Biderivations and commuting linear maps on Hom-Lie algebras
aa r X i v : . [ m a t h . R A ] M a y BIDERIVATIONS AND COMMUTING LINEAR MAPSON HOM-LIE ALGEBRAS
BING SUN, YAO MA AND LIANGYUN CHENA
BSTRACT . The purpose of this paper is to determine skew-symmetric biderivations Bider s ( L , V ) and commuting linear maps Com ( L , V ) on a Hom-Lie algebra ( L , α ) having their ranges in an ( L , α ) -module ( V , ρ , β ) , which are both closely related to Cent ( L , V ) , the centroid of ( V , ρ , β ) .Specifically, under appropriate assumptions, every δ ∈ Bider s ( L , V ) is of the form δ ( x , y ) = β − γ ([ x , y ]) for some γ ∈ Cent ( L , V ) , and Com ( L , V ) coincides with Cent ( L , V ) . Besides, wegive the algorithm for describing Bider s ( L , V ) and Com ( L , V ) respectively, and provide severalexamples. Keywords:
Biderivation, commuting linear map, centroid, Hom-Lie algebra.
MSC(2020):
1. I
NTRODUCTION
The notion of biderivations appeared in different areas. Maksa used biderivations to study realHilbert space [25]. Vukman investigated symmetric biderivations in the prime and semiprimerings [30]. The well-known result that every biderivation on a noncommutative prime ring A isof the form λ [ x , y ] for some λ belonging to the extended centroid of A , was discovered indepen-dently by Bresar et al [5], Skosyrskii [28], and Farkas and Letzter [13], where biderivations wereconnected with noncommutative Jordan algebras by Skosyrskii and with Poisson algebras byFarkas and Letzter, respectively. Besides their wide applications, biderivations are interestingin their own right and have been introduced to Lie algebras [32], which were studied by manyauthors recently. In particular, biderivations are closely related to the theory of commuting lin-ear maps that has a long and rich history, and we refer to the survey [4] for the development ofcommuting maps and their applications. It is worth mentioning that Breˇsar and Zhao consid-ered a general but simple approach for describing biderivations and commuting linear maps ona Lie algebra L that having their ranges in an L -module [6], which covered most of the resultsin [11, 14, 22, 29, 31], and inspires us to generalize their method to Hom-Lie algebras.Hom-Lie algebras are a generalization of Lie algebras, the notion of which was initiallyintroduced in [15], motivated by the study of quantum deformations or q -deformations of theWitt and the Virasoro algebras via twisted derivations, after several investigations of variousquantum deformations of Lie algebras [2, 8, 12, 16, 18]. Due to the close relation to discreteand deformed vector fields and differential calculus, this kind of algebraic structures have beenstudied extensively during the last decade, which we mention here just a few: representations, *Corresponding author (Y. Ma): [email protected] by NNSF of China (Nos. 11901057, 11801066, 11771069 and 11771410), NSF of Jilin province(No. 20170101048JC), Natural Science Foundation of Changchun Normal University. (co)homology and deformations of Hom-Lie algebras [1,3,23,27,34]; structure theory of simpleand semisimple Hom-Lie algebras [9,17,33]; Hom-Lie structures on Kac-Moody algebras [24];extensions of Hom-Lie algebras [7, 19]; geometric generalization of Hom-Lie algebras [21, 26];integration of Hom-Lie algebras [20].Hence it would be natural to generalize known theories from Lie algebras to Hom-Lie al-gebras, and we are interested in determining biderivations and commuting linear maps on aHom-Lie algebra follows from [6]. The paper is organized as follows.In Section 2, after recalling some preliminaries on Hom-Lie algebras, we introduce the notionof Hom-Lie algebra module homomorphisms, and give the Schur’s lemma for adjoint Hom-Liealgebra modules.In Section 3, we define biderivations on a Hom-Lie algebra ( L , α ) having their ranges in an ( L , α ) -module ( V , ρ , β ) as well as the centroid of ( V , ρ , β ) , and derive that skew-symmetricbiderivations arise from the centroid provided that ( L , α ) is perfect, α is surjective, β is in-vertible and Z V ( L ) = { } (see Theorem 3.6). In particular, every skew-symmetric biderivationon a simple ( L , α ) with α invertible having its range in the α k -adjoint ( L , α ) -module is of theform λα k ([ x , y ]) (see Theorem 3.7). We also give an algorithm to find all skew-symmetricbiderivations and apply it to several examples.In Section 4, we give the definition of commuting linear maps from a Hom-Lie algebra ( L , α ) to an ( L , α ) -module ( V , ρ , β ) , which coincides with the centroid of ( V , ρ , β ) if α is surjective, β is invertible and Z V ( L ′ ) = { } (see Theorem 4.3). An algorithm to describe all commutinglinear maps on ( L , α ) is also provided.Throughout this paper, all the vector spaces are over a fixed field F such that char ( F ) = HE S CHUR ’ S LEMMA FOR ADJOINT MODULES OVER A H OM -L IE ALGEBRA
We begin with some definitions concerning Hom-Lie algebras.
Definition 2.1. [27] A
Hom-Lie algebra ( L , α ) is an F -vector space L endowed with a bilinearmap [ − , − ] : L × L → L and a linear homomorphism α : L → L satisfying that for any x , y , z ∈ L , [ x , y ] = − [ y , x ] , (skew-symmetry) [ α ( x ) , [ y , z ]] + [ α ( y ) , [ z , x ]] + [ α ( z ) , [ x , y ]] = . (Hom-Jacobi identity)In particular, a Hom-Lie algebra ( L , α ) is called multiplicative , if α is an algebra homomor-phism, i.e., α ([ x , y ]) = [ α ( x ) , α ( y )] , for all x , y ∈ L .For the structural study of ( L , α ) in this paper, we need the stability properties that the multi-plicative condition of α offers, so in the sequel, we will say Hom-Lie algebras when referringto multiplicative Hom-Lie algebras .Moreover, a subspace I of ( L , α ) is called an ideal , if [ IL ] ⊆ I and α ( I ) ⊆ I . If ( L , α ) has noproper ideals and is not abelian, then we call it simple . ( L , α ) is called perfect if L ′ : = [ L , L ] = L . IDERIVATIONS AND COMMUTING LINEAR MAPS ON HOM-LIE ALGEBRAS 3
The center of a Hom-Lie algebra ( L , α ) , denoted by Z ( L ) , is the set { z ∈ L | [ z , L ] = } . Formore details of these definitions, we refer to [7, 9, 24]. Definition 2.2. [27, 34] For a Hom-Lie algebra ( L , α ) , a triple ( V , ρ , β ) consisting of a vectorspace V , a linear map ρ : L → End ( V ) and β ∈ End ( V ) is said to be a representation of ( L , α ) or an ( L , α ) - module , if for all x , y ∈ L , the following equalities are satisfied β ◦ ρ ( x ) = ρ ( α ( x )) ◦ β , (2.1) ρ ([ x , y ]) ◦ β = ρ ( α ( x )) ◦ ρ ( y ) − ρ ( α ( y )) ◦ ρ ( x ) . (2.2)For brevity of notation, we usually put xv = ρ ( x )( v ) , ∀ x ∈ L , v ∈ V , just like the case in Liealgebras. A subspace W of V is called a submodule of ( V , ρ , β ) if W is both β -invariant and L -invariant, i.e., β ( W ) ⊂ W and xW ⊂ W , ∀ x ∈ L . Example 2.3. (1) For any integer k , ( L , ad k , α ) with ad k ( x )( y ) : = [ α k ( x ) , y ] is an ( L , α ) -moduleby [27, Lemma 6.2], called the α k -adjoint ( L , α ) -module. Note that α need to be invertible incase k is negative.(2) If ( V , ρ , β ) is an ( L , α ) -module, then ( V , ρ k : = ρα k , β ) is also an ( L , α ) -module, whichfollows from the equalities that βρ k ( x )( v ) = βρ ( α k ( x ))( v ) ( . ) = ρ ( α k + ( x )) β ( v ) = ρ k ( α ( x )) β ( v ) , ρ k ([ x , y ]) β ( v ) = ρα k ([ x , y ]) β ( v ) = ρ ([ α k ( x ) , α k ( y )]) β ( v ) (by (2.2)) = ρ ( α k + ( x )) ρ ( α k ( y ))( v ) − ρ ( α k + ( y )) ρ ( α k ( x ))( v )= ρ k ( α ( x )) ρ k ( y )( v ) − ρ k ( α ( y )) ρ k ( x )( v ) . Remark 2.4. If I is a submodule of ( L , ad k , α ) and α k ( L ) = L , then I is an ideal of L . In fact, α ( I ) ⊂ I and [ L , I ] = [ α k ( L ) , I ] = ad k ( L )( I ) ⊂ I . Next we introduce the notion of homomorphisms between representations of a Hom-Liealgebra ( L , α ) and show that the Schur’s Lemma holds for finite-dimensional adjoint ( L , α ) -modules. Definition 2.5.
