k , i < . . . < i k ≤ n ,1 ≤ j ≤ . . . ≤ j k − k ≤ m . Note that H ( f i ∧ . . . ∧ f i k ∧ g j ∧ . . . ∧ g j k − k ) = k ( n + 1) + ( k − k )( m + 1) − k X a =1 i a + k − k X b =1 j b !! f i ∧ . . . ∧ f i k ∧ g j ∧ . . . ∧ g j k − k . From Eq. (3 . . (cid:16)V k I ∗ (cid:17) and (cid:16)V k I ∗ (cid:17) .We consider the following cases: Case 1 : k ( n + 1) + ( k − k )( m + 1) is even. Then k ( n + 1) + ( k − k )( m + 1) − k X a =1 i a + k − k X b =1 j b !! = 0 , if and only if k P a =1 i a + k − k P b =1 j b = k ( n + 1) + ( k − k )( m + 1)2 . Case 2 : k ( n + 1) + ( k − k )( m + 1) is odd. Then k ( n + 1) + ( k − k )( m + 1) − k X a =1 i a + k − k X b =1 j b !! = 1 , if and only if k P a =1 i a + k − k P b =1 j b = k ( n + 1) + ( k − k )( m + 1) − 12 .Thus, the conclusion holds. Theorem 2. Suppose k ≥ 0. Thendim H k ( L n,m ) = k X k =0 f n,m ( k , k − k ) + k − X k =0 f n,m ( k , k − k − . n cohomology of filiform Lie superalgebras Proof. It follows from Lemmas 3.1 and 3.2.In order to characterize the superalgebra structure of H • ( L n,m ), we make a Z -gradationof V k I ∗ for any k ≥ ≤ s ≤ k , let l such that α k,s ≤ l ≤ β k,s , where α k,s = s ( s +1)2 + k − s , β k,s = s (2 n − s +1)2 + ( k − s ) m .Let (cid:16)V k I ∗ (cid:17) ls be the space spanned by X ∗ i ∧ . . . ∧ X ∗ i s ∧ Y ∗ α ∧ . . . ∧ Y ∗ αm m , where 1 ≤ i < . . . < i s ≤ n, and α , . . . , α m ≥ , satisfying that m X j =1 α j = k − s, s X a =1 i a + m X b =1 α b b = l. Let D k,ls : k ^ I ∗ ! ls −→ k ^ I ∗ ! l − s ,x D k ( x ) . Obviously, k ^ I ∗ = k M s =0 β k,s M l = α k,s k ^ I ∗ ! ls . Since the linear mapping D k is compatible with this graduation, we have:Ker D k = k M s =0 β k,s M l = α k,s Ker D k,ls , (3.1.2)Im D k = k M s =0 β k,s M l = α k,s Im D k,ls , (3.1.3)where α k,s = s ( s +1)2 + k − s , β k,s = s (2 n − s +1)2 + ( k − s ) m .We describe the superalgebra structures of the cohomology for some low-dimensionalfiliform Lie superalgebras by the decomposition in Theorem 1. Example 1. The following Z -graded superalgebra isomorphism holds:H • ( L , ) ∼ = U , ⋉ V , , where U , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α ,i , α i | i ≥ } , satisfying that | α ,i | = | α i | = i (mod 2), k α ,i k = i + 1, k α i k = i , and the multiplication isgiven by α i α j = α i + j , α ,i α j = α ,i + j , i, j ≥ . n cohomology of filiform Lie superalgebras V , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α ,i , α , ,i | i ≥ } , satisfying that | α ,i | = | α , ,i | = i (mod 2), k α ,i k = i + 1, k α , ,i k = i + 2, and the trivialmultiplication. The multiplication between U , and V , is graded-supercommutative, andthe multiplication is given by α ,i α , = ( − i α , ,i , α ,i α = α ,i , α , ,i α = α , ,i , i ≥ . Proof. Note that k ^ I ∗ ! ls = F Y ∗ k − l ∧ Y ∗ l − k , s = 0; F X ∗ ∧ Y ∗ k − l − ∧ Y ∗ l − k , s = 1;0 , else.Moreover, we have Ker D k,ls = F Y ∗ k , s = 0 , l = k ; F X ∗ ∧ Y ∗ k − , s = 1 , l = k ;0 , else.Im D k,ls = F Y ∗ k − l +1 ∧ Y ∗ l − k − , s = 0 , k + 1 ≤ l ≤ k ; F X ∗ ∧ Y ∗ k − l ∧ Y ∗ l − k − , s = 1 , k + 1 ≤ l ≤ k − , else.From Theorem 1 and Eqs. (3.1.2), (3.1.3), H • ( L , ) has a basis: Y ∗ i , X ∗ ∧ Y ∗ i , X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ Y ∗ i , i ≥ . The conclusion can be obtained by a direct calculation. Example 2. The following Z -graded superalgebra isomorphism holds:H • ( L , ) ∼ = U , ⋉ V , , where U , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α i , α ,i , α , ,i | i ≥ } , satisfying that | α i | = | α ,i | = | α , ,i | = i (mod 2), k α i k = i , k α ,i k = i + 1, k α , ,i k = i + 2,and the multiplication is given by α i α j = α i + j , α ,i α j = α ,i + j , α , ,i α j = α , ,i + j , i, j ≥ . V , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α ,i , α , ,i , α , , ,i | i ≥ } , satisfying that | α ,i | = | α , ,i | = | α , , ,i | = i (mod 2), k α ,i k = i + 1, k α , ,i k = i + 2, k α , , ,i k = i + 3, and the trivial multiplication. The multiplication between U , and V , is graded-supercommutative, and the multiplication is given by α ,i α j = α ,i + j , α , ,i α j = α , ,i + j , α , , ,i α j = α , , ,i + j ,α , ,i α ,j = ( − i +1 α , , ,i + j , α ,i α , ,j = α , , ,i + j , i, j ≥ . n cohomology of filiform Lie superalgebras Proof. Note that k ^ I ∗ ! ls = F Y ∗ l , s = 0; F X ∗ ∧ Y ∗ l − , s = 1 , l = k ; F X ∗ ∧ Y ∗ l − , s = 1 , l = k + 1; F X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 2 , l = k + 1;0 , else.Moreover, we haveKer D k,ls = F Y ∗ l , s = 0; F X ∗ ∧ Y ∗ l − , s = 1 , l = k ; F X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 2 , l = k + 1;0 , else.Im D k,ls = (cid:26) F X ∗ ∧ Y ∗ l − , s = 1 , l = k + 1;0 , else.From Theorem 1 and Eqs. (3.1.2), (3.1.3), H • ( L , ) has a basis: Y ∗ i , X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ Y ∗ i , X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ X ∗ ∧ Y ∗ i , i ≥ . The conclusion can be obtained by a direct calculation. Example 3. The following Z -graded superalgebra isomorphism holds:H • ( L , ) ∼ = U , ⋉ V , , where U , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α i,j , α ,i,j , | i, j ≥ } , satisfying that | α i,j | = | α ,i,j | = 2 j + i (mod 2), k α i,j k = 2 j + i , k α ,i,j k = 2 j + i + 1, andthe multiplication is given by α i,j α i ′ ,j ′ = α i + i ′ ,j + j ′ , α ,i,j α i ′ ,j ′ = α ,i + i ′ ,j + j ′ , i, i ′ , j, j ′ ≥ . V , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α ,i,j , α , ,i,j | i, j ≥ } , satisfying that | α ,i,j | = | α , ,i,j | = 2 i + j (mod 2), k α ,i,j k = 2 i + j + 1, k α , ,i,j k =2 i + j + 2, and the trivial multiplication. The multiplication between U , and V , isgraded-supercommutative, and the multiplication is given by α ,i,j α i ′ ,j ′ = ρ i,j,i ′ ,j ′ α ,i + i ′ + j ′ ,j − i ′ ,α ,i,j α ,i ′ ,j ′ = ( − i + j α , ,i,j α i ′ ,j ′ = ( − i + j ρ i,j,i ′ ,j ′ α ,i + i ′ + j ′ ,j − i ′ , i, j ≥ , where ρ i,j,i ′ ,j ′ = i ′ − Q s =0 s − j i + j ′ + s )+1 + i ′ + j ′ P t = i ′ +1 t − i ′ Q a =1 2( a − − j ′ ) a t − Q b =0 b − j − t + i ′ i + i ′ + j ′ − t + b )+1 ! . n cohomology of filiform Lie superalgebras Proof. Note that k ^ I ∗ ! ls = (cid:4) k − l (cid:5)L i =2 k − l F Y i ∧ Y k − l − i ∧ Y i + l − k , s = 0 , k ≤ l ≤ k ; (cid:4) k − l (cid:5)L i =0 F Y i ∧ Y k − l − i ∧ Y i + l − k , s = 0 , k + 1 ≤ l ≤ k (cid:4) k − l (cid:5) − L i =2 k − l − F X ∧ Y i ∧ Y k − l − i − ∧ Y i + l − k +13 , s = 1 , k ≤ l ≤ k − (cid:4) k − l (cid:5) − L i =0 F X ∧ Y i ∧ Y k − l − i − ∧ Y i + l − k +13 , s = 1 , k ≤ l ≤ k − , else.From a direct computation, we have the following conclusion:If 2 k + 1 ≤ l ≤ k , or k ≤ l ≤ k and 3 k − l is odd, we have Ker D k,l = 0.If k ≤ l ≤ k , and 3 k − l is even, we have dim Ker D k,l = 1 and Ker D k,l has a basis: Y ∗ k − l ∧ Y ∗ l − k + k − l X i =2 k − l +1 λ k,li Y ∗ i ∧ Y ∗ k − l − i ∧ Y ∗ i + l − k , where λ k,li = i − k + l Q j =1 2( j − k − lj . Moreover, we have k ^ I ∗ ! l − / Im D k,l = (cid:26) F Y ∗ k − l +1 ∧ Y ∗ l − k − , k + 1 ≤ l ≤ k and 3 k − l is odd;0 , else.From Theorem 1 and Eqs. (3.1.2), (3.1.3), H • ( L , ) has a basis: Y ∗ i ∧ Y ∗ j + i + j X s = i +1 λ j + i, j + is Y ∗ s ∧ Y ∗ i + j − s ) ∧ Y ∗ s − i ,X ∗ ∧ Y ∗ i ∧ Y ∗ j + i + j X s = i +1 λ j + i, j + is X ∗ ∧ Y ∗ s ∧ Y ∗ i + j − s ) ∧ Y ∗ s − i ,X ∗ ∧ Y ∗ i ∧ Y ∗ j , X ∗ ∧ X ∗ ∧ Y ∗ i ∧ Y ∗ j , i, j ≥ . The conclusion can be obtained by a direct calculation. Example 4. The following Z -graded superalgebra isomorphism holds:H • ( L , ) ∼ = U , ⋉ V , , where U , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α i , α ,i , α , ,i , α , , ,i | i ≥ } , n cohomology of filiform Lie superalgebras | α i | = | α ,i | = | α , ,i | = | α , , ,i | = i (mod 2), k α i k = i , k α ,i k = i + 1, k α , ,i k = i + 2, k α , , ,i k = i + 3, and the multiplication is given by α i α j = α i + j , α ,i α j = α ,i + j , α , ,i α j = α , ,i + j , α , , ,i α j = α , , ,i + j , i, j ≥ . V , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α ,i , α , ,i , α , , ,i , α , , , ,i | i ≥ } , satisfying that | α ,i | = | α , ,i | = | α , , ,i | = | α , , , ,i | = i (mod 2), k α ,i k = i + 1, k α , ,i k = i +2, k α , , ,i k = i +3, k α , , , ,i k = i +4, and the trivial multiplication. The multiplicationbetween U , and V , is graded-supercommutative, and the multiplication is given by α ,i α j = α ,i + j , α , ,i α j = α , ,i + j , α , , ,i α j = α , , ,i + j , α , , , ,i α j = α , , , ,i + j ,α , , ,i α ,j = α ,i α , , ,j = ( − i α , ,i α , ,j = ( − i α , , , ,i + j , i, j ≥ . Proof. Note that k ^ I ∗ ! ls = F Y ∗ l , s = 0; F X ∗ ∧ Y ∗ l − , s = 1 , l = k ; F X ∗ ∧ Y ∗ l − , s = 1 , l = k + 1; F X ∗ ∧ Y ∗ l − , s = 1 , l = k + 2; F X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 2 , l = k + 1; F X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 2 , l = k + 2; F X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 2 , l = k + 3; F X ∗ ∧ X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 3 , l = k + 3;0 , else.Moreover, we haveKer D k,ls = F Y ∗ l , s = 0; F X ∗ ∧ Y ∗ l − , s = 1 , l = k ; F X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 2 , l = k + 1; F X ∗ ∧ X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 3 , l = k + 3;0 , else. . Im D k,ls = F X ∗ ∧ Y ∗ l − , s = 1 , l = k + 1; F X ∗ ∧ Y ∗ l − , s = 1 , l = k + 2; F X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 2 , l = k + 2; F X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 2 , l = k + 3;0 , else. . From Theorem 1 and Eqs. (3.1.2), (3.1.3), H • ( L , ) has a basis: Y ∗ i , X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ X ∗ ∧ Y ∗ i ,X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ X ∗ ∧ X ∗ ∧ Y ∗ i , i ≥ . The conclusion can be obtained by a direct calculation. n cohomology of filiform Lie superalgebras Example 5. The following Z -graded superalgebra isomorphism holds:H • ( L , ) ∼ = U , ⋉ V , , where U , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α i , α ,i , α , ,i , β i | i ≥ } , satisfying that | α i | = | α ,i | = | α , ,i | = i (mod 2), | β i | = i + 1 (mod 2), k α i k = i , k α ,i k = i + 1, k α , ,i k = k β i k = i + 2, and the multiplication is given by α i α j = α i + j , α ,i α j = α ,i + j , α , ,i α j = α , ,i + j ,β i α j = β i + j , α ,i β j = ( − i +1 α , ,i + j +1 , i, j ≥ . V , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α ,i , α , ,i , α , ,i , α , , ,i | i ≥ } , satisfying that | α ,i | = | α , ,i | = | α , , ,i | = i (mod 2), | α , ,i | = i + 1 (mod 2), k α ,i k = i + 1, k α , ,i k = i + 2, k α , ,i k = k α , , ,i k = i + 3, and the trivial multiplication. Themultiplication between U , and V , is graded-supercommutative, and the multiplicationis given by α ,i α = α ,i , α , ,i α = α , ,i , α , ,i α = α , ,i , α , , ,i α = α , , ,i ,α , ,i α , = ( − i +1 α ,i α , , = ( − i +1 α , , ,i ,α , ,i β = ( − i +1 α , , ,i +1 , α ,i +1 α , = ( − i +1 α , ,i ,α ,i β = ( − i i + 2 i + 1 α , ,i , α , ,i α = − i + 1 α , ,i , i ≥ . Proof. Note that k ^ I ∗ ! ls = F Y ∗ k − l ∧ Y ∗ l − k , s = 0; F X ∗ ∧ Y ∗ k − l − ∧ Y ∗ l − k L F X ∗ ∧ Y ∗ k − l ∧ Y ∗ l − k − , s = 1; F X ∗ ∧ X ∗ ∧ Y ∗ k − l − ∧ Y ∗ l − k − , s = 2;0 , else.Moreover, we haveKer D k,ls = F Y ∗ l , s = 0 , l = k ; F X ∗ ∧ Y ∗ l − , s = 1 , l = k ; F (cid:16) X ∗ ∧ Y ∗ l − ∧ Y ∗ − X ∗ ∧ Y ∗ l − (cid:17) , s = 1 , l = k + 1; F X ∗ ∧ X ∗ ∧ Y ∗ l − , s = 2 , l = k + 1;0 , else. . Im D k,ls = F Y ∗ k − l +1 ∧ Y ∗ l − k − , s = 0 , k + 1 ≤ l ≤ k ; F X ∗ ∧ Y ∗ l − , s = 1 , l = k + 1; F X ∗ ∧ Y ∗ k − l ∧ Y ∗ l − k − L F X ∗ ∧ Y ∗ k − l +1 ∧ Y ∗ l − k − , s = 1 , k + 2 ≤ l ≤ k ; F X ∗ ∧ X ∗ ∧ Y ∗ k − l ∧ Y ∗ l − k − , s = 2 , k + 2 ≤ l ≤ k − , else. . n cohomology of filiform Lie superalgebras • ( L , ) has a basis: Y ∗ i , X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ Y ∗ i , X ∗ ∧ Y ∗ i ∧ Y − X ∗ ∧ Y ∗ i +1 ,X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ Y ∗ i +1 , X ∗ ∧ X ∗ ∧ Y ∗ i , X ∗ ∧ X ∗ ∧ X ∗ ∧ Y ∗ i , i ≥ . The conclusion can be obtained by a direct calculation. In this section, we describe the cohomology of F , , F , by using the Hochschild-Serrespectral sequence. Lemma 3.3. For F t s s, , s = 1 , t = 1 , , t = 1 , , , 5, let I t s s, = [ F t s s, , F t s s, ]. For k ≥ ≤ i ≤ k , the following conclusions hold:(1) E k − i,i = V k − i (cid:16) F t s s, / I t s s, (cid:17) ∗ N V i (cid:16) I t s s, (cid:17) ∗ = ⇒ H k (cid:16) F t s s, (cid:17) . (2) E k − i,i ∞ = E k − i,i . Proof. (1) Since I t s s, ⊆ C (cid:16) F t s s, (cid:17) , the action of F t s s, / I t s s, on (cid:16) I t s s, (cid:17) ∗ is trivial. Moreover,from Eq. (2.0.3), we haveE k − i,i = H k − i (cid:16) F t s s, / I t s s, , H i (cid:16) I t s s, (cid:17)(cid:17) = k − i ^ (cid:16) F t s s, / I t s s, (cid:17) ∗ O i ^ (cid:16) I t s s, (cid:17) ∗ = ⇒ H k (cid:16) F t s s, (cid:17) . (2) From (1), we have E k − i,ir = E k − i,i , r ≥ . Moreover, E k − i,i ∞ = E k − i,i . Theorem 3. The following Z -graded superalgebra isomorphisms hold:(1) H • ( F , ) ∼ = U , ⋉ V , , where U , is an infinite-dimensional Z -graded superalgebrawith a Z -homogeneous basis { α i , α ,i | i ≥ } , satisfying that | α i | = | α ,i | = i (mod 2); k α i k = i, k α ,i k = i + 1 , and the multiplication is given by α i α j = α i + j , α ,i α j = α ,i + j , i, j ≥ , V , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α i , , α , ,i | i ≥ } , satisfying that | α i , | = | α , ,i | = i (mod 2); k α i , k = i + 1 , k α , ,i k = i + 2 , n cohomology of filiform Lie superalgebras U , and V , is graded-super-commutative, and the multiplication is given by α i , α = α i , , α , ,i α = α , ,i , α , α i , = α , ,i , i ≥ . In particular, dim H k ( F , ) = , k = 0;3 , k = 1;4 , k ≥ . (2) H • ( F , ) ∼ = U , , where U , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α , α , α ,i , β i | i ≥ } , satisfying that | α | = 0 (mod 2) , | α | = 1 (mod 2) , | α ,i | = | β i | = i (mod 2); k α k = 0 , k α k = 1 , k α ,i k = i + 1 , k β i k = i + 2 , and the multiplication is given by α α = α , α α = α , α α ,i = α ,i , α β i = β i , i ≥ . In particular, dim H k ( F , ) = (cid:26) , k = 0;2 , k ≥ . (3) H • ( F , ) ∼ = U , , where U , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α , α , β , α , , α ,i , β ,i , α , ,i , β , ,i | i ≥ } , satisfying that | α | = | α | = 0 (mod 2) , | β | = | α , | = 1 (mod 2) , | α ,i | = | α , ,i | = | β ,i | = | β , ,i | = i (mod 2); k α k = 0 , k α k = k β k = 1 , k α , k = 2 , k α ,i k = i +1 , k α , ,i k = k β ,i k = i +2 , k β , ,i k = i +3 , and the multiplication is given by α α = α , α α = α , α β = β , α α , = α , ,α α ,i = α ,i , α α , ,i = α , ,i , α β ,i = β ,i , α β , ,i = β , ,i ,α β = α , , α α ,i = − α , ,i , α β ,i = β , ,i , β β = 2 α , , ,β β ,i = 2 α , ,i +1 , β ,i β ,j = 2 α , ,i + j +2 , i ≥ . In particular, dim H k ( F , ) = , k = 0;3 , k = 1;4 , k ≥ . n cohomology of filiform Lie superalgebras • ( F , ) ∼ = U , ⋉ V , , where U , is an infinite-dimensional Z -graded superalgebrawith a Z -homogeneous basis { α i , β i | i ≥ } , satisfying that | α i | = i (mod 2) , | β i | = i + 1 (mod 2); k α i k = i, k β i k = i + 2 , and the multiplication is given by β i α = β i , α i α j = α i + j , i, j ≥ . V , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α, β, α ,i , α , ,i , α , ,i , α , , ,i , β ,i , β ,i | i ≥ } , satisfying that | α | = | β | = 0 (mod 2) , | α ,i | = | α , ,i | = | α , , ,i | = i (mod 2) , | α , ,i | = | β ,i | = | β ,i | = i + 1 (mod 2); k α k = 1 , k β k = 2 , k α ,i k = i + 1 , k α , ,i k = k β ,i k = i + 2 , k α , ,i k = k α , , ,i k = k β ,i k = i + 3 , and the multiplication is given by α , β ,i = α , , ,i +1 , α , β = α , , , , α , β , = α , , − α , , ,α ,i β ,j +1 = ( − i α , ,i + j +2 , α ,i +1 β ,j = ( − i +1 α , ,i + j +2 ,α , , α = − α , , , , αβ , = − α , , − α , , ,α , , β , = α , , , , β ,i α = ( − i α , , ,i +1 ,β ,i β , = ( − i +1 α , , ,i +2 , αβ ,i +1 = − α , ,i +2 i, j ≥ . The multiplication between U , and V , is graded-supercommutative, and the multipli-cation is given by α , ,i α = α , ,i , α , , ,i α = α , , ,i , β ,i α = β ,i , αα = α, βα = β,α , β i = α , ,i , α ,i α j = α ,i + j , α , ,i α j = α , ,i + j , β i α = ( − i +1 α , ,i ,β ,i α = − i + 2 α , ,i +1 , αα i +1 = − α ,i +1 , α i β ,j = ( − i β ,i + j ,α , , β i = − α , , ,i +1 , β i β = α , , ,i +1 , β i β , = ( − i i + 3 i + 2 α , ,i +1 ,βα = − α , , − α , , , βα i +2 = − α , ,i +2 , i, j ≥ . In particular, dim H k ( F , ) = , k = 0;3 , k = 1;6 , k = 2;8 , k ≥ . n cohomology of filiform Lie superalgebras • ( F , ) ∼ = U , , where U , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α , α , α , ,i , α i , , β ,i , β ,i | i ≥ } , satisfying that | α | = | α | = 0 (mod 2) , | α , ,i | = | α i , | = i (mod 2) , | β ,i | = | β ,i | = i + 1 (mod 2); k α k = 0 , k α k = 1 , k α i , k = k β ,i k = i + 1 , k α , ,i k = k β ,i k = i + 2 , and the multiplication is given by α α = α , α α = α , α α , ,i = α , ,i , α α i , = α i , , α β ,i = β ,i , α β ,i = β ,i ,α α i , = α , ,i , α β ,i = β ,i , β ,i β ,i = 2 α , ,i + i , i, i , i ≥ . In particular, dim H k ( F , ) = , k = 0;3 , k = 1;4 , k ≥ . Proof. (1) For k ≥ 0, 0 ≤ i ≤ k , consider the mapping d k − i,i : k − i ^ ( F , / F Y ) ∗ O F Y ∗ i −→ k − i +2 ^ ( F , / F Y ) ∗ O F Y ∗ i − ,f ⊗ Y ∗ i ( − k f k if ∧ d ( Y ∗ ) ⊗ Y ∗ i − , where d ( Y ∗ ) = ( X ∗ + X ∗ ) ∧ Y ∗ . By Lemma 3.3, we haveE k − i,i ∞ = V k ( F , / F Y ) ∗ V k − ( F , / F Y ) ∗ V F d ( Y ∗ ) , i = 0; F ( X ∗ + X ∗ ) ∧ Y ∗ k − i − ⊗ Y ∗ i L F X ∗ ∧ X ∗ ∧ Y ∗ k − i − ⊗ Y ∗ i V k − − i ( F , / F Y ) ∗ V F d ( Y ∗ ) N F Y ∗ i , ≤ i ≤ k − F ( X ∗ + X ∗ ) N Y ∗ k − , i = k − , i = k .From H k ( F , ) = k L i =0 E k − i,i ∞ , we can obtain the conclusion.The proofs of (2) , (3) , (4) , (5) are similar to (1). Lemma 3.4. For F , , let I = span F { X , X , Y , Y } .(1) For k ≥ 0, the following conclusion holds:H k ( F , ) ∼ = H (cid:16) F X , H k ( I ) (cid:17) M H (cid:16) F X , H k − ( I ) (cid:17) . (2) For k ≥ 2, H k ( I ) has a basis: Y ∗ k , X ∗ ∧ Y ∗ k − − X ∗ ∧ Y ∗ ∧ Y ∗ k − . n cohomology of filiform Lie superalgebras k ≥ 2, H (cid:0) F X , H k ( I ) (cid:1) has a basis: Y ∗ k , X ∗ ∧ Y ∗ k − − X ∗ ∧ Y ∗ ∧ Y ∗ k − . (4) For k ≥ 3, H (cid:0) F X , H k − ( I ) (cid:1) has a basis: X ∗ ∧ Y ∗ k − , X ∗ ∧ X ∗ ∧ Y ∗ k − − X ∗ ∧ X ∗ ∧ Y ∗ ∧ Y ∗ k − . Proof. (1) Note I is an ideal of F , . From Eq. (2.0.3), we haveE i,k − i ∞ ∼ = H (cid:0) F X , H k ( I ) (cid:1) , i = 0;H (cid:0) F X , H k − ( I ) (cid:1) , i = 1;0 , else.Moreover, we haveH k ( F , ) = k M i =0 E i,k − i ∞ ∼ = H (cid:16) F X , H k ( I ) (cid:17) M H (cid:16) F X , H k − ( I ) (cid:17) . (2) We use the Hochschild-Serre spectral sequence relative to the ideal I = [ I, I ]. Notethat I ⊆ C ( I ), we haveE k − i,i = (cid:26) V k − i ( I/I ) ∗ N V i I ∗ , ≤ i ≤ , else.Moreover, we haveE k − i,i ∞ = E k − i,i = F Y ∗ k , i = 0; F (cid:16) X ∗ ∧ Y ∗ k − − X ∗ ∧ Y ∗ ∧ Y ∗ k − (cid:17) , i = 1;0 , else.Form H k ( I ) = k L i =0 E k − i,i ∞ , we can obtain the conclusion.(3) By the definitions of the low cohomology (see [10]), we haveH (cid:16) F X , H k ( I ) (cid:17) = H k ( I ) . (4) From (3), we have H (cid:16) F X , H k − ( I ) (cid:17) = F X ∗ ^ H k − ( I ) . Theorem 4. The following Z -graded superalgebra isomorphism holds:H • ( F , ) ∼ = U , , where U , is an infinite-dimensional Z -graded superalgebra with a Z -homogeneous basis { α , α , α ,i , β , β ,i , β ,i , β ,i | i ≥ } , n cohomology of filiform Lie superalgebras | α | = 0 (mod 2) , | α | = | β | = 1 (mod 2) , | α ,i | = | β ,i | = | β ,i | = i (mod 2) , | β ,i | = i + 1 (mod 2); k α k = 0 , k α k = 1 , k β k = 2 , k α ,i k = i + 1 , k β ,i k = i + 2 , k β ,i k = k β ,i k = i + 3 , and the multiplication is given by α α = α , α α = α , α α ,i = α ,i , α β = β , α β ,i = β ,i , α β ,i = β ,i ,α β ,i = β ,i , β ,i α = 2 β ,i , α ,i β = ( − i β ,i , α ,i β ,j = α ,i + j +2 ,α ,i β ,j = ( − i β ,i + j +1 , β ,i β = ( − i β ,i +1 , β ,i β ,j = β ,i + j +2 ,β ,i β ,j = β ,i + j +2 , β ,i β ,j = β ,i + j +2 , i, j ≥ . In particular, dim H k ( F , ) = , k = 0;2 , k = 1;3 , k = 2;4 , k ≥ . Proof. For k ≥ 3, by Lemma 3.4, letΦ : H (cid:16) F X , H k ( I ) (cid:17) −→ H k ( F , ) ,Y ∗ k Y ∗ k − kX ∗ ∧ X ∗ ∧ Y ∗ k − ,X ∗ ∧ Y ∗ k − − X ∗ ∧ Y ∗ ∧ Y ∗ k − X ∗ ∧ Y ∗ k − − X ∗ ∧ Y ∗ ∧ Y ∗ k − + 2( k − X ∗ ∧ X ∗ ∧ X ∗ ∧ Y ∗ k − , and Ψ : H (cid:16) F X , H k − ( I ) (cid:17) −→ H k ( F , ) ,X ∗ ∧ Y ∗ k − X ∗ ∧ Y ∗ k − ,X ∗ ∧ X ∗ ∧ Y ∗ k − − X ∗ ∧ X ∗ ∧ Y ∗ ∧ Y ∗ k − X ∗ ∧ X ∗ ∧ Y ∗ k − − X ∗ ∧ X ∗ ∧ Y ∗ ∧ Y ∗ k − . Then Φ and Ψ are injective linear mappings, and Im Φ ∩ Im Ψ = 0. Moreover, we haveH k ( F , ) = Φ (cid:16) H (cid:16) F X , H k ( I ) (cid:17)(cid:17) M Ψ (cid:16) H (cid:16) F X , H k − ( I ) (cid:17)(cid:17) . Thus, H • ( F , ) has a basis:1 , Y ∗ , X ∗ ∧ Y ∗ i , X ∗ ∧ Y ∗ − X ∗ ∧ Y ∗ , Y ∗ i +2 − i + 2) X ∗ ∧ X ∗ ∧ Y ∗ i ,X ∗ ∧ X ∗ ∧ Y ∗ i +1 − X ∗ ∧ X ∗ ∧ Y ∗ ∧ Y ∗ i ,X ∗ ∧ Y ∗ i +2 − X ∗ ∧ Y ∗ ∧ Y ∗ i +1 + 2( i + 1) X ∗ ∧ X ∗ ∧ X ∗ ∧ Y ∗ i , i ≥ . The conclusion can be obtained by a direct calculation. n cohomology of filiform Lie superalgebras p > Throughout this section the ground field F is an algebraically closed field of characteristic p > Lemma 4.1. For F t s s, , s = 1 , t = 1 , , t = 1 , , , 5, let I t s s, = [ F t s s, , F t s s, ]. For k ≥ ≤ i ≤ k , the following conclusions hold:(1) E k − i,i = O k − i (cid:16)(cid:16) F t s s, / I t s s, (cid:17) ∗ ; t (cid:17) N O i (cid:16)(cid:16) I