Local Automorphisms and Local Superderivations of Model Filiform Lie Superalgebras

In this paper, we give the forms of local automorphisms (resp. superderivations) of model fliform Lie superalgebra L n,m in the matrix version. Linear 2-local automorphisms (resp. superderivations) of L n,m are also characterized. We prove that each linear 2-local automorphism of L n,m is an automorphism.


Introduction and Basics
As a signifcant class of nilpotent Lie algebras, fliform Lie algebras were introduced by Vergne [1] and have been studied extensively, see [2][3][4][5][6] and references in them.Model fliform Lie algebra L n is the simplest fliform Lie algebra.Vergne proved that each fliform Lie algebra can be obtained by deformations of model fliform Lie algebra (see [1]).Similarly, model fliform Lie superalgebra L n,m is the simplest fliform Lie superalgebra.
In this paper, we will use matrices to study local automorphisms (resp.superderivations) of model fliform Lie superalgebra L n,m .We will give concrete forms of local automorphisms (resp.superderivations) of L n,m .For fnitedimensional nilpotent Lie algebra L with dim L ⩾ 2. In [23], Ayupov and Kudaybergenov proved that there is a 2-local automorphism of L which is not an automorphism.Ten, it is impossible that every 2-local automorphism of Lie superalgebra L n,m is an automorphism.But if a 2-local automorphism is linear, then we can prove that it must be an automorphism.So, we add an additional linear condition in the defnition of 2-local automorphism, we call it linear 2-local automorphism.We will prove that all linear 2-local automorphisms of L n,m are automorphisms.But for 2-local superderivation of L n,m , the situation is diferent.We also add an additional linear condition in the defnition of 2local superderivation, and we call it linear 2-local superderivation.In this paper, we will show that not all linear 2local superderivations of L n,m are superderivations, but they is very close to a superderivations.Te same situation also occurs in 2-local automorphisms (resp.derivations) of model fliform Lie algebra L n .We fnd that not all linear 2local automorphisms (resp.derivations) of Lie algebra L n are automorphisms (resp.derivations), and the linear 2-local automorphisms (resp.derivations) which are not automorphisms (resp.derivations) are very close to automorphisms (resp.derivations).
Model fliform Lie superalgebra L n,m is a superalgebra with multiplication where is the homogeneous basis and the other brackets vanished.If we only consider the Lie algebra with x 0 , x 1 , • • • , x n   a basis, their multiplication are same to (1), then it is the model fliform Lie algebra L n .
For a Lie superalgebra ( Denote the group consisting of all automorphisms of G by Aut(G).Suppose D : G ⟶ G is a linear map of degree α, we call D a superderivation of degree α if Troughout the paper, we assume that 3 ⩽ n ⩽ m.All mappings mentioned in this paper are linear.Te matrices of mappings of L n,m are all with respect to the homogeneous basis

Denote all superderivations of degree α by Der
, and the matrices of mappings of L n are all with respect to the basis x 0 , x 1 , • • • , x n  .F stands for an arbitrary feld of characteristic zero, F * is the set of all nonzero elements of F, and F n is the n-dimensional column vector space over F. E ij and e i represent the matrix unit and unit vector, respectively.
Denote block matrices by A ⊕ B and C � ⊕ D, respectively.
Ten, for any (2) Let B be an m × m invertible lower triangular matrix.
Ten, for any a ∈ F * and X ∈ F m , there exists where u ∈ F * , β ∈ F n and U is an n × n invertible lower triangular matrix.For any We will prove that there exist α ∈ F n and B(a, a 1 , Ten, it is easy to see that there exists α ∈ F n such that (7) holds.Case 2. x 0 � 0. Assume that the frst nonzero component of vector X 1 is the r− th.Put a � 1, α � 0, a n− r � • • • � a n � 0. Ten, it is easy to prove that there exist a 1 ∈ F * and a 2 , • • • , a n− r− 1 ∈ F such that (7) holds.
(2) In a similarly way to the proof of (1), one can come to the conclusion.