Let ( V , ρ , β ) and ( V , ρ , β ) be two ( L , α ) -modules. A linear map f : V → V is said to be a homomorphism of ( L , α ) -modules, if β ◦ f = f ◦ β and f ◦ ρ ( x ) = ρ ( α ( x )) ◦ f , ∀ x ∈ L , or, in terms of two commutative diagrams, V f (cid:15) (cid:15) β / / V f (cid:15) (cid:15) V β / / V , V f (cid:15) (cid:15) ρ ( x ) / / V f (cid:15) (cid:15) V ρ α ( x ) / / V . Note that the identity map is no longer a natural homomorphism on a Hom-Lie algebra mod-ule in general, which is actually twisted by α in some way. BING SUN, YAO MA AND LIANGYUN CHEN
Example 2.6.
For any integers k , s , it follows that α s + : ( L , ad k , α ) → ( L , ad k + s , α ) is a homo-morphism of ( L , α ) -modules, since α ◦ α s + = α s + ◦ α and α s + ad k ( x )( y ) = α s + [ α k ( x ) , y ] = [ α k + s + ( x ) , α s + ( y )] = ad k + s ( α ( x )) α s + ( y ) . In particular, α = id L is a homomorphism from ( L , ad , α ) to ( L , ad − , α ) in the case that α isinvertible.By direct calculations, we have the following property. Proposition 2.7.
Let ( V , ρ , β ) and ( V , ρ , β ) be two ( L , α ) -modules and f : V → V amodule homomorphism. Then ker f is a submodule of ( V , ρ , β ) . Theorem 2.8 (Schur’s lemma) . Let ( L , α ) be a Hom-Lie algebra with α invertible, which iscalled regular in [27]. Let the base field F be algebraically closed. If ( L , α ) is simple andfinite-dimensional, and f : ( L , ad k , α ) → ( L , ad k + s , α ) is a homomorphism of ( L , α ) -modules,then f = λα s + , for some λ ∈ F . Proof.
We know that f α − s − : L → L is a linear map. Then use the facts that ( L , α ) is finite-dimensional and F is algebraically closed to find an eigenvector 0 = x ∈ L of f α − s − for someeigenvalue, say λ ∈ F , i.e., f α − s − ( x ) = λ x . Since f ◦ α = α ◦ f , f ( x ) = λα s + ( x ) .By Example 2.6, λα s + : ( L , ad k , α ) → ( L , ad k + s , α ) is an ( L , α ) -module homomorphism,which implies that f − λα s + is also an ( L , α ) -module homomorphism. Then ker ( f − λα s + ) is a non-zero submodule of ( L , ad k , α ) , so is a non-zero ideal of ( L , α ) , by Remark 2.4. Henceker ( f − λα s + ) = L by the simplicity of ( L , α ) , that is, f = λα s + . (cid:3)
3. B
IDERIVATIONS OF H OM -L IE ALGEBRAS
Let ( V , ρ , β ) be an ( L , α ) -module. A bilinear map δ : L × L → V is said to be a biderivation if βδ ( x , y ) = δ ( α ( x ) , α ( y )) , (3.1) δ ( α ( z ) , [ x , y ]) = α ( x ) δ ( z , y ) − α ( y ) δ ( z , x ) , (3.2) δ ([ x , y ] , α ( z )) = α ( x ) δ ( y , z ) − α ( y ) δ ( x , z ) , (3.3)for all x , y , z ∈ L . We say that a biderivation δ is skew-symmetric , if(3.4) δ ( x , y ) = − δ ( y , x ) . If this is the case, Eqs. (3.2) and (3.3) coincide. Hence, a bilinear map δ : L × L → V is askew-symmetric biderivation provided that δ satisfies (3.4), (3.1), and any one of (3.2) or (3.3).Denote by Bider ( L , V ) (resp. Bider s ( L , V ) ) the set of all biderivations (resp. skew-symmetricbiderivations) δ : L × L → V . In particular, we write Bider s ( L , ad k ) as the set of all skew-symmetric biderivations on ( L , α ) for the adjoint module ( L , ad k , α ) . IDERIVATIONS AND COMMUTING LINEAR MAPS ON HOM-LIE ALGEBRAS 5
Using the Hom-Jacobi identity, it is easy to see that a skew-symmetric biderivation δ : L × L → V is a generalization of the Hom-Lie bracket [ − , − ] : L × L → L , where the last L is viewedas ( L , ad , α ) .Now we would like to explain the terminology “biderivation”. Note that the notion of deriva-tions on a Hom-Lie algebra ( L , α ) was introduced in [27], where a linear map D : L → L iscalled an α k - derivation of ( L , α ) , if D ◦ α = α ◦ D and D [ x , y ] = [ α k ( x ) , D ( y )] − [ α k ( y ) , D ( x )] , ∀ x , y ∈ L . Under the above definition, for any nonnegative integer k and any z ∈ L satisfying α ( z ) = z , alinear map D k ( z ) : L → L defined by D k ( z )( x ) = [ z , α k ( x )] , ∀ x ∈ L becomes an α k + -derivation, which implies that [ z , − ] and [ − , z ] are α -derivations, then it isreasonable to call [ − , − ] a biderivation. Finally, its generalization δ can be called by the samename.We’ve already seen that [ − , − ] ∈ Bider s ( L , ad ) . To provide more examples of biderivations,we define the centroid of ( V , ρ , β ) asCent ( L , V ) = { γ : L → V | γ ([ x , y ]) = α ( x ) γ ( y ) and β ◦ γ = γ ◦ α } . Clearly, Cent ( L , V ) is just the space of ( L , α ) -module homomorphisms from ( L , ad , α ) to ( V , ρ , β ) . In particular, we write Cent ( L , ad k ) as the centroid of the adjoint module ( L , ad k , α ) . Example 3.1.
