The Centroid of a Lie Triple Algebra

Abstract

General results on the centroids of Lie triple algebras are developed. Centroids of the tensor product of a Lie triple algebra and a unitary commutative associative algebra are studied. Furthermore, the centroid of the tensor product of a simple Lie triple algebra and a polynomial ring is completely determined.

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Recommended by Emrullah Yasar

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received 21 June 2013; Accepted 11 July 2013

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In recent years, Lie triple algebras (i.e., Lie-Yamaguti algebras or general Lie triple systems) have attracted much attention in Lie theories. They contain Lie algebras and Lie triple systems as special cases ([1-6]). So, it is of vital importnce to study some properties of Lie triple algebras. The concept of Lie triple algebra has been introduced, originally, by Yamaguti as general Lie triple system by himself, Sagle, and others. Since Lie triple algebras are generalization of Lie algebras and Lie triple system, it is natural for us to imagine whether or not some results of Lie algebras and Lie triple system hold in Lie triple algebras. Now, as a generalization of Lie triple algebra, Hom-Lie-Yamaguti was introduced by Lister in [7].

Benkart and Neher studied centroid of Lie algebras in [8], and Melville investigated centroid of nilpotent Lie algebras in [9]. It turns out that result on the centroid of Lie algebras is a key ingredient in the classification of extended affine Lie algebras. The centroids of Lie triple systems were mentioned by Benito et al. [10]. Now, some results on centroids of Lie triple system and n -Lie algebras were developed in [11, 12].

In this paper we present new results concerning the centroids of Lie triple algebras and give some conclusion of the tensor product of a Lie triple algebra and a unitary commutative associative algebra. Furthermore, we completely determine the centroid of the tensor product of a simple Lie triple algebra and a polynomial ring. The organization of the rest of this paper is as follows. Section 1 is for basic notions and facts on Lie triple algebras. Section 2 is devoted to the structures and properties of the centroids of Lie triple algebras. Section 3 describes the structures of the centroids of tensor product of Lie triple algebras.

2. Preliminaries

Definition 1 (see [1]).

A Lie triple algebra (also called a Lie-Yamaguti algebra or a general Lie triple system) ... is a vector space over an arbitrary field F with a bilinear map denoted by x y of ... × ... into ... and a ternary map denoted by [ x , y , z ] of ... × ... × ... into ... satisfying the following axioms:

(a) x x = 0 ,

(b) [ x , x , y ] = 0 ,

(c) ( x y ) z + ( y z ) x + ( z x ) y + [ x , y , z ] + [ y , z , x ] + [ z , x , y ] = 0 ,

(d) [ x y , z , w ] + [ y z , x , w ] + [ z x , y , w ] = 0 ,

(e) [ x , y , z w ] = ( [ x , y , z ] ) w + z ( [ x , y , w ] ) ,

(f) ( [ x , y , v ] ) ( [ z , w , v ] ) = [ [ x , y , z ] , w , v ] + [ z [ x , y , w ] , v ] ,

where x , y , z , w , v ∈ ... .

Remark 2.

Any Lie algebra is a Lie triple algebra relative to x y = [ x , y ] and D ( x , y ) z = [ [ x , y ] , z ] , for x , y , z ∈ ... . If D ( x , y ) z = 0 , for all x , y , z ∈ ... , the axioms stated earlier reduce to that of Lie algebra, and if x y = 0 , for all x , y ∈ ... , the axioms stated earlier reduce to that of Lie triple system. In this sense, the Lie triple algebra is a more general concept than that of the Lie algebra and Lie triple system.

Definition 3 (see [1, 2]).

A derivation of a Lie triple algebra ... is a linear transformation D of ... into ... satisfying the following conditions:

(1) D ( x y ) = ( D x ) y + x ( D y ) ,

(2) D ( [ x , y , z ] ) = [ D x , y , z ] + [ x , D y , z ] + [ x , y , D z ] , for all x , y , z ∈ ... .

Let Der ( ... ) be the set of all derivation of ... ; then Der ( ... ) is regarded as a subalgebra of the general Lie algebra gl ( ... ) and is called the derivation algebra of ... .

Definition 4.

A Lie triple subalgebra I of ... is called an ideal if ... I ⊆ I and [ ... , I , ... ] ⊆ I .

Definition 5.

A Lie triple algebra ... is perfect if ... = ... ... = [ ... , ... , ... ] .

Definition 6.

Let I be a nonempty subset of ... ; we call _{Z ...} ( I ) = { x ∈ ... |" x a = [ x , a , y ] = 0 , for all a ∈ I , y ∈ ... } the centralizer of I in ... . In particular, Z ( ... ) = { x ∈ ... |" x ... = [ x , ... , ... ] = 0 } is the center of ... .

Definition 7.

Suppose that I is an ideal of the Lie triple algebra ... , on the quotient vector space [figure omitted; refer to PDF] Define the operator: D ( x ¯ , y ¯ ) ( z ¯ ) = D ( x , y ) z ¯ , x ¯ y ¯ = x y ¯ , where x , y , z ∈ g . Then g / I is also a Lie triple algebra, and it is called a quotient algebra of ... by I .

3. The Centroids of Lie Triple Algebras

Definition 8.

Let ... be a Lie triple algebra over a field F . The centroid of ... is the transform on ... given by Γ ( ... ) = { ψ ∈ _{End F} ( ... ) |" ψ ( [ x , y , z ] ) = [ ψ ( x ) , y , z ] , ψ ( x y ) = ( ψ ( x ) ) y , for all x , y , z , ∈ ... } .

