aa r X i v : . [ m a t h . R A ] M a r CLASSIFICATION PROBLEM OF SIMPLE HOM-LIE ALGEBRAS
YOUNESS EL KHARRAF
Abstract.
First, we construct some families of nonsolvable anticommutative alge-bras, solvable Lie algebras and even nilpotent Lie algebras, that can be endowed withthe structure of simple Hom-Lie algebras. This situation shows that a classificationof simple Hom-Lie algebras would be unrealistic without any further restrictions.Therefore, we introduce the class of strongly simple Hom-Lie algebras , which is theclass of anticommutative algebras that are simple Hom-Lie with respect to all theirtwisting maps. We show some of its properties, provide a characterisation and ex-plore some of its subclasses. Then, we classify completely regular simple Hom-Liealgebras over any arbitrary field. Furthermore, we establish that every simple anti-commutative algebra of dimension 3 turns out to be a simple Lie algebra where itsLie bracket is deformed by a bijective linear map, and also we determine all the sim-ple Hom-Lie algebras in dimension 2, that were wrongly claimed to be nonexistentin [5]. Introduction
For a long time, classification problems have arisen as natural quests with thewide expansion of mathematics. Such investigation often leads to a better and adeeper comprehension of the subject and frequently gives birth to new ideas andtechniques, and even in some cases, allows breakthroughs on some hard questions,e.g
Two Generators Theorem of simple Lie algebras in arbitrary characteristics , cf.[3]. If we quoted some examples here of achieved classifications, there would be two :One, the classification of finite simple groups , which is considered as one of the greatintellectual achievements of humanity, and the other is the classification of simpleLie algebras over fields of characteristic different from and
3, which is a landmarkachievement of modern mathematics. There is no need to recall the well-known rolesand the importance that they play in almost every area of mathematics, especially ingeometry and physics.
Date : March 2, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Classification, Simple Hom-Lie algebras, Simple anticommutative, Reg-ular simple Hom-Lie, Arbitrary field.
Hoping to do the same for the class of simple Hom-Lie algebras, even partially.Unfortunately, we show the existence of many aberrant cases, which makes a classifi-cation seems to be out of reach once it is considered without any further restrictions.In 2016, there was an attempt by X. Chen and W. Han, in [5], to classify
Multi-plicative simple Hom-Lie algebras , but their classification revealed to be incompleteas they only classified regular simple Hom-Lie algebras over an algebraically closedfield of characteristic zero. So, we complete it here and also we relax the restrictivecondition on the field to any arbitrary one. In addition, we determine all the simpleHom-Lie algebras in dimension 2 that were wrongly claimed to be nonexistent in [5].On the other hand, we introduce a new interesting class that we call
Strongly simpleHom-Lie algebras , it is defined as the class of simple Hom-Lie algebras with respect toevery twisting map. In this work, we determine the structure of this class, we describesome of its properties and we characterize it along with some of its subclasses.The article is organized as follows. In section 2, we summarize the definitions. Insection 3, we determine all the 2-dimensional simple Hom-Lie algebras. In section4, we revisit the classification of multiplicative simple Hom-Lie algebras and extendit to any arbitrary field. In section 5, we construct many families of nonsimpleanticommutative algebras that can be endowed with the structure of simple Hom-Liealgebras, then we introduce the class of strongly simple Hom-Lie algebras and wecharacterize it. The last section is devoted to some interesting subclasses of stronglysimple Hom-Lie algebras.Throughout this article, there is no restriction on the field F and all the algebrasare considered to be finite dimensional, unless it is stated otherwise.2. Preliminaries
The notion of Hom-Lie algebras was first introduced by J. T. Hartwig, D. Larsson,S. D. Silvestrov in [6], as it naturally emerged from the deformations of Witt andVirasoro algebras based on σ -derivations.Recall that an anticommutative algebra is an algebra such that its multiplicationsatisfies the anticommutative condition, i.e. x = 0 , for all x . LASSIFICATION PROBLEM OF SIMPLE HOM-LIE ALGEBRAS 3
Definition 2.1. A Hom-Lie algebra is a triple ( A, [ · , · ] , σ ) consisting of an anticom-mutative algebra ( A, [ · , · ]) and a linear map σ satisfying+ (cid:9) x,y,z [ σ ( x ) , [ y, z ]] = 0 , ∀ x, y, z ∈ A, (the Hom-Jacobi identity)The linear map σ is called the twisting map of Hom-Lie algebra A . The set of alltwisting maps on A is noted by HS( A ). Definition 2.2.