Let ( V , ρ , β ) be a representation of a Hom-Lie algebra ( L , α ) with β invertible.If γ ∈ Cent ( L , V ) , then δ ( x , y ) = β − γ ([ x , y ]) , ∀ x , y ∈ L , is a skew-symmetric biderivation.In fact, δ is clearly skew-symmetric; (3.1) holds since δ ( α ( x ) , α ( y )) = β − γ ([ α ( x ) , α ( y )]) = β − γα ([ x , y ]) = γ ([ x , y ]) = βδ ( x , y ) , ∀ x , y ∈ L ;(3.3) holds since δ ([ x , y ] , α ( z )) = β − γ ([[ x , y ] , α ( z )]) (by Hom-Jacobi identity) = β − γ ([ α ( x ) , [ y , z ]]) − β − γ ([ α ( y ) , [ x , z ]])= β − ( α ( x ) γ ([ y , z ]) − β − α ( y ) γ ([ x , z ]) (by (2.1)) = α ( x ) β − γ ([ y , z ]) − α ( y ) β − γ ([ x , z ])= α ( x ) δ ( y , z ) − α ( y ) δ ( x , z ) . From the example above we could see that for an ( L , α ) -module ( V , ρ , β ) such that β isinvertible, each γ ∈ Cent ( L , V ) induces a skew-symmetric biderivation δ ( x , y ) = β − γ ([ x , y ]) .Later, we will show that, under appropriate assumptions, all biderivations are of this form, i.e.,Theorem 3.6, to prove which we need the following technical lemmas. BING SUN, YAO MA AND LIANGYUN CHEN
In Lemmas 3.2, 3.3, 3.4 and 3.5, ( V , ρ , β ) is a representation of a Hom-Lie algebra ( L , α ) and δ ∈ Bider s ( L , V ) . Moreover, we denote by Z V ( S ) = { v ∈ V | sv = , ∀ s ∈ S } the subspace of V that is killed by any subset S of L . Note that for an adjoint module ( L , ad k , α ) with α surjective, Z L ( L ) is an ideal of ( L , α ) that coincides with Z ( L ) , the center of ( L , α ) . Lemma 3.2. β ([ x , y ] δ ( z , w ) + [ z , w ] δ ( x , y )) = , ∀ x , y , z , w ∈ L . Proof.
Consider the following quaternary linear map ϕ : L × L × L × L → V , ( x , y , z , w ) [ x , y ] δ ( z , w ) + [ z , w ] δ ( x , y ) . Then it suffices to prove βϕ ( x , y , z , w ) = , ∀ x , y , z , w ∈ L .Clearly, ϕ satisfies ϕ ( x , y , z , w ) = ϕ ( z , w , x , y ) and(3.5) ϕ ( x , y , z , w ) = − ϕ ( y , x , z , w ) by the definition.We’ll prove βϕ ( x , y , z , w ) = βϕ ( z , y , x , w ) , by computing δ ([ x , y ] , [ z , w ]) in two ways. For any x , y , z , w ∈ L , we have δ ( α ([ x , y ]) , α ([ z , w ]))= δ ([ α ( x ) , α ( y )] , α ([ z , w ]))= α ( x ) δ ( α ( y ) , [ z , w ]) − α ( y ) δ ( α ( x ) , [ z , w ])= α ( x )( α ( z ) δ ( y , w ) − α ( w ) δ ( y , z )) − α ( y )( α ( z ) δ ( x , w ) − α ( w ) δ ( x , z ))= α ( x ) α ( z ) δ ( y , w ) − α ( x ) α ( w ) δ ( y , z ) − α ( y ) α ( z ) δ ( x , w ) + α ( y ) α ( w ) δ ( x , z ) . On the other hand, δ ( α ([ x , y ]) , α ([ z , w ]))= δ ( α ([ x , y ]) , [ α ( z ) , α ( w )])= α ( z ) δ ([ x , y ] , α ( w )) − α ( w ) δ ([ x , y ] , α ( z ))= α ( z )( α ( x ) δ ( y , w ) − α ( y ) δ ( x , w )) − α ( w )( α ( x ) δ ( y , z ) − α ( y ) δ ( x , z ))= α ( z ) α ( x ) δ ( y , w ) − α ( z ) α ( y ) δ ( x , w ) − α ( w ) α ( x ) δ ( y , z ) + α ( w ) α ( y ) δ ( x , z ) . Comparing above relations and using (2.2), we have [ α ( x ) , α ( z )] βδ ( y , w ) + [ α ( y ) , α ( w )] βδ ( x , z ) = [ α ( y ) , α ( z )] βδ ( x , w ) + [ α ( x ) , α ( w )] βδ ( y , z ) . By (2.1), it follows that βϕ ( x , z , y , w ) = βϕ ( y , z , x , w ) , which implies(3.6) βϕ ( x , y , z , w ) = βϕ ( z , y , x , w ) , ∀ x , y , z , w ∈ L . Then(3.7) βϕ ( x , y , z , w ) ( . ) = βϕ ( z , y , x , w ) ( . ) = − βϕ ( y , z , x , w ) ( . ) = − βϕ ( x , z , y , w ) . IDERIVATIONS AND COMMUTING LINEAR MAPS ON HOM-LIE ALGEBRAS 7
It follows that βϕ ( x , y , z , w ) ( . ) = − βϕ ( y , x , z , w ) ( . ) = βϕ ( y , z , x , w ) ( . ) = − βϕ ( z , y , x , w ) ( . ) = − βϕ ( x , y , z , w ) . The proof is complete. (cid:3)
Lemma 3.3. (cid:9) x , y , z δ ([ x , y ] , α ( z )) = ( α ( z ) δ ( x , y ) − δ ( α ( z ) , [ x , y ])) , ∀ x , y , z ∈ L, where (cid:9) x , y , z denotes the sum over cyclic permutations of x , y , z.Proof. Note that (cid:9) x , y , z δ ([ x , y ] , α ( z ))= δ ([ x , y ] , α ( z )) + δ ([ y , z ] , α ( x )) + δ ([ z , x ] , α ( y ))= α ( x ) δ ( y , z ) − α ( y ) δ ( x , z ) + α ( y ) δ ( z , x ) − α ( z ) δ ( y , x ) + α ( z ) δ ( x , y ) − α ( x ) δ ( z , y )= ( α ( x ) δ ( y , z ) − α ( y ) δ ( x , z ) + α ( z ) δ ( x , y ))= ( α ( z ) δ ( x , y ) − δ ( α ( z ) , [ x , y ])) . Hence the lemma follows. (cid:3)
Lemma 3.4. If β is invertible, then δ ( α ( u ) , [ x , y ]) − α ( u ) δ ( x , y ) ∈ Z V ( L ′ ) , ∀ x , y , u ∈ L . Proof.