By (a)-(f), we can conclude that if ψ ∈ Γ ( ... ) , then we have ψ ( [ x , y , z ] ) = [ ψ ( x ) , y , z ] = [ x , ψ ( y ) , z ] = [ x , y , ψ ( z ) ] , ψ ( x y ) = ( ψ ( x ) ) y = x ( ψ ( y ) ) , for all x , y , z , ∈ ... .

From the definition, it is clear that the scalars will always be in the centroid.

Proposition 9.

If ... is perfect, then the centroid Γ ( ... ) is commutative.

Proof.

For all x , y , z ∈ ... , [varphi] , ψ ∈ Γ ( ... ) , we have [varphi] ψ ( [ x , y , z ] ) = [ ψ ( x ) , [varphi] ( y ) , z ] = ψ [varphi] ( [ x , y , z ] ) , [varphi] ψ ( x y ) = ψ ( x ) [varphi] ( y ) = ψ [varphi] ( x y ) .

Proposition 10.

Let ... be a Lie triple algebra over a field F and B a subset of ... . Then;

(1) _{Z ...} ( B ) is invariant under Γ ( ... ) ,

(2) every perfect ideal of ... is invariant under Γ ( ... ) .

Proof.

( 1 ) For any ψ ∈ Γ ( ... ) , x ∈ _{Z ...} ( B ) , y ∈ B , z ∈ ... , we have [ ψ ( x ) , y , z ] = [ ψ ( x ) , y , z ] = ψ [ x , y , z ] = 0 , ( ψ ( x ) ) z = ψ ( x z ) = 0 , and [ z , y , ψ ( x ) ] = ψ [ z , y , x ] = 0 , z ( ψ ( x ) ) = ψ ( z x ) = 0 . Therefore, ψ ( x ) ∈ Γ ( ... ) , which implies that _{Z ...} ( B ) is invariant under Γ ( ... ) .

( 2 ) Let J be any perfect ideal of ... ; then J = J J = [ J , J , J ] . For any y ∈ J , there exist a , b , c , d , e ∈ J , such that y = a b = [ c , d , e ] , and then we have ψ ( y ) = ψ ( a b ) = ( ψ ( a ) ) b ∈ J ... ⊆ J , ψ ( y ) = ψ ( [ c , d , e ] ) = [ c , ψ ( d ) , e ] ∈ [ J , ... , ... ] ⊆ J . Hence J is invariant under Γ ( ... ) .

Definition 11.

Let [straight phi] ( ... ) ⊆ Z ( ... ) and [straight phi] ( ... ... ) = [straight phi] ( [ ... , ... , ... ] ) = 0 ; then [straight phi] is called a central derivation.

The set of all central derivation of ... is denoted by C ( ... ) . Clearly, C ( ... ) ⊆ Γ ( ... ) and C ( ... ) is an ideal of Γ ( ... ) . A more precise relationship is summarized as follows.

Next, we will develop some general results on centroids of Lie triple algebra.

Proposition 12.

If ... has no nonzero ideals I and J with [ ... , I , J ] = 0 , I J = 0 , then Γ ( ... ) is an integral domain.

Proof.

Clearly, id ∈ Γ ( ... ) . If there exist ψ , [straight phi] ∈ Γ ( ... ) , ψ ...0; 0 , [varphi] ...0; 0 such that ψ [straight phi] = 0 , then there exist x , y ∈ ... such that ψ ( x ) ...0; 0 and [varphi] ( y ) ...0; 0 . Then, [ ... , ψ ( x ) , [varphi] ( y ) ] = ψ [varphi] ( [ ... , x , y ] ) = 0 , ( ψ ( x ) ) ( [varphi] ( y ) ) = ψ [varphi] ( x y ) = 0 . Therefore, ψ ( x ) and [varphi] ( y ) can span two nonzero ideals I , J of ... such that [ ... , I , J ] = I J = 0 , which is a contradiction. Hence, Γ ( ... ) has no zero divisor; it is an integral domain.

Theorem 13.

If ... is a simple Lie triple algebra over an algebraically closed field F , then Γ ( ... ) = F ... id .

Proof.

Let [varphi] ∈ Γ ( ... ) ⊆ _{End F} ( ... ) . Since F is algebraically closed, [varphi] has an eigenvalue λ . We denote the corresponding eigenspace to be _{E λ} ( [varphi] ) = { v ∈ ... |" [varphi] ( v ) = λ ( v ) } , so _{E λ} ( [varphi] ) ...0; 0 , for any v ∈ _{E λ} ( [varphi] ) , x , y ∈ ... , and we have [varphi] ( [ v , x , y ] ) = [ [varphi] ( v ) , x , y ] = λ [ v , x , y ] , [varphi] ( ( v x ) ) = [varphi] ( v ) x = ( λ v ) x = λ ( v x ) , so [ v , x , y ] ∈ _{E λ} ( [varphi] ) , v x ∈ _{E λ} ( [varphi] ) . It follows that _{E λ} ( [varphi] ) is an ideal of ... . But, ... is simple, so _{E λ} ( [varphi] ) = ... ; that is, _{E λ} ( [varphi] ) = λ i _{d ...} . This proves the theorem.

When Γ ( ... ) = F id , the Lie triple algebra ... is said to be central. Furthermore, if ... is simple, ... is said to be central simple. Every simple Lie triple algebra is central simple over its centroid.

Proposition 14.

Let ... be a Lie triple algebra over a field F ; then

(1) ... is indecomposable if and only if Γ ( ... ) does not contain idempotents except 0 and id ;

(2) if ... is perfect, then every [varphi] ∈ Γ ( ... ) is symmetric ( f ( [ a , b , c ] , d ) = f ( a [ d , c , b ] ) , f ( a b , c ) = f ( a , b c ) ) with respect to any invariant form on ... .