□
Theorem 2. Let φ be a linear mapping of L n,m .Ten, φ ∈ Aut(L n,m ) if and only if the matrix of φ is of the form For any 1 where 1 If a 10 ≠ 0, then by (11), we have (10), they contradict the invertibility of A. Terefore, a 10 � 0. Consequently, we have a 0 ≠ 0 since A is invertible.For any 1 ⩽ k ⩽ n − 1, according to (10) and (11), we have a i+1,k+1 � a 0 a ik and a nk � 0. Denote Proof.Assume that the matrix of φ is where where A X ⊕ B X is of the form (8). We let 14) is arbitrary, we see that D � 0.
Similarly, we let k j � 0 for all j ∈ 1, 14) is arbitrary, we see that C � 0.
If A is not invertible, then there exists a nonzero vector α ∈ F n+1 such that Aα � 0. Substituting X � (α T , O) T into (14), we have A X α � 0, which contradicts the invertibility of A X .Tus, A is invertible.Similarly, B is also invertible.
Substituting (14) where By Teorem 2, for any X, Y ∈ F n+m+1 , there exist A XY and B XY such that where But for the sake of simplicity, we still denote them in this way without causing confusion.Substituting X � e s , Y � e s+1 into (16), we have Ten, substituting X � e t , Y � e t+1 + e t+2 into (16), we have Similarly, we can conclude that d is Next, substituting X � e 2 + e n+2 , Y � e 3 + e n+3 into (16), we have l � k 2 .Finally, substituting X � e 1 + e n+2 , Y � e n+3 into (16), we obtain l � c.Tus, by Teorem 2, we have φ ∈ Aut(L n,m ).
From the proof of the above theorems, we get the following conclusions immediately. where Corollary 6 where Corollary 7. Let φ be a linear mapping of L n .
(1) Te automorphism group of L n is where where where σ is a linear mapping of L n whose matrix is Proof.From the proof of the above theorems, (1), (2), and the necessity of (3) hold immediately.Next, we only need to prove the sufciency of (3).Assume the matrix of φ is A(a, a 1 , • • • , a n , α) + kE 11 .For any X, Y ∈ F n+1 , we will fnd appropriate A XY such that (24) where , and, therefore, (24) holds.
First, we fnd a ′ � a + k.Ten, it is easy to fnd

Local Superderivations and Linear 2-Local Superderivations of L n,m
Suppose a, a i , c i , As early as 1996, Goze and Khakimdjanov had characterized derivations of L n in [31].Te following lemma comes from [31].

Lemma 12
From this lemma, we can easily get the following conclusion.

Journal of Mathematics
Corollary 13.Let φ be a linear mapping of L n .Ten, φ ∈ Der(L n ) if and only if there exist a, a 1 , a 2 , • • • , a n ∈ F and α ∈ F n such that the matrix of φ is A(a, a 1 , • • • , a n , α).
Next, we will characterize the matrix form of the superderivation of L n,m .Theorem 14.Let D be a linear mapping of L n,m , then D is a superderivation if and only if its matrix is of the form where A, D, C, and B are in the forms of ( 28)- (31), respectively.
Proof.Clearly, a direct verifcation can prove the sufciency, and so we only need to prove the necessity of the theorem.
If D ∈ Der(L n,m ), we can assume that the matrix of D is First, we will deduce the form of the matrix of even derivation.
By Corollary 13, Tat is, Next, we will deduce the form of the matrix of odd derivation.Similar to the above process, substituting (x, y) � (x 0 , x i ), (x 0 , x n ), (y 1 , y k ), (x 0 , y m ), (x 0 , y j ) into the next equation successively, then we can get the following equations in turn: Ten, where |x| refers the degree of x, respectively.
Proof.First, we prove the necessity of the theorem.If φ ∈ LDer(L n,m ), then for any , there exist A X , B X , C X , and D X such that where A X , B X , C X , and D X are of the forms A, B, C, and D in Teorem 14, respectively.If we let l 1 � • • • � l m � 0, then by the arbitrariness of k 0 , k 1 , • • • , k n and (43) and in a similar way to the proof of Teorem 3, we obtain that A and D have the required forms.
Similarly, if we let k 0 � k 1 � • • • � k n � 0, then by the arbitrariness of l 1 , • • • , l m in (43) and in a similar way to the proof of Teorem 3, we deduce that C and B have the required forms.
Next, we will prove the sufciency of the theorem.
For any X 1 ∈ F n+1 , in a similar way to prove Lemma 1, we have A X 1 and D X 1 such that AX 1 � A X 1 X 1 and DX 1 � D X 1 X 1 , where A X 1 and D X 1 are of the forms A and D in Teorem 14, respectively.
Similarly, for any X 2 ∈ F m and a ∈ F which is (1, 1)-entry of A X 1 , there exist C X 2 and B X 2 such that CX 2 � C X 2 X 2 and BX 2 � B X 2 X 2 , where C X 2 and B X 2 are of the forms C and B in Teorem 14, respectively.