A homomophism of Hom-Lie algebras ϕ : ( A, [ · , · ] A , σ A ) −→ ( B, [ · , · ] B , σ B )is a linear map such that ϕ ◦ [ · , · ] A = [ · , · ] B ◦ ( ϕ ⊗ ϕ ) and ϕ ◦ σ A = σ B ◦ ϕ . Definition 2.3.
Let ( A, [ · , · ] , σ ) be a Hom-Lie algebra. It is called to be(1) Multiplicative , if σ ([ x, y ]) = [ σ ( x ) , σ ( y )] , ∀ x, y ∈ A .(2) Regular , if σ is an automorphism of anticommutative algebras.To avoid ambiguity, we introduce the notion of Hom-ideal. Definition 2.4.
A Hom-ideal I is an ideal in the usual sense, which is, in addition,left stable by the twisting map, i.e [ I, A ] ⊂ I and σ ( I ) ⊂ I . Definition 2.5.
A Hom-Lie algebra ( A, [ · , · ] , σ ) is called simple, if it is not abelianand has no proper Hom-ideals. It is called semisimple if its Hom-radical is zero, wherea Hom-radical is the maximal solvable Hom-ideal.3. Classification of 2-dimensional simple Hom-Lie algebras
Here, we determine all 2-dimensional simple Hom-Lie algebras, but first, we de-scribe the phenomena that over some nonalgebraically closed fields there exists abelianalgebras of dimension greater than one such that, with respect to some twisting maps,their only Hom-ideals are the trivial ones.
Proposition 3.1.
There is infinitely many fields over which the abelian two-dimensional algebra a has only trivial Hom-ideals with respect to some twisting map.Proof. Every proper nontrivial ideal of the 2-dimensional abelian algebra a is spannedby a nonzero vector. Thus, a twisting map σ ∈ End F ( a ), which makes the onlyHom-ideals of a to be the trivial ones, must have no eigenvectors. Hence, F is notalgebraically closed. It is obvious that HS( a ) = End( a ) for dimensional reason. Let YOUNESS EL KHARRAF σ be such that B = { e , σ ( e ) } is a basis for some vector e ∈ a . Hence, the matrixof σ with respect to this basis is of the form M B ( σ ) = (cid:18) a a (cid:19) Clearly a = 0, because if not σ ( e ) would be an eigenvector of σ . Let’s suppose thatthere exists a nonzero vector x = αe + βσ ( e ) such that ∃ λ ∈ F , σ ( x ) = λx , then (cid:26) λα = a βλβ = a β + α (3.1.1)So αβ = 0 and so do λ = 0 which in addition must satisfy the following equation λ − a λ − a = 0 (3.1.2)For finite fields, a simple counting argument suffices. Indeed, there is | F | = q monicpolynomials of degree 1, X − c , c ∈ F , and similarly there is q monic polynomialsof degree 2, X + c X + c ∈ F [ X ]. By the commutativity there is only q + q distinctproducts of monic polynomials of degree 1. Hence, there is (cid:0) q (cid:1) irreducible monicquadratic polynomials and therefore there exists a and a such that (3 . .
2) has nosolution. For infinite fields, a has a twisting map with no eigenvectors if and only ifthe field F admits at least an irreducible quadratic polynomial. In both cases, suchfields are infinitely many and in characteristic zero they are uncountably many. (cid:3) Corollary 3.2.
Over algebraically closed fields, any abelian algebra of dimensiongreater than one, has proper nontrivial Hom-ideals with respect to any linear map.
Remark 3.3.
In the definition of simple anticommutative algebras or simple Liealgebras, the condition to be not abelian, has for unique purpose to avoid that the1-dimensional algebra to be considered as simple. But, for the case of simple Hom-Liealgebras, the situation is slightly different as this condition excludes, effectively, allthe abelian algebras, cf. the above Proposition 3.1.
Theorem 3.4.