Proof.

( 1 ) If there exists [varphi] ∈ Γ ( ... ) which is an idempotent and satisfies [varphi] ...0; 0 , id , then ^{[varphi] 2} ( x ) = [varphi] ( x ) , for all x ∈ ... . We assert that Ker [varphi] and Im [varphi] are ideals of ... . In fact, for any x ∈ Ker [varphi] and y , z ∈ ... , we have [varphi] ( [ x , y , z ] ) = [ [varphi] ( x ) , y , z ] = 0 , [varphi] ( x y ) = ( [varphi] ( x ) ) y = 0 y = 0 , which implies [ x , y , z ] ∈ Ker [varphi] , x y ∈ Ker [varphi] . For any x ∈ Im [varphi] , there exists a ∈ ... such that x = [varphi] ( a ) . Then, we have [figure omitted; refer to PDF] This proves our assertion. Moreover, Ker [varphi] ∩ Im [varphi] = 0 . Indeed, if x ∈ Ker [varphi] ∩ Im [varphi] , then there exists y ∈ ... such that x = [varphi] ( y ) and 0 = [varphi] ( x ) = ^{[varphi] 2} ( y ) = [varphi] ( y ) = x . We have a decomposition x = [varphi] ( x ) + y , for all x ∈ ... , where [varphi] ( y ) = 0 . So, we have ... = Ker [varphi] [ecedil]5; Im [varphi] , which is a contradiction.

On the other hand, suppose ... has a decomposition ... = _{... 1} [ecedil]5; _{... 2} . Then for any x ∈ ... , we have x = _{x 1} + _{x 2} , _{x i} ∈ _{... i} , i = 1,2 . We choose [varphi] ∈ Γ ( ... ) such that [varphi] ( _{x 1} ) = _{x 1} and [varphi] ( _{x 2} ) = 0 . Then, ^{[varphi] 2} ( x ) = [varphi] ( _{x 1} ) = _{x 1} = [varphi] ( x ) . Hence, [varphi] is an idempotent. By assumption, we have [varphi] = 0 or [varphi] = id . If [varphi] = 0 , then _{x 1} = 0 , implying ... = _{... 2} . If [varphi] = id , then _{x 2} = 0 , implying ... = _{... 1} .

( 2 ) Let f be an invariant F -bilinear form on ... . Then f ( [ a , b , c ] , d ) = f ( a [ d , c , b ] ) , f ( a b , c ) = f ( a , b c ) , for all a , b , c , d ∈ ... . Since ... is perfect, for [varphi] ∈ Γ ( ... ) , we have [figure omitted; refer to PDF]

The result follows.

Remark 15.

Set L ( x ) ( y ) = x y , L ( x , y ) ( z ) = [ x , y , z ] , R ( x ) ( y ) = y x , R ( x , y ) ( z ) = [ z , x , y ] , for all x , y , z , ∈ ... ... ... ; then we call L ( x ) , and L ( x , y ) the left multiplication operator of ... ; R ( x ) and R ( x , x ) right multiplication operator of ... . Denote by Mult ( ... ) = _{Mult F} ( ... ) the subalgebra of _{End F} ( ... ) generated by the left and right multiplication operators of ... . Then, Γ ( ... ) is the centralizer of Mult ( ... ) .

Theorem 16.

Let π : _{... 1} [arrow right] _{... 2} be an epimorphism of Lie triple algebra; for any f ∈ _{End ... F} ( _{... 1} ; Ker π ) : = { g ∈ _{End ... F} ( _{... 2} ) |" g ( Ker π ) ⊆ Ker π } , there exists a unique f ¯ ∈ _{End ... F} ( _{... 2} ) satisfying π [composite function] f = f ¯ [composite function] π . Moreover, the following results hold.

(1) The map _{π End} : _{End ... F} ( _{... 1} ; Ker π ) [arrow right] _{End ... F} ( _{... 2} ) , f ... f ¯ is a homomorphism with the following properties:

: _{π End} ( Mult ( _{... 1} ) ) = Mult ( _{... 2} ) , _{π End} ( Γ ( _{... 1} ) ∩ _{End ... F} ( _{... 1} ; Ker π ) ) ⊆ Γ ( _{... 2} ) . There is a homomorphism _{π Γ} : Γ ( _{... 1} ) ∩ _{End ... F} ( _{... 1} ; Ker π ) [arrow right] Γ ( _{... 2} ) , f ... f ¯ .

If Ker π = Z ( _{... 1} ) , then every [varphi] ∈ Γ ( _{... 1} ) leaves Ker π invariant; that is, _{π Γ} is defined on all of Γ ( _{... 1} ) .

(2) Suppose _{... 1} is perfect and Ker π ⊆ Z ( _{... 1} ) ; then _{π Γ} : Γ ( _{... 1} ) ∩ _{End ... F} ( _{... 1} ; Ker π ) [arrow right] Γ ( _{... 2} ) , f ... f ¯ is injective.

(3) If _{... 1} is perfect, Z ( _{... 2} ) = 0 , and Ker π ⊆ Z ( _{... 1} ) , then _{π Γ} : Γ ( _{... 1} ) [arrow right] Γ ( _{... 2} ) is a monomorphism.

Proof.

