Linear and Multilinear Algebra
ISSN: 0308-1087 (Print) 1563-5139 (Online) Journal homepage: http://www.tandfonline.com/loi/glma20
Biderivations and linear commuting maps on the restricted Cartan-type Lie algebras and Yuan Chang & Liangyun Chen To cite this article: Yuan Chang & Liangyun Chen (2018): Biderivations and linear commuting maps on the restricted Cartan-type Lie algebras and , Linear and Multilinear Algebra, DOI: 10.1080/03081087.2018.1465525 To link to this article: https://doi.org/10.1080/03081087.2018.1465525
Published online: 25 Apr 2018.
Submit your article to this journal
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=glma20
LINEAR AND MULTILINEAR ALGEBRA, 2018 https://doi.org/10.1080/03081087.2018.1465525
Biderivations and linear commuting maps on the restricted Cartan-type Lie algebras W(n; 1) and S(n; 1) Yuan Chang and Liangyun Chen School of Mathematics and Statistics, Northeast Normal University, Changchun, China
ABSTRACT
ARTICLE HISTORY
Let W and S denote the restricted Cartan-type Lie algebras W(n, 1) and S(n, 1) in a field of characteristic p > 3, respectively. We prove that each anti-symmetric biderivation of W and S is inner. As applications of the anti-symmetric biderivations, it is shown that commuting maps on W and S are scalar multiplication maps. Moreover, we prove that the commuting automorphisms and dertivations of W and S are the identity mappings and zero mappings, respectively.
Received 22 September 2017 Accepted 12 April 2018 COMMUNICATED BY
B. Kuzma KEYWORDS
Restricted Cartan-type Lie algebras; biderivations; linear commuting maps AMS SUBJECT CLASSIFICATIONS
17B05; 17B40; 17B50
1. Introduction The research on the commuting map in [1] was brought to our attention, where the author surveyed the development of the theory of commuting maps on associative algebras (or rings) and their applications by discussing the following topics: (1) (2) (3) (4) (5)
Various generalization of the notion of commuting maps; Commuting additive maps; Commuting trace of multiadditive maps; Commuting derivations; Applications of results of commuting maps to different areas, in particular to Lie theory.
Let A be an associative algebra (or a ring). A map ψ : A → A is called a commuting map if ψ(x)x = xψ(x) ∀x ∈ A.
(1.1)
The associative algebra A induces a Lie algebra A− with the product of the elements x, y ∈ A given by commutator [x, y] = xy − yx. Accordingly, we can write [ψ(x), x] = 0 instead of the equality (1.1). It is trivial to show the identity mapping and zero mapping are both commuting maps. CONTACT Liangyun Chen
[email protected]
© 2018 Informa UK Limited, trading as Taylor & Francis Group
2
Y. CHANG AND L. CHEN
In order to determine the commuting maps of the restricted Lie algebras, we need to study their anti-symmetric biderivations. In recent years, biderivations have attracted many scholars’ interests. In [1,2], Brešar showed that all biderivations on commutative prime rings are inner and described the biderivations of semiprime rings. In [3], the authors introduced the notion of biderivation of Lie algebras. In [4–6], the authors proved that all anti-symmetric biderivations on the Schrödinger-Virasoro algebra, the simple generalized Witt algebra and the infinite-dimensional Galilean conformal algebra are inner, respectively. But not all anti-symmetric biderivations are inner. In [7] the authors determined all the skew-symmetric biderivations of W(a, b) and found that there exist non-inner biderivations. The notion of the anti-symmetric biderivation was generated to the super-anti-symmetric biderivation on some super-algebras in [8,9]. Note that almost all the above papers on Lie (super-)algebras assume that the biderivation is (super-)anti-symmetric. In [10,11], the authors characterized the biderivations without the anti-symmetric condition of the finite dimensional complex simple Lie algebra and some W-algebras, and presented some classes of non-inner biderivations. We notice that all the above papers on biderivations assumed the fields have characteristic zero. Hence, we want to consider the biderivations and commuting maps on modular Lie algebras (i.e. Lie algebras under a field of characteristic p). In this paper, we study the anti-symmetric biderivations of the restricted generalized Cartan-type Lie algebras W and S on a field of characteristic p > 3 and prove that each anti-symmetric biderivation is inner. As applications of the anti-symmetric biderivations, we also show that their commuting maps are scalar multiplication maps. Let us recall the definition of the derivation and biderivation of a Lie algebra as follows. Suppose that L is a Lie algebra with Lie product [−, −]. A linear map φ : L → L is called a derivation if it satisfies φ([x, y]) = [φ(x), y] + [x, φ(y)] for all x, y ∈ L. For any x ∈ L, it is easy to see that φx : L → L, y → adx(y) = [x, y], for all y ∈ L, is a derivation of L, which is called an inner derivation. A bilinear map ϕ : L × L → L is called a biderivation of L if it is a derivation with respect to both components, meaning that ϕ([x, y], z) = [x, ϕ(y, z)] + [ϕ(x, z), y],
(1.2)
ϕ(x, [y, z]) = [y, ϕ(x, z)] + [ϕ(x, y), z],
(1.3)
for all x, y, z ∈ L. A bilinear map ϕ is called anti-symmetric if ϕ(x, y) = −ϕ(y, x) for all x, y ∈ L. The biderivation ϕλ : L × L → L for λ ∈ F, satisfying ϕλ (x, y) = λ[x, y], is called inner.