In dimension , the only nonabelian anticommutative algebra, up toisomorphism, is the affine Lie algebra aff ( F ) , which can be endowed with the structureof simple Hom-Lie algebra whenever the twisting map does not leave stable its one-dimensional proper ideal.Proof. It’s an easy task to show that every 2-dimensional nonabelian anticommutativealgebra is isomorphic to the affine Lie algebra aff ( F ) : [ x, y ] = y . Then, since y spans the only proper ideal, to have the structure of a simple Hom-Lie algebra, itis reduced to take a twisting map σ such that σ ( y ) / ∈ F y , which is possible sinceHS( aff ( F )) = End( aff ( F )). (cid:3) LASSIFICATION PROBLEM OF SIMPLE HOM-LIE ALGEBRAS 5
Corollary 3.5.
There is no multiplicative simple Hom-Lie algebra of dimension .Proof. Let I is a 1-dimensional ideal, then [ I, aff ( F )] = F y = I . Thus, σ ( I ) = σ ([ I, aff ( F )]) = [ σ ( I ) , σ ( aff ( F ))] ⊂ F y = I , and hence I is a proper nontrivial Hom-ideal of aff ( F ). (cid:3) Classification of Multiplicative Simple Hom-Lie algebras
Despite the title in [5],
Classification of Multiplicative Simple Hom-Lie Algebras , theauthors only classified regular simple Hom-Lie algebras over an algebraically closedfield of characteristic zero. Here, we extend the classification to any arbitrary field F and also we give a characterisation of the multiplicative nonregular case. Lemma 4.1.
Let ( A, [ · , · ] , σ ) be a multiplicative simple Hom-Lie algebra, then σ ∈ Aut( A ) or σ = 0 .Proof. By multiplicativity Ker( σ ) is a Hom-ideal. Hence, by simplicity, it is eitherzero or equal to A . (cid:3) Corollary 4.2.
A multiplicative simple Hom-Lie algebra is either a regular simpleHom-Lie algebra or a simple anticommutative algebra when the twisting map is thezero map.
Lemma 4.3.
Let ( A, [ · , · ] , σ ) be a regular Hom-Lie algebra, then ( A, [ · , · ] σ − := σ − ◦ [ · , · ]) is its induced Lie algebra and σ is an automorphism of Lie algebras.Proof. We have + (cid:9) x,y,z [ σ ( x ) , [ y, z ]] = + (cid:9) x,y,z σ (cid:0) σ − ◦ [ x, σ − ◦ [ y, z ]] (cid:1) = 0 implies+ (cid:9) x,y,z [ x, [ y, z ] σ − ] σ − = 0. In addition, σ is automatically an automorphism of Liealgebras. (cid:3) Lemma 4.4. ϕ : ( A, [ · , · ] A , σ A ) −→ ( B, [ · , · ] B , σ B ) is a homomorphism of regularHom-Lie algebras, if and only if ϕ : ( A, [ · , · ] Aσ − A ) −→ ( B, [ · , · ] Bσ − B ) is a homomorphismof Lie algebras and ϕ ◦ σ A = σ B ◦ ϕ . YOUNESS EL KHARRAF
Proof. ϕ : ( A, [ · , · ] A , σ A ) −→ ( B, [ · , · ] B , σ B ) ⇐⇒ (cid:26) ϕ ◦ [ · , · ] A = [ · , · ] B ◦ ϕ ⊗ ϕϕ ◦ σ A = σ B ◦ ϕ ⇐⇒ (cid:26) ϕ ◦ σ − A ◦ [ · , · ] A ◦ σ A ⊗ σ A = σ − B ◦ [ · , · ] B ◦ σ B ⊗ σ B ◦ ϕ ⊗ ϕϕ ◦ σ A = σ B ◦ ϕ ⇐⇒ ( ϕ ◦ [ · , · ] Aσ − A ◦ σ A ⊗ σ A = [ · , · ] Bσ − B ◦ ϕ ⊗ ϕ ◦ σ A ⊗ σ A ϕ ◦ σ A = σ B ◦ ϕ ⇐⇒ ( ϕ ◦ [ · , · ] Aσ − A = [ · , · ] Bσ − B ◦ ϕ ⊗ ϕϕ ◦ σ A = σ B ◦ ϕ ⇐⇒ ( ϕ : ( A, [ · , · ] Aσ − A ) −→ ( B, [ · , · ] Bσ − B ) ϕ ◦ σ A = σ B ◦ ϕ (cid:3) Remark 4.5.