By Lemma 3.2 and (2.1), we have [ α ( x ) , α ( y )] βδ ( z , w ) + [ α ( z ) , α ( w )] βδ ( x , y ) = . Then 0 = (cid:9) x , u , y ([[ α ( x ) , α ( u )] , α ( y )] βδ ( z , w ) + [ α ( z ) , α ( w )] βδ ([ x , u ] , α ( y )))= (cid:9) x , u , y [[ α ( x ) , α ( u )] , α ( y )] βδ ( z , w )+ (cid:9) x , u , y [ α ( z ) , α ( w )] βδ ([ x , u ] , α ( y ))= − [ α ( z ) , α ( w )] β ( (cid:9) x , y , u δ ([ x , y ] , α ( u )))= β ([ z , w ]( δ ( α ( u ) , [ x , y ]) − α ( u ) δ ( x , y )) , where the last equality uses Lemma 3.3 and (2.1). Hence the lemma follows from the fact that β is invertible and char F = (cid:3) Lemma 3.5. If β is invertible and ( L , α ) is a perfect Hom-Lie algebra with α surjective, then δ ( z , [ x , y ]) = z δ ( x , y ) , ∀ x , y , z ∈ L . Proof.
Since ( L , α ) is a perfect Hom-Lie algebra with α surjective, for any x , y , z ∈ L , there exist u i , v i ( ≤ i ≤ m ) , s , t ∈ L such that z = ∑ mi = [ α ( u i ) , α ( v i )] and [ x , y ] = [ α ( s ) , α ( t )] . Then δ ( z , [ x , y ]) − z δ ( x , y )= m ∑ i = ( δ ([ α ( u i ) , α ( v i )] , [ α ( s ) , α ( t )]) − [ α ( u i ) , α ( v i )] δ ( α ( s ) , α ( t )))= m ∑ i = ( δ ([ α ( u i ) , α ( v i )] , α ([ s , t ])) − [ α ( u i ) , α ( v i )] βδ ( s , t )) BING SUN, YAO MA AND LIANGYUN CHEN = m ∑ i = ( α ( u i ) δ ( α ( v i ) , [ s , t ]) − α ( v i ) δ ( α ( u i ) , [ s , t ]) − α ( u i ) α ( v i ) δ ( s , t ) + α ( v i ) α ( u i ) δ ( s , t ))= m ∑ i = ( α ( u i )( δ ( α ( v i ) , [ s , t ]) − α ( v i ) δ ( s , t )) − α ( v i )( δ ( α ( u i ) , [ s , t ]) − α ( u i ) δ ( s , t ))) . By Lemma 3.4, we obtain that the last equation above is equal to zero. (cid:3)
Theorem 3.6.
Let ( L , α ) be a perfect Hom-Lie algebra with α surjective and ( V , ρ , β ) an ( L , α ) -module with β invertible such that Z V ( L ) = . Then for all δ ∈ Bider s ( L , V ) , thereexists γ ∈ Cent ( L , V ) such that δ ( x , y ) = β − γ ([ x , y ]) , ∀ x , y ∈ L.Proof.
Since L = L ′ , for any z ∈ L , we have z = ∑ mi = [ z ′ i , z ′′ i ] for some z ′ i , z ′′ i ∈ L . Define γ : L → V by γ (cid:16) m ∑ i = [ z ′ i , z ′′ i ] (cid:17) = m ∑ i = βδ ( z ′ i , z ′′ i ) . Then it suffices to show that γ is well-defined as well as γ ([ x , y ]) = α ( x ) γ ( y ) , ∀ x , y ∈ L . Indeed,If ∑ mi = [ z ′ i , z ′′ i ] =
0, then for any u ∈ L ,0 = δ (cid:0) u , m ∑ i = [ z ′ i , z ′′ i ] (cid:1) = m ∑ i = δ ( u , [ z ′ i , z ′′ i ]) = u m ∑ i = δ ( z ′ i , z ′′ i ) , and hence ∑ mi = δ ( z ′ i , z ′′ i ) = Z V ( L ) =
0. So ∑ mi = βδ ( z ′ i , z ′′ i ) =
0, which impliesthat γ is well-defined. Furthermore, suppose y ′ i , y ′′ i ∈ L such that y = ∑ ki = [ y ′ i , y ′′ i ] , then γ ([ x , y ]) = βδ ( x , y ) = δ ( α ( x ) , α ( y )) = δ (cid:0) α ( x ) , k ∑ i = [ α ( y ′ i ) , α ( y ′′ i )] (cid:1) = k ∑ i = α ( x ) δ ( α ( y ′ i ) , α ( y ′′ i ))= k ∑ i = α ( x ) βδ ( y ′ i , y ′′ i ) = k ∑ i = α ( x ) γ ([ y ′ i , y ′′ i ]) = α ( x ) γ ( y ) . Therefore, γ ∈ Cent ( L , V ) . (cid:3) Now we turn to Bider s ( L , ad k ) , which is easy to determine in the case that ( L , α ) is centerlessand perfect, or in particular, ( L , α ) is simple. Theorem 3.7. If ( L , α ) is a centerless and perfect Hom-Lie algebra with α invertible, thenevery skew-symmetric biderivation δ : L × L → ( L , ad k , α ) is of the form δ ( x , y ) = α − γ ([ x , y ]) , where γ ∈ Cent ( L , ad k ) . Moreover, if ( L , α ) is simple and finite-dimensional, then δ ( x , y ) = λα k ([ x , y ]) , for some λ ∈ F . Proof.
The first assertion is an immediate corollary to Theorem 3.6 when ( L , α ) is centerlessand perfect. Now let ( L , α ) be simple and finite-dimensional.If δ =
0, then take λ = δ =
0. Using Theorem 3.6, there exists γ ∈ Cent ( L , ad k ) such that δ ( x , y ) = α − γ ([ x , y ]) .Note that γ : ( L , ad , α ) → ( L , ad k , α ) is a module homomorphism over a finite-dimensional IDERIVATIONS AND COMMUTING LINEAR MAPS ON HOM-LIE ALGEBRAS 9 simple Hom-Lie algebra ( L , α ) together with the fact that α is invertible. By Theorem 2.8, γ = λα k + and δ ( x , y ) = λα k ([ x , y ]) . (cid:3) In general, on a Hom-Lie algebra ( L , α ) such that α is invertible, we would like to give analgorithm to find all skew-symmetric biderivations in Bider s ( L , ad k ) . To this end, we need thenotions of central and special biderivations. Definition 3.8.
Let ( L , α ) be a Hom-Lie algebra and δ ∈ Bider s ( L , ad k ) . δ is called central if δ ( L , L ) ⊂ Z ( L ) = Z L ( L ) ; δ is called special if δ ( L , L ) ⊂ Z L ( L ′ ) and δ ( L ′ , L ′ ) = . Denote by CBider s ( L , ad k ) and SBider s ( L , ad k ) the sets consisting of all central and specialskew-symmetric biderivations in Bider s ( L , ad k ) , respectively. Remark 3.9.
Clearly, each δ ∈ CBider s ( L , ad k ) satisfies δ ( L , L ′ ) =
0. Then it follows thatCBider s ( L , ad k ) ⊆ SBider s ( L , ad k ) , and any skew-symmetric bilinear map δ satisfying βδ ( x , y ) = δ ( α ( x ) , α ( y )) , ∀ x , y ∈ L , δ ( L , L ′ ) = , δ ( L , L ) ⊆ Z ( L ) automatically belongs to CBider s ( L , ad k ) . Example 3.10.