2. Preliminaries Recall some basic notation and results about the restricted Cartan-type Lie algebras W(n; 1) and S(n; 1). Throughout this paper, let F be an algebraically closed field of characteristic p > 3 and N the set of natural numbers including 0. Given a positive integer n ≥ 3 and m = {m1 , . . . , mn } ∈ Nn , we put τ (m) = (pm1 − 1, . . . , pmn − 1), specially τ := τ (1) = (p − 1, . . . , p − 1). Let (n, F) denote the divided (α+β)algebra over F with power an F-basis {x (α) | α ∈ Nn } with the formula x (α) x (β) = α+β and consider the α x
LINEAR AND MULTILINEAR ALGEBRA
3
subalgebra (n; m) := spanF {x (α) | 0 ≤ α ≤ τ (m)}. For convenience, put I = {1, . . . , n} and εi = (δi1 , . . . , δin ), where δij denotes the sign of Kronecker, then we abbreviate x (εi ) to xi . For i ∈ I we consider Di ∈ Der((n; m)) given by Di (x (α) ) = x (α−εi ) . Consider the set n (α) (α) W(n; m) = xi Di | xi ∈ (n; m) . i=1
The algebra W(n; m) of special derivations of (n; m) is called the generalized JacobsonWitt algebra. Lemma 2.1 [12]: The following statements hold. (1) Given x (α) Di , x (β) Dj ∈ W(n; m), then we have [x (α) Di , x (β) Dj ] = x (α) Di (x (β) )Dj − x (β) Dj (x (α) )Di . (2) W(n; m) is simple unless n = 1 and p > 2. (3) W(n; m) is restricted if and only if mi = 1, 1 ≤ i ≤ n. So W(n; 1) = spanF {x (α) Di | 0 ≤ α ≤ τ , i ∈ I} is a finite restricted Witt algebra. For n ≥ 3, W(n; 1) is simple and finitely generated by the set {Di , x (2εi ) Dj | i,
j ∈ I}.
Fix i, j ∈ I, we introduce the mapping Dij : (n; m) → W(n; m), Dij (x (α) ) = Dj (x (α) )Di − Di (x (α) )Dj .
(2.1)
Note that Dii = 0 and Dij = −Dji , for i, j ∈ I. Let S(n, m) = spanF {Dij (x (α) ) | x (α) ∈ (n; m), i, j ∈ I}. Then S(n, m) is a subalgebra of W(n, m), which is called the special algebra. Lemma 2.2 [12]: Suppose n ≥ 3, then the following statements hold, (1) Given Ds1 s2 (x (α) ), Dt1 t2 (x (β) ) ∈ S(n, m), we have [Ds1 s2 (x (α) ), Dt1 t2 (x (β) )] = Ds1 t1 (Ds2 (x (α) )Dt2 (x (β) )) + Ds1 t2 (Ds2 (x (α) )Dt1 (x (β) )) + Ds2 t1 (Ds1 (x (α) )Dt2 (x (β) )) + Ds2 t2 (Ds1 (x (α) )Dt1 (x (β) )). (2) [Dk , Dij (x (α) )] = Dij (Dk (x (α) )) ∀x (α) ∈ (n, m). (3) S(n; m) is simple. (4) S(n; m) is restricted if and only if mi = 1, 1 ≤ i ≤ n. Then S(n; 1) = spanF {Dij (x (α) ) | 0 ≤ α ≤ τ , i, j ∈ I} is a finite restricted special algebra. For convenience, W(n; 1) and S(n; 1) are denoted by W and S, respectively.