( 1 ) It is easy to see that _{π End} is a homomorphism. Since Ker π is an ideal of _{... 1} and all left and right multiplication operators of _{... 1} leave Ker π invariant, Mult ( _{... 1} ) ⊆ En _{d F} ( _{... 1} ; Ker π ) . Furthermore, for the left multiplication operator L ( x , y ) , L ( x ) on _{... 1} , we have π [composite function] L ( x , y ) = L ( π ( x ) , π ( y ) ) [composite function] π , π [composite function] L ( x ) = L ( π ( x ) ) [composite function] π , so _{π End} ( L ( x , y ) ) = L ( π ( x ) , π ( y ) ) , _{π End} ( L ( x ) ) = L ( π ( x ) ) . For the right multiplication, we have the analogous formula _{π End} ( R ( x , y ) ) = R ( π ( x ) , π ( y ) ) , _{π End} ( R ( x ) ) = R ( π ( x ) ) . Moreover, π is an epimorphism, so _{π End} ( Mult ( _{... 1} ) ) = Mult ( _{... 2} ) . Now, we show that _{π End} ( Γ ( _{... 1} ) ∩ En _{d F} ( _{... 1} ; Ker π ) ) ⊆ Γ ( _{... 2} ) . Let [varphi] ∈ Γ ( _{... 1} ) ∩ En _{d F} ( _{... 1} ; Ker π ) . For any ^{x [variant prime]} , ^{y [variant prime]} , ^{z [variant prime]} ∈ _{... 2} , there exist x , y , z ∈ _{... 1} such that π ( x ) = ^{x [variant prime]} , π ( y ) = ^{y [variant prime]} , π ( z ) = ^{z [variant prime]} . Then, we have [figure omitted; refer to PDF] which proves [varphi] ¯ ∈ Γ ( _{... 2} ) .

( 2 ) If [varphi] ¯ = 0 for, [varphi] ∈ Γ ( _{... 1} ) ∩ En _{d F} ( _{... 1} ; Ker π ) , then π ( [varphi] ( _{... 1} ) ) = [varphi] ¯ ( π ( _{... 1} ) ) = 0 , which means that [varphi] ( _{... 1} ) ⊆ Ker π ⊆ Z ( _{... 1} ) . Hence, [varphi] ( [ x , y , z ] ) = [ [varphi] ( x ) , y , z ] = 0 , [varphi] ( x y ) = ( [varphi] ( x ) y ) = 0 , for all x , y , z ∈ _{... 1} . Furthermore, since _{... 1} is a perfect ideal, we can get [varphi] = 0 .

( 3 ) We can see that π ( Z ( _{... 1} ) ) ⊆ Z ( _{... 2} ) = 0 , which follows that Z ( _{... 1} ) ⊆ Ker π . So, Ker π ⊆ Z ( _{... 1} ) . From (1), we know that _{π Γ} : Γ ( _{... 1} ) [arrow right] Γ ( _{... 2} ) is a well-defined homomorphism, which is an injection by (2).

Proposition 17.

If the characteristic of F is not 2 , then [figure omitted; refer to PDF]

Proof.

If ψ ∈ Γ ( ... ) ∩ Der ( ... ) , then by the definition of Γ ( ... ) and Der ( ... ) , for all x , y , z ∈ ... , we have ψ ( x y ) = ( ψ ( x ) ) y + x ( ψ ( y ) ) , ψ ( [ x , y , z ] ) = [ ψ ( x ) , y , z ] + [ x , ψ ( y ) , z ] + [ x , y , ψ ( z ) ] , and ψ ( x y ) = ( ψ ( x ) ) y = x ( ψ ( y ) ) , ψ ( [ x , y , z ] ) = [ ψ ( x ) , y , z ] = [ x , ψ ( y ) , z ] = [ x , y , ψ ( z ) ] , so ψ ( ... ... ) = ψ ( [ ... , ... , ... ] ) = 0 and ψ ( ... ) ⊆ C ( ... ) . It follows easily that Γ ( ... ) ∩ Der ( ... ) ⊆ C ( ... ) .

To show the inverse inclusion, let ψ ∈ C ( ... ) ; then 0 = ψ ( [ x , y , z ] ) = [ ψ ( x ) , y , z ] = [ x , ψ ( y ) , z ] = [ x , y , ψ ( z ) ] , 0 = ψ ( x y ) = ( ψ ( x ) ) y = x ( ψ ( y ) ) . Thus, ψ ∈ Γ ( ... ) ∩ Der ( ... ) .

This implies that C ( ... ) = Γ ( ... ) ∩ Der ( ... ) .

Lemma 18.

Let I be a nonzero Γ ( ... ) -invariant ideal of ... , V ( I ) = { ψ ∈ Γ ( ... ) ψ ( I ) = 0 } , and let Hom ( ... / I , _{Z ...} ( I ) ) be the vector space of all linear maps from ... / I to _{Z ...} ( I ) over F . Define T ( I ) = { f ∈ Hom ( ... / I , _{Z ...} ( I ) ) |" f ( [ x ¯ , y ¯ , z ¯ ] ) = [ f ( x ¯ ) , y , z ] = [ x , f ( y ¯ ) , z ] = [ x , y , f ( z ¯ ) ] , f ( x ¯ y ¯ ) = ( f ( x ¯ ) ) y = x ( f ( y ¯ ) ) } , where x ¯ , y ¯ , z ¯ ∈ ... / I . Then one has the following.

(1) T ( I ) is a subspace of Hom ( ... / I , _{Z ...} ( I ) ) and V ( I ) [congruent with] T ( I ) as vector spaces.

(2) If Γ ( I ) = F ... _{id ... I} , then Γ ( ... ) = F ... _{id ... ...} [ecedil]5; V ( I ) as vector spaces.

Proof.