Every Hom-Lie algebra ( L , α ) with nontrivial center and such that the codi-mension of L ′ in L is no less than 2 has nonzero special biderivations. Indeed, by the codimen-sion assumption, there exists a nonzero skew-symmetric bilinear form ω : L × L → F such that ω ( L , L ′ ) =
0, and taking any nonzero z ∈ Z ( L ) , we have that δ ( x , y ) : = ω ( x , y ) z is a nonzerocentral biderivation as well as a nonzero special biderivation.Suppose that ( L , α ) is a Hom-Lie algebra with α surjective and δ ∈ Bider s ( L , ad k ) . Themotivation for our algorithm is the following observation.Note that for any x , y ∈ L , z ∈ Z ( L ) we have0 = δ ([ z , x ] , α ( y )) = [ α k + ( z ) , δ ( x , y )] − [ α k + ( x ) , δ ( z , y )] = − [ α k + ( x ) , δ ( z , y )] , which implies δ ( Z ( L ) , L ) ⊂ Z ( L ) . Since Z ( L ) is an ideal of ( L , α ) , we consider the quo-tient Hom-Lie algebra ( ¯ L , ¯ α ) and its quotient adjoint module ( ¯ L , ad k , ¯ α ) , where ¯ L : = L / Z ( L ) ,¯ α ( ¯ x ) : = α ( x ) , and ad k ( ¯ x )( ¯ y ) : = [ α k ( x ) , y ] , for all x , y ∈ L . Then we can define a skew-symmetricbiderivation ¯ δ ∈ Bider s ( ¯ L , ad k ) by¯ δ ( ¯ x , ¯ y ) = δ ( x , y ) , ∀ ¯ x , ¯ y ∈ ¯ L . If δ and δ satisfy ¯ δ = ¯ δ , then ˆ δ : = δ − δ is a skew-symmetric biderivation with ˆ δ ( L , L ) ⊂ Z ( L ) , so ˆ δ is a central biderivation on ( L , α ) . Then we derive the following lemma. Lemma 3.11.
Let ( L , α ) be a Hom-Lie algebra with α surjective. Up to CBider s ( L , ad k ) , themap δ → ¯ δ is a - map from Bider s ( L , ad k ) to Bider s ( ¯ L , ad k ) . Hence, Bider s ( L , ad k ) are determined by CBider s ( L , ad k ) and Bider s ( ¯ L , ad k ) . We define asequence of quotient Hom-Lie algebras(3.8) ( L ( ) , α ( ) ) = ( L , α ) , · · · , ( L ( r + ) , α ( r + ) ) = ( L ( r ) / Z ( L ( r ) ) , α ( r ) ) , and write ( L ( r ) , ad ( r ) k , α ( r ) ) for the quotient adjoint module of ( L ( r ) , α ( r ) ) . If there exists r ∈ N such that Z ( L ( r ) ) = { } , then Bider s ( L , ad k ) are characterized by CBider s ( L ( i ) , ad ( i ) k )( ≤ i ≤ r − ) and Bider s ( L ( r ) , ad ( r ) k ) , where Bider s ( L ( r ) , ad ( r ) k ) can be determined by Theorem 3.7 incase ( L ( r ) , α ( r ) ) is perfect and α is invertible.However, we still don’t know how to determine Bider s ( L ( r ) , ad ( r ) k ) if ( L ( r ) , α ( r ) ) is not perfect,which inspires us to consider skew-symmetric biderivations on derived Hom-Lie algebras.Now let ( L , α ) be an arbitrary centerless Hom-Lie algebra such that α is invertible and δ ∈ Bider s ( L , ad k ) . Then δ satisfies(3.9) δ ( α ( u ) , [ x , y ]) = [ α k + ( x ) , δ ( u , y )] − [ α k + ( y ) , δ ( u , x )] ∈ L ′ , ∀ x , y , u ∈ L . Thus we could restrict δ to L ′ × L ′ to get a skew-symmetric biderivation δ ′ ∈ Bider s ( L ′ , ad k ) ,which makes sense since L ′ is α -invariant. If δ , δ ∈ Bider s ( L , ad k ) such that δ | L ′ × L ′ = δ | L ′ × L ′ , then ˜ δ : = δ − δ ∈ SBider s ( L , ad k ) .In fact, it suffices to prove ˜ δ ( L , L ) ⊂ Z L ( L ′ ) , since ˜ δ ( L ′ , L ′ ) = δ ( L , L ′ ) ⊂ Z L ( L ′ ) holds by taking u , y ∈ L ′ in (3.9). Then for any x , y , z , u , v ∈ L , we have [[[ x , y ] , ˜ δ ( u , v )] , α ( z )] = [[[ x , y ] , z ] , α ˜ δ ( u , v )] + [ α ([ x , y ]) , [ ˜ δ ( u , v ) , z ]]= [[[ x , y ] , z ] , ˜ δ ( α ( u ) , α ( v ))] − [[ α ( x ) , α ( y )] , [ z , ˜ δ ( u , v )]] ( α is isomorphic) = [ α k ([[ ˜ x , ˜ y ] , ˜ z ]) , ˜ δ ( α ( u ) , α ( v ))] − [[ α k + ( ˜ x ) , α k + ( ˜ y )] , [ α k ( ˜ z ) , ˜ δ ( u , v )]] (by Lemmas 3.2 and 3.4) = − [[ α k + ( u ) , α k + ( v )] , ˜ δ ([ ˜ x , ˜ y ] , ˜ z )] − [[ α k + ( ˜ x ) , α k + ( ˜ y )] , ˜ δ ( ˜ z , [ u , v ])]( by ˜ δ ( L , L ′ ) ⊂ Z L ( L ′ )) = . It follows that ˜ δ ( L , L ) ⊂ Z L ( L ′ ) , since ( L , α ) is centerless. Thus, ˜ δ ∈ SBider s ( L , ad k ) , and wededuce the following lemma. Lemma 3.12.
Suppose that ( L , α ) is a centerless Hom-Lie algebra with α invertible. Then any δ ∈ Bider s ( L , ad k ) , up to SBider s ( L , ad k ) , is an extension of a unique δ ′ ∈ Bider s ( L ′ , ad k ) . Hence, for centerless but not perfect Hom-Lie algebra ( L ( r ) , α ( r ) ) , Bider s ( L ( r ) , ad ( r ) k ) are de-termined by SBider s ( L ( r ) , ad ( r ) k ) and Bider s ( L ( r ) ′ , ad ( r ) k ) . Taking ( L , α ) = ( L ( r ) ′ , α ( r ) ) and re-peating the above arguments based on (3.8), then we continue the algorithms.Now we apply our method to several concrete examples. Example 3.13.