□ Corollary 16
Der L n,m   � span adx i , ady j , a k , b l , c t , d s , g, h, t, u, v where Corollary 17 where Journal of Mathematics where A XY , D XY , C XY , and B XY are of the forms of ( 28)-( 31), respectively.
Put a ′ � a.For any a t , c t , n − s + 2 ⩽ t ⩽ n, it is easy to choose appropriate then we can choose α ′ , β ′ , c ′ such that (50) holds.
Subcase 23. i � 1 and s > n.Similar to the proof in Subcase 22, we can achieve the goal.From the proof of the above theorems, we get the following conclusion immediately.
Corollary 24.Let φ be a linear mapping of L n .Ten, (1) a linear 2-local automorphism (resp.superderivation), if for any x, y ∈ G, there exists σ xy ∈ Aut(G) (resp.Der(G)) such that σ(x) � σ xy (x) and σ(y) � σ xy (y).Denote the group consisting of all local automorphisms of G by LAut(G) and the superalgebra consisting of all local superderivations of G by LDer(G), respectively.

Theorem 3 .
L n,m by verifcation.□Let φ be a linear mapping of L n,m .Ten, φ ∈ LAut(L n,m ) if and only if the matrix of φ is of the form A ⊕ B, where A and B are (n + 1) × (n + 1) and m × m invertible lower triangular matrices, respectively.

Theorem 1 .
, we complete the proof of the necessity of the theorem.□Let φ be a linear mapping of L n,m whose matrix is(A ⊕ B) + (D � ⊕ C),where B is an m × m matrix.Ten, φ ∈ LDer(L n,m ) if and only if A and B are both lower triangular matrices, and D and C are of the form

)where 1
If m − n � 1, substituting X � e 2 + e n+3 , Y � e 3 + e n+4 into (50), we have k � l.Else if m − n ≠ 1, substituting X � e 2 + e n+2 , Y � e 3 + e n+3 into (50), we have k � l.Tus, φ is desired.Next, we will prove the sufciency of the theorem.Assume the matrix of φ is(A ⊕ B) + (D � ⊕ C),where C, D, and A are the same as in (51) and (52), respectively, andB � B(k, b 1 , • • • , b n ).For any X, Y ∈ F n+m+1 , we want to fnd A XY , B XY , C XY and D XY such that (50) holds, where A XY , D XY , C XY , and B XY are of the forms (29)-(31), respectively.□ Case 19.If X and Y are linear dependent, then by Teorem 18, the existence of A XY , B XY , C XY , and D XY is obvious.Case 20.If X and Y are linear independent, then without loss of generality, we only need to consider the case ofX � e i +  ⩽ s ⩽ n + m + 1 − i, 1 ⩽ i ⩽ n + m.Subcase 21. i > 1.Put A XY � A − (a − k)E 11 B XY � B,C XY � C and D XY � D, and, therefore, (50) holds.Subcase 22. i � 1 and s ⩽ n.Let in turn, by the arbitrariness of k i , • • • , k n+1 , we obtain that A is a lower triangular matrix.Similarly, B is also a lower triangular matrix.Conversely, if the matrix of φ is A ⊕ B, where A and B are (n + 1) × (n + 1) and m × m invertible lower triangular matrices respectively, then by Lemma 1, φ is a local automorphism of L n,m .If φ is a linear 2-local Lie superalgebra automorphism of L n,m , then φ ∈ LAut(L n,m ).By Teorem 3, the matrix of φ is of the form A ⊕ B, where A and B are (n + 1) × (n + 1) and m × m invertible lower triangular matrices, respectively.Denote all related to X and Y.If X and Y are linear dependent, then by Teorem 3, the existence of A XY is obvious.
) If φ is a linear 2-local superderivation of L n,m , then by Teorem 18, we can assume that the matrix of L n,m is (A ⊕ B) + (D � ⊕ C), where A and B are both lower triangular matrices, and D and C are of the forms (41) and (42), respectively.Tus, for any X, Y ∈ F n+m+1 , there exist A XY , D XY , C XY , and B XY such that φ ∈ Der(L n ) if and only if there exist a, a 1 , a 2 , • • • , a n ∈ F and α ∈ F n such that the matrix of φ is A(a, a 1 , • • • , a n , α); (2) φ ∈ LDer(L n ) ifand only if the matrix of φ is a lower triangular matrix; (3) φ is a linear 2-local automorphism of L n if and only if there exist a, a 1 , a 2 , • • • , a n , k ∈ F and α ∈ F n such that the matrix of φ is A(a, a 1 , • • • , a n , α) + kE 11 .