t s s, (cid:17) ∗ ; t (cid:17) = ⇒ DPH k ( F t s s, ) . (2) E k − i,i ∞ = E k − i,i . Proof. The proof is similar to Lemma 3.3. F , Theorem 5. The following Z -graded superalgebra isomorphisms hold:(1) DPH • ( F , ) ∼ = U , ( t ) , where U , ( t ) is a p t − (4 p t + 2 p − Z -gradedsuperalgebra with a Z -homogeneous basis { α i,j , α ,k , α k , α i,j , , α s,t , α s,t | ≤ i ≤ p t − − , ≤ j ≤ p t − , ≤ k ≤ p t − , ≤ s ≤ p t − , ≤ t ≤ p t − } , satisfying that | α i,j | = | α i,j , | = i + j (mod 2) , | α k | = | α ,k | = k (mod 2) , | α s,t | = | α s,t | = s + t − k α i,j k = ip + j +1 , k α i,j , k = ip + j +2 , k α k k = k, k α ,k k = k +1 , k α s,t k = sp + t − , k α s,t k = sp + t, and the multiplication is given by α i ,j α ,i p = ( − i + j (cid:18) ( i + i ) pi p (cid:19) α i + i ,j , , α i ,j α i p = (cid:18) ( i + i ) pi p (cid:19) α i + i ,j ,α ,k α k = (cid:18) k + k k (cid:19) α ,k + k , α ,ip α s,t = (cid:18) ( i + s ) p − ip (cid:19) α i + s,t ,α k α k = (cid:18) k + k k (cid:19) α k + k , α i ,j , α i p = (cid:18) ( i + i ) pi p (cid:19) α i + i ,j , ,α s,t α ip = (cid:18) ( i + s ) p − ip (cid:19) α i + s,t , α ip α s,t = (cid:18) ( i + s ) p − ip (cid:19) α i + s,t , where 0 ≤ i, i , i ≤ p t − − , ≤ j ≤ p t − , ≤ k , k ≤ p t − , ≤ s ≤ p t − , ≤ t ≤ p t − . (2) DPH • ( F , ) ∼ = U , ( t ) , where U , ( t ) is a p t − (4 p t + 2 p − Z -gradedsuperalgebra with a Z -homogeneous basis { α i,j , α s,k , , α i,j , α l , α l , α , ,s,k | ≤ i ≤ p t − , ≤ j ≤ p t − , ≤ s ≤ p t − − , ≤ k ≤ p t − , ≤ l ≤ p t − } , n cohomology of filiform Lie superalgebras | α i,j | = | α i,j | = i + j − , | α s,k , | = | α , ,s,k | = s + k (mod 2) , | α l | = | α l | = l (mod 2); k α i,j k = ip + j, k α i,j k = ip + j − , k α s,k , k = sp + k +1 , k α , ,s,k k = sp + k +2 , k α l k = l +1 , k α l k = l, and the multiplication is given by α i,j α sp = (cid:18) ( i + s ) p − sp (cid:19) α i + s,j , α sp α i,j = (cid:18) ( i + s ) p − sp (cid:19) α i + s,j ,α i,j α sp = (cid:18) ( i + s ) p − sp (cid:19) α i + s,j , α s p α s ,k , = ( − s (cid:18) ( s + s ) ps p (cid:19) α , ,s + s ,k ,α s ,k , α s p = (cid:18) ( s + s ) ps p (cid:19) α s + s ,k , , α , ,s ,k α s p = (cid:18) ( s + s ) ps p (cid:19) α , ,s + s ,k ,α l α l = (cid:18) l + l l (cid:19) α l + l , α l α l = (cid:18) l + l l (cid:19) α l + l , where 1 ≤ i ≤ p t − , 1 ≤ j ≤ p t − 1, 0 ≤ s, s , s ≤ p t − − , ≤ k ≤ p t − , ≤ l, l , l ≤ p t − . (3) DPH • ( F , ) ∼ = U , ( t ) , where U , ( t ) is a p t − (4 p t + 1)-dimensional Z -graded su-peralgebra with a Z -homogeneous basis { α i,j , , α i , α i , α i,k , α s,k , α ,s , α s,t | ≤ i ≤ p t − − , ≤ j ≤ p t − , ≤ k ≤ p t − , ≤ s ≤ p t − , ≤ t ≤ p t − } , satisfying that | α i,j , | = i + j (mod 2) , | α i | = i (mod 2) , | α i | = i + 1 (mod 2) , | α i,k | = i + k (mod 2) , | α s,k | = s + k − , | α ,s | = s (mod 2) , | α s,t | = s + t − k α i,j , k = ip + j + 2 , k α i k = ip, k α i k = ip + 1 , k α i,k k = ip + k + 1 , k α s,k k = sp + k, k α ,s k = sp − , k α s,t k = sp + t − , and the multiplication is given by α i ,j , α i = (cid:18) ( i + i ) pi p (cid:19) α i + i ,j , , α i α i = (cid:18) ( i + i ) pi p (cid:19) α i + i ,α i α i = (cid:18) ( i + i ) pi p (cid:19) α i + i , α i ,k α i = (cid:18) ( i + i ) pi p (cid:19) α i + i ,k ,α s,k α i = (cid:18) ( i + s ) p − ip (cid:19) α i + s,k , α ,s α i = (cid:18) ( i + s ) p − ip (cid:19) α ,i + s ,α s,t α i = (cid:18) ( i + s ) p − ip (cid:19) α i + s,t , α ,s α i = (cid:18) ( i + s ) p − ip + 1 (cid:19) α i + s, ,α i,k α ,s = ( − i + k +1 (cid:18) ( i + s ) p − ip (cid:19) α i + s,k +1 , where 0 ≤ i, i , i ≤ p t − − , ≤ j ≤ p t − , ≤ k ≤ p t − , ≤ s ≤ p t − , ≤ t ≤ p t − . n cohomology of filiform Lie superalgebras Proof. (1) For k ≥ 0, 0 ≤ i ≤ k , consider the mapping d k − i,i : O k − i (( F , / F Y ) ∗ ; t ) O F Y ∗ ( i ) −→ O k − i +2 (( F , / F Y ) ∗ ; t ) O F Y ∗ ( i − ,f ⊗ Y ∗ ( i ) ( − k f k f ∧ d ( Y ∗ ) ⊗ Y ∗ ( i − , where d ( Y ∗ ) = X ∗ ∧ Y ∗ . By Lemma 4.1, we haveE k − i,i ∞ = O k (( F , / F Y ) ∗ ; t ) O k − (( F , / F Y ) ∗ ; t ) V F d ( Y ∗ ) , i = 0; Ker d k − i,i ∞ O k − − i (( F , / F Y ) ∗ ; t ) V F d ( Y ∗ ) N F Y ∗ ( i )2 , ≤ i ≤ k − F X ∗ N Y ∗ ( k − , i = k − , i = k .where Ker d k − i,i ∞ = F X ∗ ∧ Y ∗ ( k − i − ⊗ Y ∗ ( i ) M F X ∗ ∧ X ∗ ∧ Y ∗ ( k − i − ⊗ Y ∗ ( i ) M F δ k − i X ∗ ∧ Y ∗ ( k − i − ⊗ Y ∗ ( i ) M F δ k +1 − i Y ∗ ( k − i ) ⊗ Y ∗ ( i ) , where δ a = 1 when a = 0(mod p ) and δ a = 0 otherwise. From DPH k ( F , ) = k L i =0 E k − i,i ∞ ,we can obtain the conclusion.The proofs of (2) , (3) are similar to (1). F , Theorem 6. The following Z -graded superalgebra isomorphisms hold:(1) DPH • ( F , ) ∼ = U , ( t ) , where U , ( t ) is a p t − (8 p t + 4 p − Z -gradedsuperalgebra with a Z -homogeneous basis { α i , α s,j , α i , α s,j , , α s,t , , α s,j , , , α u,t , α u,v , α u,t , , β u,h , α i , , β i | ≤ i ≤ p t − , ≤ j ≤ p t − , ≤ s ≤ p t − − , ≤ t ≤ p t − , ≤ u ≤ p t − , ≤ v ≤ p t − , ≤ h ≤ p t − } , satisfying that | α i | = | α i | = | α i , | = i (mod 2) , | β i | = i + 1 (mod 2) , | α s,j | = | α s,j , | = | α s,j , , | = s + j (mod 2) , | α s,t , | = s + t (mod 2) , | α u,t | = | α u,t , | = u + t − , | α u,v | = u + v − , | β u,h | = u + h − k α i k = i, k α i k = i + 1 , k α i , k = k β i k = i + 2 , k α s,j k = sp + j + 1 , k α s,j , k = sp + j + 2 , k α s,j , , k = sp + j + 3 , k α s,t , k = sp + t + 2 , k α u,t k = up + t − , k α u,t , k = up + t + 1 , k α u,v k = up + v, k β u,h k = up + h, n cohomology of filiform Lie superalgebras α i α i = (cid:18) i + i i (cid:19) α i + i , α i α i = (cid:18) i + i i (cid:19) α i + i ,α i , α i = (cid:18) i + i i (cid:19) α i + i , , β i α i = (cid:18) i + i i (cid:19) β i + i ,α s ,j α s p = (cid:18) ( s + s ) ps p (cid:19) α s + s ,j , α s ,j , α s p = (cid:18) ( s + s ) ps p (cid:19) α s + s ,j , ,α s ,t , α s p = (cid:18) ( s + s ) ps p (cid:19) α s + s ,t , , α s ,j , , α s p = (cid:18) ( s + s ) ps p (cid:19) α s + s ,j , , ,α s ,j , α s p +1 = − (cid:18) ( s + s ) ps p (cid:19) α s + s ,j +10 , , α s ,j α s p = ( − s + j (cid:18) ( s + s ) ps p (cid:19) α s + s ,j , ,α s ,j α s p , = (cid:18) ( s + s ) ps p (cid:19) α s + s ,j , , , α s p α s ,j , = − (cid:18) ( s + s ) ps p (cid:19) α s + s ,j , , ,α sp α u,t = (cid:18) ( s + u ) p − sp (cid:19) α s + u,t , α u,v α sp = (cid:18) ( s + u ) p − sp (cid:19) α s + u,v ,α u,t , α sp = (cid:18) ( s + u ) p − sp (cid:19) α s + u,t , , β u,h α sp = (cid:18) ( s + u ) p − sp (cid:19) β s + u,h ,β u,h α sp +1 = − (cid:18) ( s + u ) p − sp (cid:19) α s + u,h +11 , α sp α u,t = (cid:18) ( s + u ) p − sp (cid:19) α s + u,t ,α sp β u,h = ( − s (cid:18) ( s + u ) p − sp (cid:19) α s + u,h , , α sp , α u,t = (cid:18) ( s + u ) p − sp (cid:19) α s + u,t , ,β sp α u,t = (cid:18) ( s + u ) p − sp (cid:19) ( t +1) α s + u,t +11 , β u,h β sp = ( − u + h (cid:18) ( s + u ) p − sp (cid:19) hα u + s,h +11 , ,α s ,j β s p = ( − s + j (cid:18) ( s + s ) ps p (cid:19) ( j + 2) α s + s ,j +10 , ,α s ,j , β s p = ( − s + j +1 (cid:18) ( s + s ) ps p (cid:19) ( j + 1) α s + s ,j +10 , , ,α i β i = ( − i +1 (cid:18) i + i + 1 i (cid:19) ( i + 1) α i + i +11 , , ≤ i , i ≤ p t − , where 0 ≤ s, s , s ≤ p t − − 1, 0 ≤ j ≤ p t − 1, 1 ≤ t ≤ p t − 1, 1 ≤ u ≤ p t − ,2 ≤ v ≤ p t − 1, 1 ≤ h ≤ p t − • ( F , ) ∼ = U , ( t ) , where U , ( t ) is a p t − (8 p t + 1)-dimensional Z -graded su-peralgebra with a Z -homogeneous basis { α i , α i, , α i,j , α i,j , , α s , , α s,j , , α ,s,h , β s , β i,t , β ,i,t , β ,s,t , γ i , γ i , γ s,h | ≤ i ≤ p t − − , ≤ j ≤ p t − , ≤ s ≤ p t − , ≤ t ≤ p t − , ≤ h ≤ p t − } , satisfying that | α i | = | γ i | = i (mod 2) , | α i, | = | γ i | = i + 1 (mod 2) , | α s , | = | β s | = s (mod 2) , n cohomology of filiform Lie superalgebras | α i,j | = | α i,j , | = i + j (mod 2) , | β i,t | = | β ,i,t | = i + t (mod 2) , | β ,s,t | = s + t − , | α s,j , | = s + j − , | α ,s,h | = | γ s,h | = s + h − k γ i k = ip, k α i k = k γ i k = ip + 1 , k α i, k = ip + 2 , k β s k = sp − , k α s , k = sp, k α i,j k = ip + j + 1 , k α i,j , k = ip + j + 2 , k β i,t k = ip + t + 2 , k β ,i,t k = ip + t +3 , k β ,s,t k = sp + t, k α s,j , k = sp + j +1 , k α ,s,h k = sp + h, k γ s,h k = sp + h − , and the multiplication is given by α ,s,h γ i = (cid:18) ( i + s ) p − ip (cid:19) α ,i + s,h , α s,j , γ i = (cid:18) ( i + s ) p − ip (cid:19) α i + s,j , ,γ s,h γ i = (cid:18) ( i + s ) p − ip (cid:19) γ i + s,h , α s , γ i = (cid:18) ( i + s ) p − ip (cid:19) α i + s , ,β s γ i = (cid:18) ( i + s ) p − ip (cid:19) β i + s , β ,s,t γ i = (cid:18) ( i + s ) p − ip + 1 (cid:19) α ,i + s,t +1 ,α s , γ i = (cid:18) ( i + s ) p − ip + 1 (cid:19) α i + s, , , β s γ i = (cid:18) ( i + s ) p − ip + 1 (cid:19) β ,i + s, ,α i β i ,t = (cid:18) ( i + i ) pi p (cid:19) β ,i + i ,t , α i,j α s , = (cid:18) ( i + s ) p − ip (cid:19) α ,i + s,j +1 ,α i,j , β s = ( − i + j +1 (cid:18) ( i + s ) p − ip (cid:19) α ,i + s,j +1 , α i γ s,h = (cid:18) ( i + s ) p − ip (cid:19) α ,i + s,h ,α i β ,s,t = ( − i (cid:18) ( i + s ) p − ip (cid:19) α i + s,t , , α i β s = ( − i (cid:18) ( i + s ) p − ip (cid:19) α i + s , ,α i, β s = ( − i +1 (cid:18) ( i + s ) p − ip + 1 (cid:19) α i + s, , , β ,s,t γ i = (cid:18) ( i + s ) p − ip (cid:19) β ,i + s,t ,α s , β i,t = (cid:18) ( i + s ) p − ip + 1 (cid:19) α i + s,t +11 , , β i,t β s = ( − i + t (cid:18) ( i + s ) p − ip + 1 (cid:19) β ,i + s,t +1 ,γ i γ i = (cid:18) ( i + i ) pi p (cid:19) γ i + i , γ i γ i = (cid:18) ( i + i ) pi p (cid:19) γ i + i ,α i ,j γ i = (cid:18) ( i + i ) pi p (cid:19) α i + i ,j , α i ,j , γ i = (cid:18) ( i + i ) pi p (cid:19) α i + i ,j , ,α i γ i = (cid:18) ( i + i ) pi p (cid:19) α i + i , α i , γ i = (cid:18) ( i + i ) pi p (cid:19) α i + i , ,β i ,t γ i = (cid:18) ( i + i ) pi p (cid:19) β i + i ,t , β ,i ,t γ i = (cid:18) ( i + i ) pi p (cid:19) β ,i + i ,t ,γ i γ i = 2 (cid:18) ( i + i ) pi p (cid:19) α i + i , , , α i γ i = (cid:18) ( i + i ) pi p (cid:19) α i + i , ,β i ,t γ i = 2 (cid:18) ( i + i ) pi p (cid:19) α i + i ,t +10 , , α i ,j α i = ( − i + j (cid:18) ( i + i ) pi p (cid:19) α i + i ,j , , n cohomology of filiform Lie superalgebras α i,j β s = ( − i + j +1 (cid:18) ( i + s ) p − ip (cid:19) γ i + s,j +1 ,β s β ,i,t = ( − s − (cid:18) ( i + s ) p − ip + 1 (cid:19) α i + s,t +11 , ,β i,t β ,s,t = ( − i + t (cid:18) ( i + s ) p − ip + 1 (cid:19) (cid:18) t + t + 1 t + 1 (cid:19) α ,i + s,t + t +2 ,β i ,t β i ,t = 2 (cid:18) ( i + i ) pi p (cid:19) (cid:18) t + t + 2 t + 1 (cid:19) α i + i ,t + t +20 , , where 0 ≤ i, i , i ≤ p t − − 1, 1 ≤ s ≤ p t − , 1 ≤ h ≤ p t − 1, 0 ≤ j ≤ p t − ≤ t, t , t ≤ p t − • ( F , ) ∼ = U , ( t ) ⋉ V , ( t ) , where U , ( t ) is a p t − (2 p + 