For multiplicative Hom-Lie algebras, the image of an ideal by thetwisting map is again an ideal.
Theorem 4.6.
An anticommutative algebra can be endowed with the structure of aregular simple Hom-Lie algebra if and only if it is simple and has an automorphictwisting map.Proof.
Let ( A, [ · , · ] , σ ) be a regular simple Hom-Lie algebra and suppose I to be aproper nontrivial minimal ideal of A . By Hom-simplicity σ ( I ) = I , and by minimality I ∩ σ ( I ) = 0 and for any i, j ≥ , σ i ( I ) ∩ σ j ( I ) = 0 or σ i − j ( I ) = I . Then, by thefinitude of the dimension of A , there exists n ≥ σ n ( I ) ⊂ L n − i =0 σ i ( I ).Hence, L n − i =0 σ i ( I ) is invariant by σ and thus equal to A . In addition, σ n ( I ) ∩ A = L n − i =0 σ n ( I ) ∩ σ i ( I ) = σ n ( I ), and therefore σ n ( I ) = I , since if σ n − j ( I ) = I for some j ≥ L n − i = j σ i ( I ) is a nontrivial proper Hom-ideal, which isabsurd. However, the Hom-Jacobi identity, for x, y, z ∈ I ,[ σ n ( x ) , [ y, z ]] + [ σ ( y ) , [ z, σ n − ( x )]] + [ σ ( z ) , [ σ n − ( x ) , y ]] = 0implies that [ I, [ I, I ]] = 0. It follows that [ A, [ A, A ]] = 0, but since [
A, A ] is a Hom-ideal and A is simple Hom-Lie, then [ A, A ] = A and so A must be zero, which is acontradiction. Thus, A is a simple anticommutative algebra, with σ is its automorphictwisting map. The converse is straightforward. (cid:3) Theorem 4.7.
A regular simple Hom-Lie algebra ( A, [ · , · ] , σ ) is a direct sum of n -copies of the same minimal Lie ideal g of a semisimple Lie algebra such that A = LASSIFICATION PROBLEM OF SIMPLE HOM-LIE ALGEBRAS 7 L ni =1 g i , ( g i , [ · , · ] i ) ≃ ( g , [ · , · ] g ) , [ · , · ] = σ ◦ ⊕ ni =1 [ · , · ] i and the twisting map is of theform σ = ⊕ ni =1 σ iρ ( i ) where σ iρ ( i ) : g i −→ g ρ ( i ) are automorphisms of the Lie algebra g and ρ is a cyclic permutation. Furthermore, if char( F ) = 0 , then g is a simple Liealgebra. Otherwise, if char( F ) = p > , then g = e S ⊗ O ( m, with e S is a simple Liealgebra and O ( m, ≃ F [ X , ..., X m ] / ( X p , · · · , X pm ) .Proof. Following the Lemma 4.3, ( A, [ · , · ] σ − ) is the induced Lie algebra of ( A, [ · , · ] , σ )and let Rad( A ) its radical. By multiplicativity, σ (Rad( A )) is also a solvable ideal,thus σ (Rad( A )) ⊂ Rad( A ), and because σ is a one-to-one map, the equality holds σ (Rad( A )) = Rad( A ). Hence, Rad( A ) is a Hom-ideal, and by Hom-simplicity eitherRad( A ) = 0 or Rad( A ) = A . But, in the multiplicative case, the derived series ofa Hom-ideal are Hom-ideals. Therefore, Rad( A ) must be zero. Hence, ( A, [ · , · ] σ − )is a semisimple Lie algebra and σ is an automorphism of Lie algebras. If char( F ) = p >
0, then, thanks to the structure theorem of semisimple Lie algebras in primecharacteristic due to R. E. Block in [2], a semisimple Lie algebra has its minimalideals in direct sum, called the socle of the semisimple Lie algebra, and a minimalideal is of the form e S ⊗ O ( m ; 1), where m ≥
0, 1 = (1 , · · · ,