( 1 ) It is easily seen that V ( I ) is an ideal of the associative algebra Γ ( ... ) . To prove (1) consider the following map α : V ( I ) [arrow right] T ( I ) given by [figure omitted; refer to PDF] where ψ ∈ V ( I ) and y ¯ = y + I ∈ ... / I . The map α is well defined. If y ¯ = _{y 1} ¯ , then y - _{y 1} ∈ I , and so ψ ( y - _{y 1} ) = 0 . It follows easily that α is injective. We now show that α is onto. For every f ∈ T ( I ) , set _{ψ f} : ... [arrow right] ... , _{ψ f} ( x ) = f ( x ¯ ) , for all x ∈ ... . It follows from the definition of T ( I ) that, for all x , y , z ∈ ... , _{ψ f} ( x y ) = f ( x ¯ y ¯ ) = ( f ( x ¯ ) ) y = x ( f y ¯ ) , _{ψ f} ( [ x , y , z ] ) = f ( [ x ¯ , y ¯ , z ¯ ] ) = [ f ( x ¯ ) , y , z ] = [ x , f ( y ¯ ) , z ] = [ x , y , f ( z ¯ ) ] . Thus, _{ψ f} ∈ Γ ( ... ) , and so _{ψ f} ∈ V ( I ) , _{ψ f} ( I ) = 0 . But, α ( _{ψ f} ) = f implies that α is onto. It is fairly easy to see that α preserves operations on vector spaces from ... / I to _{Z ...} ( I ) . This proves (1).

We now prove (2). If Γ ( I ) = F ... _{id I} , then for all ψ ∈ Γ ( ... ) , ψ _{|" I} = λ i _{d I} , for some λ ∈ F . If ψ ...0; λ _{id ...} , let [straight phi] ( x ) = λ x , for all x ∈ ... ; then ψ ∈ Γ ( ... ) and ψ - [straight phi] ∈ V ( I ) . Clearly, ψ = [straight phi] + ( ψ - [straight phi] ) . Furthermore, F i _{d ...} ∩ V ( I ) = 0 , and so (2) is proved.

Lemma 19.

Let ... be a Lie triple algebra; then [straight phi] D is a derivation for [straight phi] ∈ Γ ( ... ) , D ∈ Der ( ... ) .

Proof.

If x , y , z ∈ ... , then [figure omitted; refer to PDF] Thus, [straight phi] D is a derivation.

Theorem 20.

Let ... be a Lie triple algebra, and for any [straight phi] ∈ Γ ( ... ) , D ∈ Der ( ... ) , then one has the following.

(1) Der ( ... ) is contained in the normalizer of Γ ( ... ) in End ( ... ) .

(2) D [straight phi] is contained in Γ ( ... ) if and only if [straight phi] D is a central derivation of ... .

(3) [straight phi] D is a derivation of ... if and only if [ D , [straight phi] ] is a central derivation of ... .

Proof.

For any [straight phi] ∈ Γ ( ... ) , D ∈ Der ( ... ) , and x , y , z ∈ ... , [figure omitted; refer to PDF]

Then, we get ( D [straight phi] - [straight phi] D ) ( x y ) = ( ( D [straight phi] - [straight phi] D ) ( x ) ) y . On the other hand, we have [figure omitted; refer to PDF] So, ( D [straight phi] - [straight phi] D ) ( [ x , y , z ] ) = [ ( D [straight phi] - [straight phi] D ) ( x ) , y , z ] ; this is [ D , [straight phi] ] = D [straight phi] - [straight phi] D ∈ Γ ( ... ) . This proves (1). From Lemma 19 and (1), D [straight phi] is an element of Γ ( ... ) if and only if [straight phi] D ∈ Γ ( ... ) ∩ Der ( ... ) . Thanks to Proposition 17, we get the result (2). It follows from (1), Proposition 17, and Lemma 19 that the result holds.

Now, we study the relationship between the centroid of a decomposable Lie triple algebra and the centroid of its factors.

Theorem 21.

Suppose that ... is a Lie triple algebra over F and ... = _{... 1} [ecedil]5; _{... 2} with _{... 1} and _{... 2} being ideals of ... ; then [figure omitted; refer to PDF] as vector spaces, where C ( 1 ) = { [straight phi] ∈ Hom ( _{... 1} , _{... 1} ) ... |" ... [straight phi] ( _{... 1} ) ⊆ Z ( _{... 2} ) , [straight phi] ( _{... 2 ... 2} ) = [straight phi] [ _{... 2} , _{... 2} , _{... 2} ] = 0 } , C ( 2 ) = { [straight phi] ∈ Hom ( _{... 2} , _{... 2} ) |" [straight phi] ( _{... 2} ) ⊆ Z ( _{... 1} ) , [straight phi] ( _{... 1 ... 1} ) = [straight phi] [ _{... 1} , _{... 1} , _{... 1} ] = 0 } .

Proof.