4
Y. CHANG AND L. CHEN
3. Anti-symmetric Biderivations on W and S Lemma 3.1: Let L be a restricted Lie algebra over a field F with char(F) = p > 3, ϕ an anti-symmetric biderivation on L. Then [ϕ(x, y), [u, v]] = [[x, y], ϕ(u, v)],
(3.1)
for all x, y, u, v ∈ L. In particular, [ϕ(x, y), [x, y]] = 0.
(3.2)
The proof is similar to that of Lemma 2.3 in [5]. Lemma 3.2: Suppose that ϕ is an anti-symmetric biderivation on a restricted simple Lie algebra L. If [x, y] = 0 for x, y ∈ L, then ϕ(x, y) = 0. Proof: Since L is a simple Lie algebra, then the derived subalgebra [L, L] = L and the center Z(L) = 0. By Lemma 3.1, we have [ϕ(x, y), [u, v]] = [[x, y], ϕ(u, v)] = 0 for any u, v ∈ L. Thus ϕ(x, y) = 0. Lemma 3.3: Suppose that ϕ is an anti-symmetric biderivation on W. For any i ∈ I, there is an element λi ∈ F such that ϕ(Di , x (tεi ) Di ) = λi [Di , x (tεi ) Di ] for 1 ≤ t ≤ p − 1, where λi is independent of the choice of t. Proof: Without loss of generality, we choose a fixed i ∈ I and a nonzero integer t ∈ {1, . . . , p − 1}. Now we suppose ϕ(Di , x (tεi ) Di ) = cα,s x (α) Ds cα,s ∈ F. 0≤α≤τ , s∈I
Since [Dk , Di ] = [Dk , x (tεi ) Di ] = 0 for any k ∈ I \ {i}, we have ϕ(Dk , x (tεi ) Di ) = 0 by Lemma 3.2. So [Dk , ϕ(Di , x (tεi ) Di )] = ϕ([Dk , Di ], x (tεi ) Di ) − [ϕ(Dk , x (tεi ) Di ), Di ] = ϕ(0, x (tεi ) Di ) − [0, Di ] = 0. Then ⎡ ⎤ ⎣Dk , cα,s x (α) Ds ⎦ = cα,s x (α−εk ) Ds = 0. (3.3) 0≤α≤τ , s∈I
0≤α≤τ , s∈I
Considering the coefficients of Ds in the equality (3.3), we have cα,s x (α−εk ) = 0 for any s ∈ I and 0 ≤ α ≤ τ , which implies that there exists some m ∈ F such that α = mεi . Then we can set that ϕ(Di , x (tεi ) Di ) =
p−1 s∈I m=0
cms x (mεi ) Ds cms ∈ F.
(3.4)
LINEAR AND MULTILINEAR ALGEBRA
5
There are two cases to discuss. If t = 1, by the equalities (3.2) and (3.4) we have 0 = [[Di , xi Di ], ϕ(Di , xi Di )] = [Di ,
p−1
cms x (mεi ) Ds ] =
s∈I m=0
p−1
cms x ((m−1)εi ) Ds ,
s∈I m=1
which implies that cms = 0 for any m ∈ {1, . . . , p − 1} and s ∈ I. Then we can suppose that ϕ(Di , xi Di ) =
cs Ds cs ∈ F.
s∈I
Since [xk Dk , xi Di ] = 0 for any k ∈ I, we have ϕ(xk Dk , xi Di ) = 0 by Lemma 3.2. Dk , Di ], xi Di ) − [ϕ(xk Dk , xi Di ), Di ] = It is immediate that [xk Dk , ϕ(Di , xi Di )] = ϕ([xk ϕ(0, xi Di ) − [0, Di ] = 0. And the equality [xk Dk , s∈I cs Ds ] = −ck Dk (xk )Dk = 0 means that cs = 0 for any s ∈ I \ {i}. So we have that ϕ(Di , xi Di ) = ci [Di , xi Di ] ci ∈ F.