1) a m -tuple, e S issimple Lie algebra and O ( m ; 1) is isomorphic to the truncated polynomial algebra F [ X , ..., X m ] / ( X p , · · · , X pm ). Then, following the same arguments as the first partof the above proof of the Theorem 4.6, one can show that the socle S is σ -invariantand S = L n − i =0 σ i ( e S ⊗ O ( m ; 1)). Therefore, the socle S is a Hom-ideal and by theHom-simplicity S = A . Now, if char( F ) = 0, then A is a direct sum of simple Lieideals thanks to structure theorem in [7]. In the same fashion, A = L n − i =0 σ i ( g ) where g is a simple Lie ideal of its induced Lie algebra. Finally, we can summarize the bothcases as follows : A = L ni =1 g i where [ · , · ] = σ ◦ ⊕ ni =1 [ · , · ] i , σ n ( g i ) = g i , ≤ i ≤ n ,( g i , [ · , · ] i ) ≃ ( g , [ · , · ]) is a unique minimal Lie ideal of a semisimple Lie algebra up toisomorphism, and so σ = ⊕ ni =1 σ iρ ( i ) with σ iρ ( i ) := σ | g i : g i −→ g ρ ( i ) are automorphismsof the Lie algebra g where ρ is clearly a cyclic permutation. (cid:3) Remark 4.8.
The dimension of a regular simple Hom-Lie algebra is a multiple ofthe dimension of a minimal ideal of a semisimple Lie algebra.
Theorem 4.9.
A multiplicative simple Hom-Lie algebra ( A, [ · , · ] , σ ) is either a directsum of n -copies of the same minimal Lie ideal g of a semisimple Lie algebra upto the conjugacy of σ n | g , or a simple anticommutative algebra up to isomorphism ofanticommutative algebras, if the twisting map is zero.Proof. For σ = 0, the result is direct. For the regular case, by the structure theoremof regular simple Hom-Lie algebras, cf. the above Theorem 4.7, we have A = L ni =1 g i YOUNESS EL KHARRAF is a direct sum of the same minimal ideal g up to isomorphism, where g i ≃ g , σ | g i = σ iρ ( i ) : g i −→ g ρ ( i ) and σ = ⊕ ni =1 σ iρ ( i ) , and let B = L ni =1 h i where h i ≃ h , η | h i = η iρ ′ ( i ) : h i −→ h ρ ′ ( i ) and η = ⊕ ni =1 η iρ ′ ( i ) . So, thanks to the Lemma 4.4,( A, [ · , · ] , σ ) ϕ ≃ −→ ( B, [ · , · ] ′ , η ) ⇐⇒ (cid:26) ϕ is an isomorphism of Lie algebras ,ϕ ◦ σ = η ◦ ϕ ⇐⇒ ϕ = ⊕ ni =1 ϕ i , ϕ | g i = ϕ i : g i −→ h τ ( i ) is an isomorphism ofLie algebras, τ is a cyclic permutation ,ϕ ρ ( i ) ◦ σ iρ ( i ) = η τ ( i ) ρ ′ ◦ τ ( i ) ◦ ϕ i , ≤ i ≤ n ⇐⇒ ϕ = ⊕ ni =1 ϕ i , ϕ i : g i −→ h τ ( i ) is an isomorphism of Lie algebras ,τ is a cyclic permutation , ≤ i ≤ nϕ i ◦ σ ρ n − ( i ) i ◦ σ iρ ( i ) · · · σ ρ n − ( i ) ρ n − ( i ) = η ρ ′ n − ( τ ( i )) τ ( i ) ◦ η τ ( i ) ρ ′ ( τ ( i )) · · · η ρ ′ n − ( τ ( i )) ρ ′ n − ( τ ( i )) ◦ ϕ i ⇐⇒ ϕ = ⊕ ni =1 ϕ i , ϕ i : g i −→ h τ ( i ) is a isomorphism of Lie algebras ,τ is a cyclic permutation ,ϕ i ◦ σ n | g i = η n | ϕ i ( g i ) ◦ ϕ i , ≤ i ≤ n ⇐⇒ ∃ i, ϕ i : g i −→ h τ ( i ) an isomorphism of Lie algebras ,τ is a cyclic permutation ,ϕ i ◦ σ n | g i = η n | ϕ i ( g i ) ◦ ϕ i To convince ourselves that the last equivalence holds. It suffices to note that onecan recover ϕ from the following commutative diagram, once ϕ i is given g i g ρ ( i ) h τ ( i ) h ρ ′ ◦ τ ( i ) σ iρ ( i ) ϕ i ϕ ρ ( i ) η τ ( i ) ρ ′◦ τ ( i ) Conclusion, ( A, [ · , · ] , σ ) and ( B, [ · , · ] ′ , η ) are isomorphic