Let _{π i} : ... [arrow right] _{... i} be canonical projections for i = 1,2 ; then _{π 1} , _{π 2} ∈ Γ ( ... ) and _{π 1} + _{π 2} = i _{d ...} . So we have for [straight phi] ∈ Γ ( ... ) , [figure omitted; refer to PDF] Note that _{π i} [straight phi] _{π j} ∈ Γ ( ... ) for i , j = 1,2 . We claim that [figure omitted; refer to PDF] It suffices to show that _{π 1} Γ ( ... ) _{π 1} ∩ _{π 1} Γ ( ... ) _{π 2} (other cases are similar). For any [straight phi] ∈ _{π 1} Γ ( ... ) _{π 1} ∩ _{π 1} Γ ( ... ) _{π 2} , there exist _{f i} ∈ Γ ( ... ) , i = 1,2 , such that [straight phi] = _{π 1 f 1 π 1} = _{π 1 f 2 π 2} . Then, [straight phi] ( x ) = _{π 1 f 2 π 2} ( x ) = _{π 1 f 2 π 2} ( _{π 2} ( x ) ) = _{π 1 f 1} ( 0 ) = 0 , for all x ∈ ... , and so [straight phi] = 0 . Let Γ ( ... _{) i j} = _{π i} Γ ( ... ) _{π j} , i , j = 1,2 . We now prove that [figure omitted; refer to PDF] Since [straight phi] ( _{... 2} ) = 0 for [straight phi] ∈ Γ ( ... _{) 11} , we have [straight phi] _{|" ... 1} ∈ Γ ( _{... 1} ) . On the other hand, one can regard Γ ( _{... 1} ) as a subalgebra of Γ ( ... ) by extending any _{[straight phi] 0} ∈ Γ ( _{... 1} ) on _{... 2} being equal to zero; that is, _{[straight phi] 0} ( _{x 1} ) = _{[straight phi] 0} ( _{x 1} ) , _{[straight phi] 0} ( _{x 2} ) = 0 , for all _{x 1} ∈ _{... 1} , _{x 2} ∈ _{... 2} . Then _{[straight phi] 0} ∈ Γ ( ... ) and _{[straight phi] 0} ∈ Γ ( ... _{) 11} . Therefore, Γ ( ... _{) 11} [congruent with] Γ ( _{... 1} ) with isomorphism σ : Γ ( ... _{) 11} [arrow right] Γ ( _{... 1} ) , σ ( [straight phi] ) = [straight phi] _{|" ... 1} , for all [straight phi] ∈ Γ ( ... _{) 11} . Similarly, we have Γ ( ... _{) 22} [congruent with] Γ ( _{... 2} ) .

Next, we prove that Γ ( ... _{) 12} [congruent with] _{C 2} . If [straight phi] ∈ Γ ( ... _{) 12} there exists _{[straight phi] 0} in Γ ( ... ) such that [straight phi] = _{π 1 [straight phi] 0 π 2} . For _{x k} = ^{x k 1} + ^{x k 2} ∈ ... , where ^{x k i} ∈ _{... i} , i = 1,2 and k = 1,2 , 3 , we have [figure omitted; refer to PDF] Then, [straight phi] ( ... ) ⊆ Z ( ... ) and [straight phi] ( ... ... ) = [straight phi] [ ... , ... , ... ] = 0 . It follows that [straight phi] _{|" ... 2} ( _{... 2} ) ⊆ Z ( _{... 1} ) , [straight phi] _{|" ... 2} ( _{... 2 ... 2} ) = [straight phi] _{|" ... 2} ( [ _{... 2} , _{... 2} , _{... 2} ] ) = 0 , and so [straight phi] _{|" ... 2} ∈ _{C 2} .

Conversely, for [straight phi] ∈ _{C 2} , expending [straight phi] on ... (also denoted by [straight phi] ) or by [straight phi] ( _{... 1 ... 1} ) = [straight phi] [ _{... 1} , _{... 1} , _{... 1} ] = 0 , we have _{π 1} [straight phi] _{π 2} = [straight phi] and [straight phi] ∈ Γ ( ... _{) 12} . This proves that Γ ( ... _{) 12} is isomorphic to _{C 2} with the following isomorphism: [figure omitted; refer to PDF] for all [straight phi] ∈ Γ ( ... _{) 12} . Similarly, we can prove that Γ ( ... _{) 21} [congruent with] _{C 1} . Summarizing the aforesaid discussion, we have [figure omitted; refer to PDF]

The proof is completed.

A generalized version of Theorem 21 is stated next without proof.

Theorem 22.

Suppose that ... is a Lie triple algebra over F and with a decomposition of ideals ... = _{... 1} [ecedil]5; _{... 2} [ecedil]5; ... [ecedil]5; _{... m} . Then, one has [figure omitted; refer to PDF] as vector spaces, where _{C i j} = { [straight phi] ∈ Hom ( _{... i} , _{... j} ) |" [straight phi] ( _{... i} ) ⊆ Z ( _{... j} ) , [straight phi] ( _{... i ... i} ) = [straight phi] [ _{... i} , _{... i} , _{... i} ] = 0 , for 1 ...4; i ...0; j ...4; m } .

4. The Centroids of Tensor Product of Lie Triple Algebras

Definition 23.

Let A be an associative algebra over a field F , where the centroid of A is the space of F -linear transforms on A given by Γ ( A ) = { [straight phi] ∈ En _{d F} ( A ) [straight phi] ( a b ) = a [straight phi] ( b ) = [straight phi] ( a ) b , for all a , b ∈ A } ; then Γ ( A ) is an associative algebra of En _{d F} ( A ) . If ... is be a finite-dimensional Lie triple algebra over F , let ... [ecedil]7; A be a tensor product of the underlying vector spaces A and ... . Then, ... [ecedil]7; A over a field F with respect to the following 2-ary and 3-ary multilinear operation: [figure omitted; refer to PDF] where _{x i} ∈ ... , _{a i} ∈ A , i = 1,2 , 3 . This Lie triple algebra ... [ecedil]7; A is called the tensor product of A and ... . For f ∈ En _{d F} ( ... ) and [straight phi] ∈ En _{d F} ( A ) , there exists a unique map f [ecedil]7; ~ [straight phi] ∈ En _{d F} ( ... [ecedil]7; A ) such that [figure omitted; refer to PDF] The map should not be confused with the element f [ecedil]7; [straight phi] of the tensor product En _{d F} ( ... ) [ecedil]7; En _{d F} ( A ) . Of course, we have a canonical map as the following: [figure omitted; refer to PDF] It is easy to see that if [varphi] ∈ Γ ( ... ) and ψ ∈ Γ ( A ) , then [varphi] [ecedil]7; ~ ψ ∈ Γ ( ... [ecedil]7; A ) . Hence, Γ ( ... ) [ecedil]7; ~ Γ ( A ) ⊆ Γ ( ... [ecedil]7; A ) , where Γ ( ... ) [ecedil]7; ~ Γ ( A ) is the F -span of all endomorphisms [varphi] [ecedil]7; ~ ψ .