1)-dimensional Z -gradedsuperalgebra with a Z -homogeneous basis { α i , α ,i , α ,s | ≤ i ≤ p t − , ≤ s ≤ p t − − } , satisfying that | α i | = | α ,i | = i (mod 2) , | α ,s | = s (mod 2); k α i k = i, k α ,i k = i + 1 , k α ,s k = sp + 1 , and the multiplication is given by α i α i = (cid:18) i + i i (cid:19) α i + i , α ,i α i = (cid:18) i + i i (cid:19) α ,i + i ,α ,s α s p = (cid:18) ( s + s ) ps p (cid:19) α ,s + s , α ,s α qp + r = − (cid:18) ( s + q ) p + rsp (cid:19) α , ( s + q ) p + r , where 0 ≤ i , i ≤ p t − , ≤ s , s ≤ p t − − , ≤ s ≤ p t − − , ≤ qp + r ≤ p t − , ≤ r ≤ p − . V , ( t ) is a 2 p t − (4 p t + p − − Z -graded superalgebra witha Z -homogeneous basis { α , ,i , β , ,i , β j , ,s , β j , , ,s , α j , ,s , α k , , ,s , α , ,s , α jh , α j ,h , α j , ,h , β ρ , ,h , α , ,l | ≤ i ≤ p t − , ≤ j ≤ p t − , ≤ s ≤ p t − − , ≤ h ≤ p t − , ≤ k ≤ p t − , ≤ l ≤ p t − − , ≤ ρ ≤ p t − } , satisfying that | α , ,i | = i (mod 2) , | β , ,i | = i + 1 (mod 2) , | α , ,s | = s (mod 2) , | α , ,l | = l + 1 (mod 2) , | α k , , ,s | = s + k (mod 2) , | β ρ , ,h | = h + ρ − , | β j , ,s | = | β j , , ,s | = | α j , ,s | = s + j (mod 2) , | α jh | = | α j ,h | = | α j , ,h | = h + j − k α , ,i k = k β , ,i k = i + 2 , k α , ,s k = sp + 2 , k α , ,l k = lp + 3 , k β j , ,s k = sp + j + 1 , k β j , , ,s k = k α j , ,s k = sp + j + 2 , k α k , , ,s k = sp + k + 3 , k α jh k = hp + j − , k α j ,h k = hp + j, k α j , ,h k = hp + j + 1 , k β ρ , ,h k = hp + ρ, n cohomology of filiform Lie superalgebras α , ,sp α jh = (cid:18) ( s + h ) p − sp (cid:19) α j , ,s + h , α , ,s α jh = − (cid:18) ( s + h ) p − sp (cid:19) α j , ,s + h ,β , ,sp α jh = (cid:18) ( s + h ) p − sp (cid:19) ( j + 1) α j +10 ,s + h , β j , ,s α , ,s = (cid:18) ( s + s ) ps p (cid:19) α j , , ,s + s ,β , ,s p α , ,s = (cid:18) ( s + s ) ps p (cid:19) α , , ,s + s ,α , ,s p β j , ,s = ( − s +1 (cid:18) ( s + s ) ps p (cid:19) α j , , ,s + s ,β j , ,s β , ,s p = ( − s + j +1 (cid:18) ( s + s ) ps p (cid:19) ( j + 2) α j +10 , ,s + s ,β , ,sp β ρ , ,h = ( − s +1 (cid:18) ( s + h ) p − sp + 1 (cid:19) ρα ρ +10 , ,s + h ,β , ,s p β j , , ,s = (cid:18) ( s + s ) ps p (cid:19) ( j + 1) α j +10 , , ,s + s , where 0 ≤ s, s , s ≤ p t − − , ≤ h ≤ p t − , ≤ j ≤ p t − , ≤ ρ ≤ p t − . The multiplication between U , ( t ) and V , ( t ) is graded-supercommutative, and themultiplication is given by α , , α = − α , , − α , , , α , ,l α sp +1 = (cid:18) ( s + l ) psp (cid:19) α , ,s + l ,β , , α , = − α , , − α , , , α ,s p β j , ,s = ( − s (cid:18) ( s + s ) ps p (cid:19) α j , ,s + s ,α , ,l α hp − = (cid:18) ( h + l ) plp + 1 (cid:19) α , ,h + l , α ,hp − α , ,l = (cid:18) ( h + l ) plp + 1 (cid:19) α , , ,h + l ,α jh α sp = (cid:18) ( s + h ) p − sp (cid:19) α js + h , β j , ,s α s p = (cid:18) ( s + s ) ps p (cid:19) β j , ,s + s ,β ρ , ,h α sp = (cid:18) ( s + h ) p − sp (cid:19) β ρ , ,s + h , β ρ , ,h α sp +1 = (cid:18) ( s + h ) p − sp + 1 (cid:19) α ρ +10 ,s + h ,α j ,h α sp = (cid:18) ( s + h ) p − sp (cid:19) α j ,s + h , β j , , ,s α s p = (cid:18) ( s + s ) ps p (cid:19) β j , , ,s + s ,β j , , ,s α s p +1 = − (cid:18) ( s + s ) ps p (cid:19) α j +10 , ,s + s , α j , ,h α sp = (cid:18) ( s + h ) p − sp (cid:19) α j , ,s + h ,α j , ,s α s p = (cid:18) ( s + s ) ps p (cid:19) α j , ,s + s , α k , , ,s α s p = (cid:18) ( s + s ) ps p (cid:19) α k , , ,s + s ,α , ,i α i = (cid:18) i + i i (cid:19) α , ,i + i , α , ,s α s p = (cid:18) ( s + s ) ps p (cid:19) α , ,s + s , n cohomology of filiform Lie superalgebras α , ,s α lp +1 = (cid:18) ( s + l ) psp (cid:19) α , ,s + l , α , ,s α q p + r = − (cid:18) ( s + q ) p + r sp (cid:19) α , , ( s + q ) p + r ,α , ,l α sp = (cid:18) ( s + l ) psp (cid:19) α , ,s + l , α , ,l α q p + r = − (cid:18) ( q + l ) p + r + 1 lp + 1 (cid:19) α , , ( q + l ) p + r +1 ,β , ,i α i = (cid:18) i + i i (cid:19) β , ,i + i , α ,sp α jh = (cid:18) ( s + h ) p − sp (cid:19) α j ,s + h ,α ,s p β j , , ,s = (cid:18) ( s + s ) ps p (cid:19) α j , , ,s + s , α ,s p α , ,s = (cid:18) ( s + s ) ps p (cid:19) α , , ,s + s ,α ,s α jh = − (cid:18) ( s + h ) p − sp (cid:19) α j ,s + h , α ,s α , ,s p = − (cid:18) ( s + s ) ps p (cid:19) α , , ,s + s ,β , ,lp α ,s = ( − l +1 (cid:18) ( s + l ) psp (cid:19) α , ,s + l + ( − l (cid:18) ( s + l ) p + 1 sp (cid:19) α , ,s + l ,β , ,sp α ,l = ( − s +1 (cid:18) ( s + l ) psp (cid:19) α , ,s + l + ( − s (cid:18) ( s + l ) p + 1 lp (cid:19) α , ,s + l ,α ,sp β ρ , ,h = ( − s +1 (cid:18) ( s + h ) p − sp (cid:19) α ρ , ,s + h ,α ,i β , ,i = ( − i ( i + 1) (cid:18) i + i + 1 i (cid:19) α , ,i + i +1 ,β ρ , ,h α ,s = ( − h + ρ (cid:18) ( s + h ) p − sp (cid:19) α ρ , ,s + h ,β j , ,s α ,s = ( − s + j (cid:18) ( s + s ) ps p (cid:19) α j , ,s + s ,β j , , ,s α ,s = ( − s + j +1 (cid:18) ( s + s ) ps p (cid:19) α j , , ,s + s ,β , ,q p + r α ,s = ( − q + r +1 ( r + 1) (cid:18) ( s + q ) p + r + 1 q p + r + 1 (cid:19) α , , ( s + q ) p + r +1 , where 0 ≤ i , i , q p + r , q p + r , q p + r ≤ p t − , ≤ s, s , s ≤ p t − − , ≤ h ≤ p t − , ≤ k ≤ p t − , ≤ l ≤ p t − − , ≤ j ≤ p t − , ≤ ρ ≤ p t − , ≤ r ≤ p − , ≤ r , r ≤ p − . (4) DPH • ( F , ) ∼ = U , ( t ) , where U , ( t ) is a p t − (8 p t + 2)-dimensional Z -graded su-peralgebra with a Z -homogeneous basis { α i , α i , α i , , α i , α k,l , α i,j , , α k,j , , α k , , α k,l , β k , β i,j , , β ,i,s , β ,i,s , β ,k,j | ≤ i ≤ p t − − , ≤ j ≤ p t − , ≤ s ≤ p t − , ≤ k ≤ p t − , ≤ l ≤ p t − } , satisfying that | α i | = | α i | = i (mod 2) , | α i , | = | α i | = i + 1 (mod 2) , | α i,j , | = | β i,j , | = i + j (mod 2) , | β ,i,s | = | β ,i,s | = i + s (mod 2) , | α k,l | = | α k,l | = k + l − , n cohomology of filiform Lie superalgebras | α k,j , | = | β ,k,j | = k + j − , | α k , | = | β k | = k (mod 2); k α i k = ip, k α i k = k α i k = ip + 1 , k α i , k = ip + 2 , k α i,j , k = ip + j + 2 , k β i,j , k = ip + j + 1 , k β ,i,s k = ip + s + 2 , k β ,i,s k = ip + s + 3 , k α k,l k = kp + l, k α k,l k = kp + l − , k α k,j , k = kp + j + 1 , k β ,k,j k = kp + j, k α k , k = kp, k β k k = kp − , and the multiplication is given by α i ,j , α i = (cid:18) ( i + i ) pi p (cid:19) α i + i ,j , , α i β i ,j , = ( − i (cid:18) ( i + i ) pi p (cid:19) α i + i ,j , ,β i ,j , α i = (cid:18) ( i + i ) pi p (cid:19) β i + i ,j , , α i α i = (cid:18) ( i + i ) pi p (cid:19) α i + i ,α i α i = (cid:18) ( i + i ) pi p (cid:19) α i + i , , α i β ,i ,s = (cid:18) ( i + i ) pi p (cid:19) β ,i + i ,s ,α i , α i = (cid:18) ( i + i ) pi p (cid:19) α i + i , , α i α i = (cid:18) ( i + i ) pi p (cid:19) α i + i ,α i α i = (cid:18) ( i + i ) pi p (cid:19) α i + i , β ,i ,s α i = (cid:18) ( i + i ) pi p (cid:19) β ,i + i ,s ,β ,i ,s α i = (cid:18) ( i + i ) pi p (cid:19) β ,i + i ,s , α i,j , β k = ( − i + j +1 (cid:18) ( i + k ) p − ip (cid:19) α i + k,j +10 ,α k , α i = (cid:18) ( i + k ) p − ip (cid:19) α i + k , , α k , β i,j , = ( − k (cid:18) ( i + k ) p − ip (cid:19) α i + k,j +10 ,β k α i = (cid:18) ( i + k ) p − ip (cid:19) β i + k , β i,j , β k = ( − i + j +1 (cid:18) ( i + k ) p − ip (cid:19) α i + k,j +1 ,α i α k,l = (cid:18) ( i + k ) p − ip (cid:19) α i + k,l , α i β k = ( − i (cid:18) ( i + k ) p − ip (cid:19) α i + k , ,α k,l α i = (cid:18) ( i + k ) p − ip (cid:19) α i + k,l , α k,j , α i = (cid:18) ( i + k ) p − ip (cid:19) α i + k,j , ,α i α k,l = (cid:18) ( i + k ) p − ip (cid:19) α i + k,l , β ,k,j α i = (cid:18) ( i + k ) p − ip (cid:19) β ,i + k,j ,α i β ,k,j = ( − i (cid:18) ( i + k ) p − ip (cid:19) α i + k,j , , β k α i = (cid:18) ( i + k ) p − ip + 1 (cid:19) β ,i + k, ,α i , β k = ( − i +1 (cid:18) ( i + k ) p − ip + 1 (cid:19) α i + k, , , α k , α i = (cid:18) ( i + k ) p − ip + 1 (cid:19) α i + k, , ,α k , β ,i,s = (cid:18) ( i + k ) p − ip + 1 (cid:19) α i + k,s +10 , , β k β ,i,s = (cid:18) ( i + k ) p − ip + 1 (cid:19) β ,i + k,s +1 ,β ,i,s β k = ( − i + s (cid:18) ( i + k ) p − ip + 1 (cid:19) α i + k,s +10 , , α i α i = (cid:18) ( i + i ) p + 2 i p + 1 (cid:19) α i + i , , , n cohomology of filiform Lie superalgebras β ,i ,s α i = (cid:18) ( i + i ) p + 2 i p + 1 (cid:19) α i + i ,s +10 , , β ,k,j α i = − (cid:18) ( i + k ) p − ip + 1 (cid:19) α i + k,j +10 ,β ,i ,s β ,i ,s = (cid:18) ( i + i ) p + 2 i p + 1 (cid:19) (cid:18) s + s + 2 s + 1 (cid:19) α i + i ,s + s +20 , ,β ,k,j β ,i,s = − (cid:18) ( i + k ) p − ip + 1 (cid:19) (cid:18) j + s + 1 j (cid:19) α i + k,j + s +20 , where 0 ≤ i, i , i ≤ p t − − , ≤ j ≤ p t − , ≤ s, s , s ≤ p t − , ≤ k ≤ p t − , ≤ l ≤ p t − . Proof. The proof is similar to Theorem 5. Acknowledgment The first author was supported by the NSF of China (11501151) and the NSF of HeilongjiangProvince (A2015003, A2017005). The corresponding author was supported by the NSF of China(11471090, 11701158) and the NSF of Heilongjiang Province (A2015017). The authors appreciatethe referee for his/her valuable suggestions. References [1] M. Vergne. Cohomologie des alg`ebres de Lie nilpotentes. Application `a l’´etude de la vari´et´edes alg`ebres de Lie nilpotentes. (French) Bull. Soc. Math. France (1970) 81–116.[2] F. Grant. Armstrong and S. Sigg. On the cohomology of a class of nilpotent Lie algebras. Bull. Austral. Math. Soc. (1996) 517–527.[3] M. Bordemann. Nondegenerate invariant bilinear forms on nonassociative algebras. Acta Math.Univ. Comenian. (1997) 151–201.[4] A. Fialowski and D. Millionschikov. Cohomology of graded Lie algebras of maximal class. J.Algebra (2006) 157–176.[5] I. Tsartsaflis. On the Betti numbers of filiform Lie algebras over fields of characteristic two. Rev. Un. Mat. Argentina (2017) 95–106.[6] Y. Nikolayevsky and I. Tsartsaflis. Cohomology of N -graded Lie algebras of maximal classover Z . J. Lie Theory (2017) 529–544.[7] M. Gilg. On deformations of the filiform Lie superalgebra L n,m . Comm. Algebra (2004)2099–2115.[8] M. Bordemann, J.R. G´omez, Y. Khakimdjanov, R.M. Navarro. Some deformations of nilpotentLie superalgebras. J. Geom. Phys. (2007) 1391–1403.[9] J.R. G´omez, Y. Khakimdjanov, R.M. Navarro. Infinitesimal deformations of the Lie superal-gebra L n,m . J. Geom. Phys. (2008) 849–859.[10] Y. Khakimdjanov, R.M. Navarro. Deformations of filiform Lie algebras and superalgebras. J.Geom. Phys. (2010) 1156–1169.[11] Y. Khakimdjanov, R.M. Navarro. A complete description of all the infinitesimal deformationsof the Lie superalgebra L n,m . J. Geom. Phys. (2010) 131–141.[12] W. Liu, Y. Yang. Cohomology of model filiform Lie superalgebras. J. Algebra Appl. (4)(2018), 13 pages. n cohomology of filiform Lie superalgebras [13] I.M. Musson. Lie superalgebras and enveloping algebras. American Mathematical Society Prov-idence, Rhode Island (2013).[14] M. Gilg. Low-dimensional filiform Lie superalgebras.