if and only if there exists anisomorphism φ of the minimal ideals g and h of their induced respective Lie algebrassuch that σ n | g = φ − ◦ η n | h ◦ φ , equivalently, if there exists i and j such that σ n | g i and η n | h j are conjugate. Therefore, a regular simple Hom-Lie structure is unique up to theconjugacy of the n th power of its twisting map on any of its minimal ideals. (cid:3) LASSIFICATION PROBLEM OF SIMPLE HOM-LIE ALGEBRAS 9 Strongly Simple Hom-Lie algebras
In this section, we show that a semisimple Hom-Lie algebra needs not to be a directsum of simple Hom-Lie algebras. We provide many examples of nonsimple, solvableand nilpotent anticommutative algebras that can be endowed with the structure ofsimple Hom-Lie algebras. Also, we introduce the class of strongly simple Hom-Liealgebras and we characterize it.
Lemma 5.1.
A simple Lie algebra g has the property that for any nonzero vector x , dim[ x, g ] ≥ .Proof. Let x be a nonzero vector such that dim[ x, g ] ≤
1, and as g is centerless, then g = ker ad x ⊕ F y for some y = 0. The adjoint representation endows g with thestructure of a simple module for the universal enveloping algebra U ( g ). Letting I g be the augmentation ideal of U ( g ), we have [ y, x ] ∈ I g · x \ { } , so that I g · x = g .On the other hand, the PBW-Theorem implies I g · x = P i ≥ y i U (ker ad x ) · x = P i ≥ F (ad y ) i ( x ) ⊂ (ad y )( g ), a contradiction. (cid:3) The following proposition was inspired from the almost classical Lie algebra pgl ( n ) Proposition 5.2.
Let S be a simple Lie algebra over any arbitrary field F . Theextended anticommutative algebra A = F d ⋉ S , by setting [ d, x ] = x where x is anonzero vector, is semisimple and has a unique proper nontrivial ideal S which is asimple ideal of codimension .Proof. A = F d ⋉ S with [ d, x ] = x , so if I is a proper nontrivial ideal other than S ,then there exists y ∈ S and a subspace J ⊂ S such that I = span { d + y } ⊕ J . So, if J = 0, then there exists z ∈ [ J, S ] ⊂ S such that z / ∈ J , because S is simple. Hence, I is not an ideal and therefore J = 0. So, if I = span { d + y } , then, thanks to theabove Lemma 5.1, we must have dim[ d + y, S ] ≥
1, but as [
S, A ] = S then I cannotbe an ideal and therefore S is the unique proper nontrivial ideal of A . (cid:3) Remark 5.3.
In general, a semisimple anticommutative algebra needs not to be adirect sum of simple anticommutative algebras over any arbitrary field. The sameholds for semisimple Hom-Lie algebras.The following proposition provides a family of nonsimple anticommutative algebraswhich can have the structure of simple Hom-Lie algebras.
Proposition 5.4.
Let S n be the anticommutative algebra defined by the followingrelations, with respect to the basis { e , · · · , e n } , n ≥ (cid:26) [ e i , e i +1 ] = e i +2 , ≤ i ≤ n − , [ e n − , e n ] = e and [ e n , e ] = e Then, the extension A n +1 = F d ⋉ S n , given by [ d, e ] = e , is a simple Hom-Liealgebra with respect to the twisting map σ defined by σ ( d ) = σ ( e i ) = 0 , ≤ i ≤ n − and σ ( e n ) = d. Proof.