(3.5)
If 2 ≤ t ≤ p − 1, in order to differ the case t = 1, we suppose ϕ(Di , x
(tεi )
Di ) =
p−1
bms x (mεi ) Ds bms ∈ F.
s∈I m=0
Due to the equality [ϕ(Di , x (tεi ) Di ), [Di , xi Di ]] = [[Di , x (tεi ) Di ], ϕ(Di , xi Di )], by the equality (3.5) we have −
p−1
bms x ((m−1)εi ) Ds = −ci x ((t−2)εi ) Di .
s∈I m=1
Considering the equality (3.6), we can get that s = i, m = t − 1 and bmi = ci . Then for any t ∈ {2, . . . , p − 1}, we have ϕ(Di , x (tεi ) Di ) = ci [Di , x (tεi ) Di ],
(3.6)
where ci is in the equality (3.5) of the case t = 1. Thus, we set λi = ci and have ϕ(Di , x (tεi ) Di ) = λi [Di , x (tεi ) Di ] for 1 ≤ t ≤ p − 1, where λi is independent of the choice of t. Lemma 3.4: Suppose that ϕ is an anti-symmetric biderivation on W. There is an element λ ∈ F such that ϕ(Di , x (tεi ) Dj ) = λ[Di , x (tεi ) Dj ] for any i, j ∈ I and 1 ≤ t ≤ p − 1, where λ is independent of the choice of i, j and t. Proof: Fix i, j ∈ I and a nonzero integer t ∈ {1, . . . , p − 1}. We suppose ϕ(Di , x (tεi ) Dj ) =
0≤α≤τ , s∈I
cαs x (α) Ds cαs ∈ F.
6
Y. CHANG AND L. CHEN
Since [Dk , x (tεi ) Dj ] = 0 for any k ∈ I \ {i}, by Lemma 3.2 we have ϕ(Dk , x (tεi ) Dj ) = 0. It is obvious that [Dk , ϕ(Di , x (tεi ) Dj )] = ϕ([Dk , Di ], x (tεi ) Dj ) − [ϕ(Dk , x (tεi ) Dj ), Di ] = ϕ(0, x (tεi ) Di ) − [0, Di ] = 0. Then [Dk ,
cαs s x (αs ) Ds ] =
0≤αs ≤τ , s∈I
cαs s x (αs −εk ) Ds .
(3.7)
0≤αs ≤τ , s∈I
Considering the coefficients of Ds in the equality (3.7), we have cαs x (α−εk ) = 0 for any s ∈ I and 0 ≤ α ≤ τ , which implies that there exists some m ∈ F such that α = mεi . Then we can set that ϕ(Di , x
(tεi )
Dj ) =
p−1
cms x (mεi ) Ds cms ∈ F.
(3.8)
s∈I m=0
There are also two cases to discuss. Firstly, we consider the case t = 1. We have [[Di , xi Di ], ϕ(Di , xi Dj )] = [ϕ(Di , xi Di ), [Di , xi Dj ]] by the equality (3.1), then p−1
⎡ cms x ((m−1)εi ) Ds = ⎣Di ,
s∈I m=1
p−1
⎤ cms x (mεi ) Ds ⎦ = [λi [Di , xi Di ], Dj ] = 0,
s∈I m=0
which implies that cms = 0 for any m ∈ {1, . . . , p − 1} and s ∈ I. Then we can set that ϕ(Di , xi Dj ) =
cs Ds cs ∈ F.
(3.9)
s∈I
Since ϕ(Di , xi Dj ) = ϕ(Di , [xi Di , xi Dj ]) = [ϕ(Di , xi Di ), xi Dj ] + [xi Di , ϕ(Di , xi Dj )], by Lemma 3.3 and the equality (3.9) we have
cs Ds = [λi Di , xi Dj ] + [xi Di ,
s∈I
cs Ds ] = λi Dj − ci Di ,
s∈I
which implies cj = λi and cs = 0 for s ∈ I\{j}. Then we can get ϕ(Di , xi Dj ) = λi [Di , xi Dj ], where λi comes from Lemma 3.3. If 2 ≤ t ≤ p − 1, we suppose that ϕ(Di , x (tεi ) Dj ) =
p−1 s∈I m=0
bms x (mεi ) Ds bms ∈ F.
(3.10)
LINEAR AND MULTILINEAR ALGEBRA
7
It is immediate to obtain the equation ϕ(Di , x (tεi ) Dj ) = ϕ Di , x (tεi ) Dj , xj Dj
= ϕ(Di , x (tεi ) Dj ), xj Dj − x (tεi ) Dj , ϕ(Di , xj Dj ) ⎤ ⎡ p−1
bms x (mεi ) Ds , xj Dj ⎦ − x (tεi ) Dj , 0 =⎣ s∈I m=0 p−1
=
bmj x (mεi ) Dj .
m=0
Using Lemma 3.1, we have [[Dj , xj Di ], ϕ(Di , x (tεi ) Dj )] = [ϕ(Dj , xj Di ), [Di , x (tεi ) Dj ]], then by the equality (3.10), it is easy to get that p−1
bmj x ((m−1)εi ) Dj = λj x ((t−2)εi ) Dj .