Definition 24.

The transformation [varphi] ∈ Γ ( ... [ecedil]7; A ) is said to have finite ... -image, if for any a ∈ A , there exist finitely many _{a 1} , ... , _{a n} ∈ A such that [figure omitted; refer to PDF] It is easy to see that Γ ( ... ) [ecedil]7; ~ Γ ( A ) ⊆ { [varphi] ∈ Γ ( ... [ecedil]7; A ) |" [varphi] has finite ... - image } . In addition, [varphi] ∈ Γ ( ... [ecedil]7; A ) has finite ... -image, if [varphi] ( ... [ecedil]7; 1 ) ⊆ ... [ecedil]7; F _{a 1} + ... + ... [ecedil]7; F _{a n} for suitable _{a i} ∈ A .

Lemma 25.

Let ... be a Lie triple algebra over F and A a unitary commutative associative over F . Let { _{a r } r ∈ [real]} be a basis of A , and let [varphi] ∈ Γ ( ... [ecedil]7; A ) . Define _{[varphi] r} ∈ _{End F} ( ... ) by [figure omitted; refer to PDF] then all _{[varphi] r} ∈ Γ ( ... ) .

Proof.

For _{x 1} , _{x 2} , _{x 3} ∈ ... , we have [figure omitted; refer to PDF] Hence, one has [figure omitted; refer to PDF] for all _{x i} ∈ ... . So all _{[varphi] r} ∈ Γ ( ... ) .

Proposition 26.

Let ... be a perfect Lie triple algebra over F , and let A be a unitary commutative associative over F which is free as a F -module. Then, one has the following.

(1) If ... is perfect, then ... [ecedil]7; A is perfect too.

(2) If ... is finitely generated as a Mult ... ( ... ) -module (or as a Γ ( ... ) -module), or if ... is a central and a torsion-free F -module, then every [varphi] ∈ Γ ( ... [ecedil]7; A ) has finite ... -image.

(3) If Γ ( ... ) is a free F -module and the map ω of Definition 23 is injective, then Γ ( ... ) [ecedil]7; ~ Γ ( A ) = { [varphi] ∈ Γ ( ... [ecedil]7; A ) |" [varphi] has finite ... - i m a g e } .

Proof.

The definition of perfect shows that (1) holds. Let M = Mult ( ... ) . It follows that M [ecedil]7; ~ id ⊆ Mult ( ... [ecedil]7; A ) from A is unital. Since ... is finitely generated as a Mult ( ... ) -module, we can suppose ... = M _{x 1} + ... + M _{x n} for _{x 1} , ... , _{x n} ∈ ... . Fix [varphi] ∈ Γ ( ... [ecedil]7; A ) and a ∈ A . There exist finite families { _{x i j} } ⊆ ... and { _{a i j} } ⊆ A such that [figure omitted; refer to PDF] for 1 ...4; i ...4; n . Hence, [figure omitted; refer to PDF] Since ... [ecedil]7; A is perfect, the centroid Γ ( ... [ecedil]7; A ) is commutative. By replacing M with Γ ( ... ) we can use the same argument aforementione to show that every [varphi] ∈ Γ ( T [ecedil]7; A ) has finite ... -image, if ... is a finitely generated Γ ( ... ) -module.

Now, suppose that Γ ( ... ) = F id and that ... is a torsion-free F -module. Then there exist scalars _{k r} ∈ F such that _{[varphi] r} = _{k r} id . Hence, [figure omitted; refer to PDF] Fix x ∈ ... ; then almost all _{k r} x = 0 and almost all _{k r} = 0 , which in turn implies that [varphi] has finite ... -image.

From the aforementione discussion above, we get [figure omitted; refer to PDF] So, it suffices to prove that Γ ( ... ) [ecedil]7; ~ Γ ( A ) ⊇ { [varphi] ∈ Γ ( ... [ecedil]7; A ) |" [varphi] has finite ... -image } . We suppose that [varphi] ∈ Γ ( ... [ecedil]7; A ) has finite ... -image, and then there exists a finite subset ... ⊆ [real] such that equation in Lemma 25 becomes as follows: [figure omitted; refer to PDF] For _{x 1} , _{x 2} , _{x 3} ∈ ... and a ∈ A , we get [figure omitted; refer to PDF] Since ... is perfect, we can get [figure omitted; refer to PDF] where _{λ r} is the left multiplication in A by _{a r} . Let { _{ψ s} |" s ∈ ... } be a basis of Γ ( ... ) . Then, there exist a finite subset ... ⊆ ... and scalars _{k r s} ∈ F ( r ∈ ... , s ∈ ... ) such that [figure omitted; refer to PDF] We then get [figure omitted; refer to PDF] Now we show that _{[varphi] s} ∈ Γ ( A ) . For any _{x 1} , _{x 2} , _{x 3} ∈ ... , _{a 1} , _{a 2} ∈ A , we have [figure omitted; refer to PDF] Since [ ... ... , ... ] = ... ... = ... , we have [figure omitted; refer to PDF] for all x ∈ ... and _{a 1} , _{a 2} ∈ A . It follows that [figure omitted; refer to PDF] where _{μ s} ∈ En _{d F} ( A ) is defined by _{μ s} ( _{a 2} ) = _{[varphi] s} ( _{a 1 a 2} ) - _{[varphi] s} ( _{a 1} ) _{a 2} , for all _{a 2} ∈ A . Since ω is injective, we also have [figure omitted; refer to PDF] So by the linear independence of the _{ψ s} , we get that _{μ s} = 0 . Then _{ψ s} ∈ Γ ( A ) . Hence [varphi] ∈ Γ ( ... ) [ecedil]7; ~ Γ ( A ) .