Clearly S n is simple, and thanks to the proof of proposition 5.2, if for any x = 0, dim[ x, S n ] ≥
2, then the extension A n +1 has S n as a unique proper nontrivialideal. So, it follows from the construction of σ that it does not leave stable S n .Therefore, it suffices to check that σ is a twisting map. We observe that for any x, y ∈ { e , · · · , e n − } ∪ { d } , [ x, y ] ∈ C A n +1 ( d ) := { x ∈ A n +1 | [ x, d ] = 0 } , and so itremains + (cid:9) e n ,x,y [ σ ( e n ) , [ x, y ]] = [ d, [ x, y ]] = 0. Thus, σ ∈ HS( A n +1 ). (cid:3) Examples 5.5. (1) The Heisenberg Lie algebra h n +1 , generated by elements e i , ≤ i ≤ n + 1, with the relations :[ e i − , e i ] = e n +1 , ≤ i ≤ n is a nilpotent Lie algebra that can be endowed with the structure of a simpleHom-Lie algebra. For that, it suffices to take σ i ( e n +1 ) = e n +1 − i , ≤ i ≤ n Indeed, any linear map σ hence defined is a twisting map, since one can easilycheck that HS( h n +1 ) = End( h n +1 ). Then, as a consequence of Engel’s Theo-rem (cf. [11]), every ideal intersects nontrivially the center and so the vector e n +1 is included in every nontrivial ideal. Therefore, by the construction of σ ,a nontrivial Hom-ideal must be equal to the whole Lie algebra. Furthermore,we notice that the twisting map that provide h n +1 with the structure of asimple Hom-Lie algebra needs not to be bijective.(2) Let R n be the anticommutative algebra of dimension 2 n ≥
4, defined by : (cid:26) [ e i − , e i ] = e i +1 , ≤ i ≤ n − e n − , e n ] = e Clearly R n is solvable and a straightforward verification shows that the linearmap σ given by σ ( e i − ) = e i +2 , ≤ i ≤ n − σ ( e n − ) = e and σ = 0,is a twisting map. Finally, for a nontrivial Hom-ideal I , the brackets’ formimplies that 0 = [ I, R n ] ⊂ I ∩ [ R n , R n ], and so at least there exists a vector LASSIFICATION PROBLEM OF SIMPLE HOM-LIE ALGEBRAS 11 e j ∈ I, j ≡ σ -invariance of I we must have I = R n . Hence, R n is a simple Hom-Lie algebra.(3) The direct sum of the affine Lie algebra aff ( F ) : [ x, y ] = y with any abelianalgebra a n , can be endowed with the structure of simple Hom-Lie algebra.Indeed, it suffices with respect to a basis { e , · · · , e n } of a n , to define thedesired twisting map σ by σ ( y ) = x, σ ( x ) = e , σ ( e n ) = y and σ ( e i ) = e i +1 , ≤ i ≤ n −
1. Any nontrivial Hom-ideal , by construction, must contains y and so the whole direct sum. Remark 5.6. (1) The above examples show that a classification of simple Hom-Lie algebras is not possible without restrictions on the twisting map. Thisexactly the direction that we adopt for the rest of this paper.(2) The twisting map of a simple Hom-Lie algebra needs not to be bijective.
Definition 5.7. A strongly simple Hom-Lie algebra is a nonabelian anticommu-tative algebra where every twisting map endows it with a simple Hom-Lie structure. Theorem 5.8.
An anticommutative algebra is a strongly simple Hom-Lie algebra ifand only if it is simple.Proof.
For the zero twisting map, a strongly simple Hom-Lie algebra is just an anti-commutative algebra that has no proper ideal, and so it is simple. The converse istrivial. (cid:3)
We shall denote by SS , MS and RS , respectively, the class of strongly simpleHom-Lie algebras, the class of anticommutative algebras that have multiplicativesimple Hom-Lie structures for some twisting maps and the class of anticommutativealgebras that have regular simple Hom-Lie structures for some twisting maps. Proposition 5.9. MS = SS .Proof. Thanks to the Corollary 4.2 and the Theorem 4.6, a multiplicative simple Hom-Lie algebra is just a simple anticommutative algebra if we forget the twisting mapand by the characterisation of Theorem 5.8,
MS ⊂ SS . Reciprocally, every stronglysimple Hom-Lie algebra equipped with a zero map as a twisting map is multiplicativesimple Hom-Lie, hence
SS ⊂ MS . (cid:3) Strongly* and Pure Strongly Simple Hom-Lie algebras
In some cases, strongly simple Hom-Lie algebras may have HS = 0, as it will beencountered later in Example 6.4. So, it is natural to introduce new subclasses thatavoid this phenomena.