Let ( L , α ) belong to one class of multiplicative Heisenberg Hom-Lie algebrasin [3, Corollary 2.3]. Specifically, L = F e ⊕ F e ⊕ F e as a vector space with [ e , e ] = e , IDERIVATIONS AND COMMUTING LINEAR MAPS ON HOM-LIE ALGEBRAS 11 [ e , e ] = [ e , e ] =
0, and α = λ λ
00 0 λ for some nonzero λ ∈ F . Note that Z ( L ) = F e .As in (3.8), we have ( L ( ) , α ( ) ) = ( L , α ) , ( L ( ) , α ( ) ) = ( L / Z ( L ) , ¯ α ) = F ¯ e ⊕ F ¯ e , λ λ !! . Since ( L ( ) , α ( ) ) is abelian, ( L ( ) , α ( ) ) = ( { } , ) . Then Bider s ( L ( ) , ad ( ) k ) = { } , and we needCBider s ( L ( ) , ad ( ) k ) and CBider s ( L ( ) , ad ( ) k ) to determine Bider s ( L , ad k ) .Let δ ( ) ∈ CBider s ( L ( ) , ad ( ) k ) . Assume δ ( ) ( ¯ e , ¯ e ) = k ¯ e + k ¯ e . Note that δ ( ) satisfies α ( ) δ ( ) ( ¯ e , ¯ e ) = δ ( ) ( α ( ) ( ¯ e ) , α ( ) ( ¯ e )) , which implies k λ ¯ e + ( k + k λ ) ¯ e = λ ( k ¯ e + k ¯ e ) . Then(3.10) δ ( ) ( ¯ e , ¯ e ) = ( k ¯ e , λ = , λ = δ ( ) clearly satisfies (3.2) or (3.3) since ( L ( ) , α ( ) ) is an abelian Hom-Lie algebra.Note also that Bider s ( L ( ) , ad ( ) k ) = { } . Then any δ ∈ Bider s ( L ( ) , ad ( ) k ) must be some δ ( ) in(3.10).Let δ ( ) ∈ CBider s ( L ( ) , ad ( ) k ) . By Definition 3.8, Remark 3.9 and L ′ = Z ( L ) = F e , wecould set(3.11) δ ( ) ( e , e ) = k e , δ ( ) ( e , e ) = δ ( ) ( e , e ) = . It is straightforward to verify that the above δ ( ) satisfies (3.1)-(3.3).Now let ˆ δ ∈ Bider s ( L , ad k ) . Then ˆ δ induces a skew-symmetric biderivation that belongs toBider s ( L ( ) , ad ( ) k ) , which is of the form in (3.10). Set(3.12) ˆ δ ( e , e ) = ( k e , λ = , λ =
1, ˆ δ ( e , e ) = ˆ δ ( e , e ) = ( ∀ λ ∈ F ∗ ) . One could check that ˆ δ satisfies (3.1)-(3.3).Therefore, every δ ∈ Bider s ( L , ad k ) is of the form δ = δ ( ) + ˆ δ , for some δ ( ) in (3.11) and ˆ δ in (3.12). In particular, δ ( e , e ) = ( k e + k e , λ = k e , λ = δ ( e , e ) = δ ( e , e ) = ( ∀ λ ∈ F ∗ ) . Example 3.14.
Suppose that ( L , α ) belongs to the second class of multiplicative Hom-Lie al-gebras in [10, Theorem 4.7]. For simplicity of calculations, we consider L = F x ⊕ F y ⊕ F z as avector space with [ x , y ] = y , [ x , z ] = [ y , z ] = α = a λ b µ for a , b ∈ F and λ , µ ∈ F ∗ .Note that Z ( L ) = F z . As in (3.8), we have ( L ( ) , α ( ) ) = ( L , α ) , ( L ( ) , α ( ) ) = F ¯ x ⊕ F ¯ y , a λ !! . Then ( L ( ) , α ( ) ) is centerless but not perfect since L ( ) ′ = F ¯ y . Hence Bider s ( L ( ) , ad ( ) k ) is deter-mined by SBider s ( L ( ) , ad ( ) k ) and Bider s ( L ( ) ′ , ad ( ) k ) , where the latter is { } since dim L ( ) ′ = s ( L ( ) , ad ( ) k ) and SBider s ( L ( ) , ad ( ) k ) to determine Bider s ( L , ad k ) .Let ˜ δ ∈ SBider s ( L ( ) , ad ( ) k ) . Note that Z L ( ) ( L ( ) ′ ) = L ( ) ′ = F ¯ y . We set(3.13) ˜ δ ( ¯ x , ¯ y ) = k ¯ y , which can be verified to be a skew-symmetric biderivation. Then all skew-symmetric bideriva-tions in Bider s ( L ( ) , ad ( ) k ) are of the form given by (3.13), since there is no nonzero element inBider s ( L ( ) ′ , ad ( ) k ) .Let δ ( ) ∈ CBider s ( L ( ) , ad ( ) k ) . Since L ′ = F y and Z ( L ) = F z , we could set δ ( ) ( x , z ) = lz , δ ( ) ( x , y ) = δ ( ) ( y , z ) = . It is easy to check that the above δ ( ) is a skew-symmetric biderivation, then such δ ( ) areprecisely all elements in CBider s ( L ( ) , ad ( ) k ) .Set(3.14) δ ( x , y ) = ky , δ ( x , z ) = δ ( y , z ) = . Then δ ∈ Bider s ( L , ad k ) and δ induces a skew-symmetric biderivation in Bider s ( L ( ) , ad ( ) k ) ,which is of the form in (3.13).Therefore, every δ ∈ Bider s ( L , ad k ) is of the following form δ ( x , y ) = ky , δ ( x , z ) = lz , δ ( y , z ) = . Example 3.15.
Consider the loop algebra L = sl ⊗ F [ t , t − ] with [ x ⊗ t m , y ⊗ t n ] = [ x , y ] ⊗ t m + n and α = ˇ α ⊗ id, where ˇ α is an involution of sl such thatˇ α ( e ) = − e , ˇ α ( f ) = − f , ˇ α ( h ) = h . Then ( L , α ) is an infinite-dimensional Hom-Lie algebra (cf. [33, Corollaries 3.1 and 3.3]).Moreover, ( L , α ) is perfect and centerless, so each δ ∈ Bider s ( L , ad k ) is of the form δ ( x , y ) = α − γ ([ x , y ]) , ∀ x , y ∈ L , for some γ ∈ Cent ( L , ad k ) . IDERIVATIONS AND COMMUTING LINEAR MAPS ON HOM-LIE ALGEBRAS 13
Let γ ∈ Cent ( L , ad k ) . Then γα = αγ and for all a , b ∈ sl and m , n ∈ Z , γ ([ a ⊗ t m , b ⊗ t n ]) = [ α k + ( a ⊗ t m ) , γ ( b ⊗ t n )] Take a ⊗ t m = h ⊗
1. It follows that γ ([ h , b ] ⊗ t n ) = [ h ⊗ , γ ( b ⊗ t n )] . In particular, − γ ( f ⊗ t n ) = [ h ⊗ , γ ( f ⊗ t n )] , = [ h ⊗ , γ ( h ⊗ t n )] , γ ( e ⊗ t n ) = [ h ⊗ , γ ( e ⊗ t n )] , which implies that γ ( f ⊗ t n ) , γ ( h ⊗ t n ) and γ ( e ⊗ t n ) are eigenvectors for the operator [ h ⊗ , − ] with eigenvalues −
2, 0 and 2 respectively. Then there exist Φ nf ( t ) , Φ nh ( t ) , Φ ne ( t ) ∈ F [ t , t − ] suchthat γ ( f ⊗ t n ) = f ⊗ Φ nf ( t ) , γ ( h ⊗ t n ) = h ⊗ Φ nh ( t ) , γ ( e ⊗ t n ) = e ⊗ Φ ne ( t ) . Note that − f ⊗ Φ nf ( t ) = − γ ( f ⊗ t n ) = γ ([ h ⊗ t n , f ⊗ ])= [ h ⊗ t n , γ ( f ⊗ )] = [ h ⊗ t n , f ⊗ Φ f ( t )] = − f ⊗ t n Φ f ( t ) . Then Φ nf ( t ) = t n Φ f ( t ) , ∀ n ∈ Z . Similarly, Φ ne ( t ) = t n Φ e ( t ) . Note also that2 f ⊗ Φ nf ( t ) = γ ( f ⊗ t n ) = γ ([ f ⊗ , h ⊗ t n ])= [( − ) k + f ⊗ , γ ( h ⊗ t n )] = [( − ) k + f ⊗ , h ⊗ Φ nh ( t )] = f ⊗ ( − ) k + Φ nh ( t ) . Then we have Φ nh ( t ) = t n Φ h ( t ) and Φ f ( t ) = ( − ) k + Φ h ( t ) . Substituting f with e in the aboveequality one obtains Φ e ( t ) = ( − ) k + Φ h ( t ) . Hence ( − ) k + Φ f ( t ) = ( − ) k + Φ e ( t ) = Φ h ( t ) . Set Φ ( t ) = Φ h ( t ) and define φ ∈ End ( F [ t , t − ]) by φ ( g ( t )) = Φ ( t ) g ( t ) . It follows that γ = ˇ α k + ⊗ φ .Conversely, for any Φ ( t ) ∈ F [ t , t − ] , the γ defined as above belongs to Cent ( L , ad k ) . There-fore, Cent ( L , ad k ) = { ˇ α k + ⊗ φ | Φ ( t ) ∈ F [ t , t − ] } , and so every δ ∈ Bider s ( L , ad k ) is of the following form: δ ( x , y ) = α − ( ˇ α k + ⊗ φ )([ x , y ]) = ( ˇ α − ⊗ id )( ˇ α k + ⊗ φ )([ x , y ]) = ( ˇ α k ⊗ φ )([ x , y ]) , ∀ x , y ∈ L .