Next, we will determine the centroid of the tensor product of a simple Lie triple algebra and a polynomial ring. Here after we study the centroid of Lie triple algebra over a filed F . Let { _{e k} } be a basis of the central derivation C ( ... ) and { _{[straight phi] j} } a maximal subset of Γ ( ... ) such that { _{[straight phi] j |" ... ...} } and { _{[straight phi] j |" [ ... , ... , ... ]} } are linear independent. Then, we have the following result.

Theorem 27.

Let Ψ denote the subspace of Γ ( ... ) spanned by { _{[straight phi] j} } . Then { _{e k} , _{[straight phi] j} } is a basis of Γ ( ... ) and Γ ( ... ) = Ψ [ecedil]5; C ( ... ) as vector spaces.

Proof.

Since { _{[straight phi] j |" ... ...} } and { _{[straight phi] j |" [ ... , ... , ... ]} } are linear independent, { _{[straight phi] j} } is linear independent in Γ ( ... ) . By definition of { _{e k} , _{[straight phi] j} } , the { _{e k} , _{[straight phi] j} } is independent in Γ ( ... ) . For [straight phi] ∈ Γ ( ... ) since { _{[straight phi] j |" ... ...} } is a basis of vector spaces { [straight phi] _{|" ... ...} |" [straight phi] ∈ Γ ( ... ) } , and { _{[straight phi] j |" [ ... , ... , ... ]} } is a basis of vector spaces { [straight phi] _{|" [ ... , ... , ... ]} |" [straight phi] ∈ Γ ( ... ) } , there exist _{g s} ∈ F , s ∈ J ( J is a finite set of positive integers) such that [figure omitted; refer to PDF] We then have [figure omitted; refer to PDF] If _{y 1} , _{y 2} , _{y 3} ∈ ... , then [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] is a central derivation. So there exist _{r i} ∈ F , i ∈ I ( I is a finite set of positive integers) such that [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] The proof is completed.

Lemma 28.

Let ... be a simple Lie triple algebra, R = F [ _{x 1} , ... , _{x n} ] , and ... ~ = ... [ecedil]7; R ; then one has Γ ( ... ~ ) ⊇ Ψ [ecedil]7; R + C ( g ) [ecedil]7; _{End ... F} ( R ) .

Proof.

Let ψ ∈ Γ ( ... ) , m ∈ R . Then, we have [figure omitted; refer to PDF] Thus ψ [ecedil]7; m ∈ Γ ( ... ~ ) . For any d ∈ C ( g ) and f ∈ En _{d F} ( R ) , d [ecedil]7; f ∈ Γ ( ... ~ ) , we have [figure omitted; refer to PDF] Hence Ψ [ecedil]7; R + C ( g ) [ecedil]7; En _{d F} ( R ) ⊆ Γ ( ... ~ ) .

Theorem 29.

Let ... be a simple Lie triple algebra, R = F [ _{x 1} , ... , _{x n} ] , and ... ~ = ... [ecedil]7; R . Then Γ ( ... ~ ) = Γ ( ... ) [ecedil]7; R .

Proof.

From Lemma 28, we get Γ ( ... ) [ecedil]7; R ⊆ Γ ( ... ~ ) . Now we prove the opposite inclusion. Let { _{m i} } be a basis for R , ψ ∈ Γ ( ... ~ ) , p ∈ R , x , y , z ∈ ... , and let _{η i} ( - , p ) be suitable maps in En _{d F} ( ... ) such that [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] while [figure omitted; refer to PDF] So, for each i and p , we have _{η i} ( p [ x , y , z ] ) = [ _{η i} ( x , p ) , y , z ] = [ x , _{η i} ( y , p ) , z ] = [ x , y , _{η i} ( z , p ) ] , _{η i} ( p , ( x y ) ) = ( _{η i} ( x , p ) , y ) = ( x , _{η i} ( y , p ) ) , for all x , y , z ∈ ... . Hence, _{η i} ( - , p ) ∈ Γ ( ... ) . But Γ ( ... ) = F id . Therefore, _{η i} ( x , p ) = _{λ i} ( p ) x , for all x ∈ ... , for suitable scalars _{λ i} ( p ) .

Thus, we may write [figure omitted; refer to PDF] Since the right-hand side is in ... [ecedil]7; R , for each p , we get _{λ i} ( p ) = 0 for all except for a finite number of i . That is, [figure omitted; refer to PDF] Then, the map [figure omitted; refer to PDF] is well defined. Hence, we have ψ ( x [ecedil]7; p ) = x [ecedil]7; ρ ( p ) . Thus, [figure omitted; refer to PDF] Choose x , y , z ∈ ... such that x y ...0; 0 , [ x , y , z ] ...0; 0 . Then we can conclude that ρ ( p ) = p ρ ( 1 ) , for all p ∈ R . So ρ is determined by its action on 1. Therefore, ρ ∈ R . Thus, [figure omitted; refer to PDF] That is, ψ ∈ id [ecedil]7; R [congruent with] Γ ( ... ) [ecedil]7; R . So, we have Γ ( ... ~ ) = Γ ( ... ) [ecedil]7; R .

Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions on this paper. The works was supported by NNSF of China (no. 11171057), Natural Science Foundation of Jilin province (no. 201115006), the Fundamental Research Funds for the Central Universities (no. 12SSXT139), and Scientific Research Foundation for Returned Scholars, Ministry of Education of China.

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