Definition 6.1.
We call strongly ∗ simple Hom-Lie algebra (resp. pure stronglysimple Hom-Lie algebra ) a strongly simple Hom-Lie algebra A that has HS( A ) = 0(resp. if it has a bijective twisting map). We note respectively their classes SS ∗ and PS . Remark 6.2.
Obviously,
RS ⊂ PS ⊂ SS ∗ ⊂ SS = MS . Theorem 6.3.
In dimension 3, every simple anticommutative algebra is isomorphicto the simple Lie algebra so (3 , F ) which its Lie bracket is deformed by a bijective linearmap. In addition, RS PS = SS ∗ = SS = MS .Proof. Let { e , e , e } be a basis of the 3-dimensional simple anticommutative al-gebra ( A, [ · , · ]) and set σ be defined by [ e , e ] = σ ( e ) , [ e , e ] = σ ( e ) and[ e , e ] = σ ( e ). So, by simplicity and the number of the brackets, σ is bijective.Then, + (cid:9) e ,e ,e [ σ ( e ) , [ e , e ]] = + (cid:9) ≤ i ≤ [ σ ( e i ) , σ ( e i )] = 0. Therefore, σ is a bijective twist-ing map, and so ( A, [ · , · ] , σ ) is a pure strongly simple Hom-Lie algebra. In addition,we remark that ( A, [ · , · ] σ − ) ≃ so (3 , F ).For RS PS , we consider the anticommutative algebra defined by :[ e , e ] = e , [ e , e ] = e , [ e , e ] = e + e Its twisting map has the following form, with the coefficients a ij are in F , a a − a a a a a a a a − a Then, when considering the homomorphism condition, with the assistance of Maple ® ,we find that, in both cases of a = 0 or not, the twisting map must have a nontrivialkernel and hence we have the desired result. (cid:3) Examples 6.4.
All the checks were carried out using Maple ® , and all the coefficientsare supposed to be in an arbitrary field F . LASSIFICATION PROBLEM OF SIMPLE HOM-LIE ALGEBRAS 13 (1) Let A be the 4-dimensional anticommutative algebra defined by the followingrelations :[ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e + e Let x = P i =1 a i e i in a proper ideal I , [ x, e ] = a e − a e . If ( a , a ) = (0 , x, e ] with e or e we get e in I , which leads to I = A . So, x = a e + a e and [ x, e ] = − a e + a e in I implies that a = 0, and if a = 0, then e in I leading to I = A . Hence, A is simple. Onthe other hand, a twisting map σ has the following form, with respect to thestandard basis B = { e , e , e , e } , M B ( σ ) = a a − a a − a a a and det( M B ( σ )) = a a a Thus, A has a bijective twisting map and so A ∈ PS . In addition, we findthat σ is a homomorphism implies that a = 0 and so det( σ ) = 0. Thus, A cannot be endowed with the structure of a regular simple Hom-Lie algebra,cf. Theorem 4.6. Hence, RS PS .(2) Let A be the 4-dimensional anticommutative algebra defined by :[ e , e ] = e , [ e , e ] = e , [ e , e ] = e , [ e , e ] = e As above, one can show that A is simple. Also, we find that a twisting map σ has the form σ ( e ) = a e + a e + a e , σ ( e ) = a e and σ ( e ) = σ ( e ) = 0.Thus, HS( A ) = 0 and ∀ σ ∈ HS( A ) , det( σ ) = 0. Therefore, PS SS ∗ .(3) Let A be the 4-dimensional anticommutative algebra defined by the samerelations as A , to which we add this new entry : [ e , e ] = e . Clearly, A issimple and we find that HS( A ) = 0. So, SS ∗ SS . Proposition 6.5. RS PS SS ∗ SS = MS .Proof. Thanks to the above Example 6.4 and the Proposition 5.9. (cid:3)
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