4. C
OMMUTING LINEAR MAPS ON H OM -L IE ALGEBRAS
In this section, we consider linear maps from a Hom-Lie algebra ( L , α ) to an ( L , α ) -module ( V , ρ , β ) that are also closely related to Cent ( L , V ) . Definition 4.1.
Let ( L , α ) be a Hom-Lie algebra and ( V , ρ , β ) an ( L , α ) -module. A linear map f : L → V is called a commuting linear map if α ( x ) f ( x ) = β ◦ f = f ◦ α , ∀ x ∈ L . Denote by Com ( L , V ) the set of all commuting linear maps f : L → V . In particular, wewrite Com ( L , ad k ) as the set of all commuting linear maps on ( L , α ) for the adjoint module ( L , ad k , α ) ,Clearly, Cent ( L , V ) ⊆ Com ( L , V ) . We will show Cent ( L , V ) = Com ( L , V ) , under the assump-tion that β is invertible and Z V ( L ′ ) = Lemma 4.2.
Let ( L , α ) be a Hom-Lie algebra and ( V , ρ , β ) an ( L , α ) -module with β invertible.If f ∈ Com ( L , V ) , then [ α ( v ) , α ( w )] α ( u )( f ([ x , y ]) − α ( x ) f ( y )) = , ∀ x , y , z , u , v ∈ L . Proof.
By the definition of the commuting linear map, we get(4.1) α ( x ) f ( y ) = − α ( y ) f ( x ) , ∀ x , y ∈ L , which induces a skew-symmetric bilinear map δ : L × L → V defined by(4.2) δ ( x , y ) = β − ( α ( x ) f ( y )) , ∀ x , y ∈ L . Moreover, δ is a skew-symmetric biderivation, since βδ ( x , y ) = α ( x ) f ( y ) = α ( x ) β − β f ( y ) (by Definition 4.1) = α ( x ) β − f ( α ( y )) (by (2.1)) = β − α ( x ) f ( α ( y ))= δ ( α ( x ) , α ( y )) and δ ( α ( x ) , [ y , z ]) = β − ( α ( x ) f ([ y , z ])) (by (4.1)) = − β − ( α ([ y , z ]) f ( α ( x ))) = − β − ([ α ( y ) , α ( z )] β f ( x )) (by (2.2)) = − β − ( α ( y ) α ( z ) f ( x ) − α ( z ) α ( y ) f ( x )) (by (2.1)) = − α ( y ) β − ( α ( z ) f ( x )) + α ( z ) β − ( α ( y ) f ( x ))= − α ( y ) δ ( z , x ) + α ( z ) δ ( y , x )= α ( y ) δ ( x , z ) − α ( z ) δ ( x , y ) . Note that Lemma 3.4 says that for any v , w ∈ L , [ v , w ]( δ ( α ( u ) , [ x , y ]) − α ( u ) δ ( x , y )) =
0. Itfollows that for all x , y , u , v , w ∈ L ,0 = β ([ v , w ]( δ ( α ( u ) , [ x , y ]) − α ( u ) δ ( x , y ))) (by (2.1)) = α ([ v , w ]) β ( δ ( α ( u ) , [ x , y ]) − α ( u ) δ ( x , y )) (by (2.1)) = α ([ v , w ])( βδ ( α ( u ) , [ x , y ]) − α ( u ) βδ ( x , y ))( by (4.2) ) = [ α ( v ) , α ( w )]( α ( u ) f ([ x , y ]) − α ( u ) α ( x ) f ( y ))= [ α ( v ) , α ( w )] α ( u )( f ([ x , y ]) − α ( x ) f ( y )) , as required. (cid:3) IDERIVATIONS AND COMMUTING LINEAR MAPS ON HOM-LIE ALGEBRAS 15
Theorem 4.3.
Let ( L , α ) be a Hom-Lie algebra with α surjective and ( V , ρ , β ) an ( L , α ) -modulewith β invertible. If Z V ( L ′ ) = , then Cent ( L , V ) = Com ( L , V ) .Proof. It suffices to prove Com ( L , V ) ⊆ Cent ( L , V ) .Take f ∈ Com ( L , V ) . Note that Z V ( L ) ⊆ Z V ( L ′ ) =
0. Then f ([ x , y ]) = α ( x ) f ( y ) by Lemma4.2, so f ∈ Cent ( L , V ) . (cid:3) Now we would like to describe Com ( L , V ) , which need the following notions of central andspecial commuting linear maps. Definition 4.4.