(3.11)
m=1
Comparing the coefficients of Dj in the equality (3.11), we get m = t − 1 and bmj = λj . Then for any 2 ≤ t ≤ p − 1, ϕ(Di , x (tεi ) Dj ) = λj [Di , x (tεi ) Dj ], where λj comes from Lemma 3.3. In the following, we want to prove λi = λj for any i, j ∈ I. Fix i, j ∈ I. For any k ∈ I \ {i, j} and t ∈ {2, . . . , p − 1}, we have λj x ((t−2)εk ) Dj = [[Di , xi Dk ], ϕ(Dk , x (tεk ) Dj )] = [ϕ(Di , xi Dk ), [Dk , x (tεk ) Dj ]] = λi x ((t−2)εk ) Dj . So we get that λi = λj for any i, j ∈ I. Thus, we set λ = λ 1 = λ 2 = · · · = λn . It is obvious that λ is independent of the choice of i, j and t. Theorem 3.5: Every anti-symmetric biderivation ϕ of W is inner. Proof: By Lemma 3.4, there is an element λ ∈ F such that ϕ(Di , x (tεi ) Dj ) = λ[Di , x (tεi ) Dj ] for any i, j ∈ I. For any x, y ∈ W and 1 ≤ t ≤ p − 2, we have [ϕ(x, y), x (tεi ) Dj ] = [ϕ(x, y), [Di , x (t+1)εi Dj ]] = [[x, y], ϕ(Di , x (t+1)εi Dj )] = [λ[x, y], x (tεi ) Dj ]. So for any i, j ∈ I and 1 ≤ t ≤ p − 2, we have [ϕ(x, y) − λ[x, y], x (tεi ) Dj ] = 0.
(3.12)
Since the generator set of the restricted simple Lie algebra W is {Di , x (2εi ) Dj |i, j ∈ I}, whose elements satisfy the equality (3.12), we get that ϕ(x, y) − λ[x, y] ∈ Z(W) = {0}. So ϕ(x, y) = λ[x, y] for any x, y ∈ W. It is obvious that ϕ is an inner biderivation. Lemma 3.6 [12]: If p > 2 and n ≥ 3, the restricted special algebra S is generated by {Dij (xi ), Dij (x (εi +εk +εl ) ) | i, j, k, l ∈ I}.
8
Y. CHANG AND L. CHEN
Lemma 3.7: If p > 3 and n ≥ 3, the restricted special algebra S is generated by the set {Dij (x (tεi ) ) | i, j ∈ I, t = 1, 3}. Proof: If we want to get this conclusion, it just needs to prove that every element of the generator set in Lemma 3.6 is generated by the set {Dij (x (tεi ) ) | i, j ∈ I, t = 1, 3}. For convenience, let A denote the set {Dij (xi ), Dij (x (εi +εk +εl ) ) | i, j, k, l ∈ I}, B the set {Dij (x (tεi ) ) | i, j ∈ I, t = 1, 3}. For i, j ∈ I and i = k = l, it is obvious that Dij (xi ) and Dij (x (εi +εk +εl ) ) are in B. For j, l ∈ I \ {i} and k = i, we have the equality Dij (x (2εi +εl ) ) = −[Dij (x (3εi ) ), [Dil (xi )), Dil (x (3εl ) )]]. For j, k, l ∈ I \ {i} and k = l, we have the equality Dij (x (εi +2εk ) ) = [[Dik (xk ), Dij (x (3εi ) )], Dik (x (3εk ) )]. For j, k, l ∈ I \ {i} and k = l, we have the equality Dij (x (εi +εk +εl ) ) = [Dik (x (εi +2εk ) ), [Dil (xi ), Djl (x (3εl ) )]]. The proof is complete. Lemma 3.8: Suppose that ϕ is an anti-symmetric biderivation on S. There is an element μ ∈ F such that ϕ(Dij (xj ), Dij (x (tεi ) )) = μ[Dij (xj ), Dij (x (tεi ) )] for any i, j ∈ I and 1 ≤ t ≤ p − 1, where μ is independent of the choice of i, j and t. Proof: When t = 1, [Dij (xj ), Dij (xi )] = 0 for any i, j ∈ I, then by Lemma 3.2 we have ϕ(Dij (xj ), Dij (xi )) = 0. So the conclusion holds for t = 1. When 2 ≤ t ≤ p − 1, we can suppose
ϕ(Dij (xj ), Dij (x (tεi ) )) =
cαrs Drs (x (α) ) cαrs ∈ F.
1≤r