Let ( L , α ) be a Hom-Lie algebra and ( V , ρ , β ) an ( L , α ) -module. A commutinglinear map f ∈ Com ( L , V ) is called central if f ( L ) ⊂ Z V ( L ) ; f is called special if f ( L ) ⊂ Z V ( L ′ ) and f ( L ′ ) = . Denote by CCom ( L , V ) and SCom ( L , V ) the sets consisting of all central and special commutinglinear maps in Com ( L , V ) , respectively.Clearly, every linear map f : L → Z V ( L ) satisfying β ◦ f = f ◦ α is automatically a centralcommuting linear map.Note that Z V ( S ) is a submodule of V when S is an ideal of ( L , α ) such that α ( S ) = S . In fact,for any s ∈ S and v ∈ Z V ( S ) , there exists t ∈ S satisfying s = α ( t ) . Then Z V ( S ) is β -invariant by s β ( v ) = α ( t ) β ( v ) = β ( tv ) = Z V ( S ) is L -invariant by sxv = α ( t ) xv = [ t , x ] β ( v ) + α ( x ) tv = , ∀ x ∈ L . Now consider a Hom-Lie algebra ( L , α ) with α surjective. Since L ′ is an ideal of ( L , α ) , Z V ( L ′ ) becomes a submodule of ( V , ρ , β ) , which induces a quotient module ( V / Z V ( L ′ ) , ¯ ρ , ¯ β ) .Then for any f ∈ Com ( L , V ) , we can define ¯ f ∈ Com ( L , V / Z V ( L ′ )) by ¯ f ( x ) = f ( x ) + Z V ( L ′ ) .If f , f ∈ Com ( L , V ) such that ¯ f = ¯ f , then f : = f − f satisfies f ( L ) ⊂ Z V ( L ′ ) . Note that0 = α ([ x , y ]) f ( z ) = − α ( z ) f ([ x , y ]) , ∀ x , y , z ∈ L , which implies f ( L ′ ) ⊂ Z V ( L ) . Let ˆ f : L → Z V ( L ) ∈ CCom ( L , V ) such that ˆ f | L ′ = f | L ′ . Then˜ f : = f − ˆ f satisfies ˜ f ( L ′ ) = f ( L ) ⊂ Z V ( L ′ ) , that is, ˜ f ∈ SCom ( L , V ) . Hence f = f − f = ˆ f + ˜ f ∈ CCom ( L , V ) + SCom ( L , V ) . Then weprove the following property. Proposition 4.5.
Let ( L , α ) be a Hom-Lie algebra with α surjective and ( V , ρ , β ) an ( L , α ) -module. Up to CCom ( L , V ) + SCom ( L , V ) , the map f → ¯ f is a - map from Com ( L , V ) to Com ( L , V / Z V ( L ′ )) . Hence, in case ( V , ρ , β ) is an ( L , α ) -module with α surjective, Com ( L , V ) are determinedby CCom ( L , V ) , SCom ( L , V ) and Com ( L , V / Z V ( L ′ )) . Then we could give an algorithm fordescribing Com ( L , V ) as follows. Define a sequence of quotient modules ( V [ ] , ρ [ ] , β [ ] ) = ( V , ρ , β ) , · · · , ( V [ r + ] , ρ [ r + ] , β [ r + ] ) = ( V [ r ] / Z V [ r ] ( L ′ ) , ρ [ r ] , β [ r ] ) . If there exists r ∈ N such that Z V [ r ] ( L ′ ) = { } , then Com ( L , V ) are determined by CCom ( L , V ( i ) ) ,SCom ( L , V ( i ) ) (0 ≤ i ≤ r −
1) and Com ( L , V [ r ] ) , where Com ( L , V [ r ] ) is precisely Cent ( L , V [ r ] )in case β is required to be invertible.Now we apply our method to a concrete example. Example 4.6.
Let ( L , α ) be the multiplicative Hom-Lie algebras in Example 3.14 with α = λ
00 0 µ and ( V , ρ , β ) = ( L , ad k , α ) . Let’s determine Com ( L , ad k ) .Note that L ′ = F y and Z L ( L ′ ) = F y ⊕ F z . Then ( L [ ] , ad [ ] k , α [ ] ) = ( F ¯ x , , id ) . For f [ ] ∈ Com ( L , L [ ] ) , suppose(4.3) f [ ] ( x ) = k ¯ x , f [ ] ( y ) = k ¯ x , f [ ] ( z ) = k ¯ x . Then f [ ] clearly satisfies [ α k + ( x ) , f [ ] ( x )] = [ α k + ( y ) , f [ ] ( y )] = [ α k + ( z ) , f [ ] ( z )] = ¯0; f [ ] satisfies id ◦ f [ ] = f [ ] ◦ α , iff(4.4) k ( λ − ) = k ( µ − ) = . Hence every f [ ] ∈ Com ( L , L [ ] ) is defined by (4.3) and (4.4).Suppose that f ∈ Com ( L , ad k ) such that ¯ f = f [ ] . Set f ( y ) = k x + k ′ y + k ′′ z . Then k = = [ α k + ( y ) , f ( y )] = [ λ k + y , k x + k ′ y + k ′′ z ] = − k λ k + y . Hence f [ ] ∈ Com ( L , L [ ] ) is induced from some f ∈ Com ( L , ad k ) only if f [ ] ∈ Com ( L , L [ ] ) isdefined by (4.3) such that k ( µ − ) =
0. Define a linear map f : L → L by f ( x ) = k x , f ( y ) = , f ( z ) = k x , ( k , k ∈ F , k ( µ − ) = ) . It is straightforward to verify that f ∈ Com ( L , ad k ) .Now it remains to compute CCom ( L , ad k ) and SCom ( L , ad k ) . Let ˆ f ∈ CCom ( L , ad k ) and˜ f ∈ SCom ( L , ad k ) . Since Z L ( L ) = F z , L ′ = F y and Z L ( L ′ ) = F y ⊕ F z , setˆ f ( x ) = l z , ˆ f ( y ) = l z , ˆ f ( z ) = l z ;˜ f ( x ) = a y + a z , ˜ f ( y ) = , ˜ f ( z ) = a y + a z . By a direct computation, ˆ f is a commuting linear map iff l ( µ − ) = l ( λ − µ ) =
0; ˜ f satisfies [ α k + ( x ) , ˜ f ( x )] = [ α k + ( y ) , ˜ f ( y )] = [ α k + ( z ) , ˜ f ( z )] = a =
0; ˜ f satisfies α ◦ ˜ f = ˜ f ◦ α , iff a ( λ − ) = a ( µ − ) = a ( λ − µ ) = . Hence, ˜ f is of the form ˜ f ( x ) = a z , ˜ f ( y ) = , ˜ f ( z ) = a y + a z , IDERIVATIONS AND COMMUTING LINEAR MAPS ON HOM-LIE ALGEBRAS 17 where a , a , a ∈ F such that a ( µ − ) = a ( λ − µ ) = f ∈ Com ( L , ad k ) is of the form f ( x ) = c x + c z , f ( y ) = c z , f ( z ) = c x + c y + c z , for all c , c , c , c , c , c ∈ F such that c ( µ − ) = c ( λ − µ ) = c ( µ − ) = c ( λ − µ ) = . Finally, we would like to describe the relationship between skew-symmetric biderivationsand commuting linear maps on a Hom-Lie algebra ( L , α ) for the adjoint module ( L , ad k , α ) . Proposition 4.7.
Let ( L , α ) be a Hom-Lie algebra with α invertible and ( L , ad k , α ) an adjoint ( L , α ) -module such that every δ ∈ Bider s ( L , ad k ) is of the form δ ( x , y ) = α − γ ([ x , y ]) , where γ ∈ Cent ( L , ad k ) . Then every f ∈ Com ( L , ad k ) is of the form f = γ + µ for some γ ∈ Cent ( L , ad k ) and µ ∈ CCom ( L , ad k ) .Proof. Define δ ( x , y ) = α − [ α k + ( x ) , f ( y )] , which belongs to Bider s ( L , ad k ) , by the proof of Lemma 4.2 when ( V , ρ , β ) is taken as ( L , ad k , α ) .Thus there exists γ ∈ Cent ( L , ad k ) such that γ ([ x , y ]) = αδ ( x , y ) = [ α k + ( x ) , f ( y )] . Note that γ ([ x , y ]) = [ α k + ( x ) , γ ( y )] , which implies [ α k + ( x ) , ( f − γ )( y )] =
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