On the capablility of Hom-Lie algebras
aa r X i v : . [ m a t h . R A ] J a n ON THE CAPABLILITY OF HOM-LIE ALGEBRAS
J.M. CASAS AND X. GARCÍA-MARTÍNEZ
Abstract.
A Hom-Lie algebra p L, α L q is said to be capable if there existsa Hom-Lie algebra p H, α H q such that L – H { Z p H q . We obtain a character-isation of capable Hom-Lie algebras involving its epicentre and we use thistheory to further study the six-term exact sequence in homology and to obtaina Hopf-type formulae of the second homology of perfect Hom-Lie algebras. Introduction
Hom-Lie algebras were introduced in [27] to study the deformations of the Wittand Virasoro algebras mainly motivated by the study of quantum deformationsand discretisation of vector fields via twisted derivations. Their algebraic structureconsists in an anticommutative algebra satisfying a twisted version of the Jacobiidentity (see Definition 2.2). Since then, many authors extended this idea to manyother different algebraic structures, becoming a very prolific research area.From a categorical-algebraic point of view, the category of Hom-Lie algebrasrepresents a very interesting example that it is worth to comprehend better. Thismay get us closer to understand categorically some complicated algebraic concepts.In fact, the category of Hom-Lie algebras (even adding the property of being mul-tiplicative), it is known to be a semi-abelian category [31] which does not satisfymany of the stronger categorical-algebraic conditions, such as locally algebraicallycartesian closed [25], action representable [5], algebraic coherence [13] or
Normalityof Higgins commutators [14]. On the other hand, if we ask the twist to be an auto-morphism, then it satisfies them all, since it becomes a category of Lie objects oversome monoidal category [21, 22, 23, 24, 26]. Therefore, Hom-Lie algebras becomea not-so-complicated example of a bad-behavioured semi-abelian algebraic variety.For instance, Hom-Lie algebras were crucial to understand the conditions needed ina semi-abelian category to have a coherent universal central extension theory [12].In this article, we will study capable Hom-Lie algebras , an idea that comes fromthe concept of capable groups [1]. A group G is capable if there exists a group H such that G – H { Z p H q . In [2], the epicentre was introduced to characterise ca-pable groups: a group is capable if and only if its epicentre is trivial. Later on, avery interesting relation of the epicentre with the non-abelian exterior square wasfound [19].Capable Lie algebras were introduced in [38] and further studied in [33, 37].Nevertheless, the generalisation to the Hom-Lie case is non-trivial due to the lossof some interesting properties, such as the universal central extension condition [12],the difference between the Higgins and Huq commutator, or the fact that it is notknown whether its standard homology theory can be obtained from a T or functor.The present manuscript is organised as follows: In Section 2 we recall severalknown concepts. Section 3 is devoted to introduce the notion of non-abelian exteriorproduct of Hom-Lie algebras, and to find its relation with homology. In Section 4
This work was supported by Ministerio de Economía y Competitividad (Spain), with grantnumber MTM2016-79661-P. The second author is a Postdoctoral Fellow of the Research Founda-tion–Flanders (FWO). we obtain a six-term exact sequence involving homology and the non-abelian tensorproduct, that will be useful in Section 5 where the capability condition is studied.We define the tensor and exterior centres and relate them with the epicentre of aHom-Lie algebra. Then, we obtain a characterisation of capable Hom-Lie algebrasin terms of their epicentre. Finally, we use the work previously done to furtherstudy the six-term exact sequence in homology and to obtain a Hopf-type formulaeof the second homology of perfect Hom-Lie algebras.2.
Hom-Lie algebras
Throughout this paper we fix K as a ground field. Vector spaces are consideredover K and linear maps are K -linear maps. We write b (resp. ^ ) for the tensorproduct b K (resp. exterior product ^ K ) over K .We begin by reviewing some terminology and recalling already known notionsused in the paper. We mainly follow [27, 32, 34, 42], although with some modifica-tions.2.1. Basic definitions.Definition 2.2. A Hom-Lie algebra p L, α L q is a non-associative algebra L togetherwith a linear map α L : L Ñ L (sometimes called twist ) satisfying r x, y s “ ´r y, x s , (skew-symmetry) “ α L p x q , r y, z s ‰ ` “ α L p z q , r x, y s ‰ ` “ α L p y q , r z, x s ‰ “ , (Hom-Jacobi identity)for all x, y, z P L .In this paper we will only consider the so called multiplicative Hom-Lie alge-bras , i.e., Hom-Lie algebras p L, α L q such that α L preserves the product α L r x, y s “r α L p x q , α L p y qs for all x, y P L . Nevertheless, as it is standard in the literature, wewill omit the word multiplicative . Example . a) Taking α L “ id L , we recover exactly Lie algebras.b) Let V be a vector space and α V : V Ñ V be a linear map, then thepair p V, α V q is called Hom-vector space . A Hom-vector space p V, α V q to-gether with the trivial product r´ , ´s (i.e., r x, y s “ for any x, y P V ) is aHom-Lie algebra p V, α V q , which is called abelian Hom-Lie algebra .c) Let L be a Lie algebra, and α L : L Ñ L be a Lie algebra endomorphism.Then p L, α L q is a Hom-Lie algebra with the bracket defined by r x, y s α L “r α L p x q , α L p y qs , for all x, y P L [42].d) Any Hom-associative algebra p A, α A q can be endowed with a structure ofHom-Lie algebra by means of the bracket r a, b s “ ab ´ ba, for a, b P A [34].Hom-Lie algebras are the objects of the category HomLie , whose morphisms areLie algebra homomorphisms f : L Ñ L such that f ˝ α L “ α L ˝ f . Clearly thereis a full embedding Lie ã Ñ HomLie , L ÞÑ p L, id L q , where Lie denotes the category ofLie algebras.Since
HomLie is a variety of Ω -groups in the sense of Higgins [28] it is a semi-abelian category, therefore the ˆ -lemma and the Snake lemma automaticallyhold [3]. Below we explicitly study some categorical-algebraic notions in the partic-ular case of the category HomLie (the general definitions in the semi-abelian contextcan be found in [4, 5, 6, 7, 30]).
Definition 2.4. A subalgebra p H, α H q of a Hom-Lie algebra p L, α L q is a vectorsubspace H of L , which is closed under the bracket and invariant under α L . Insuch a case we may write α L | for α H . A subalgebra p H, α L | q of p L, α L q is said to be N THE CAPABLILITY OF HOM-LIE ALGEBRAS 3 an ideal if r x, y s P H for any x P H , y P L . A Hom-Lie algebra L is called abelian if r x, y s “ for all x, y P L .Let p H, α L | q and p K, α L | q be ideals of a Hom-Lie algebra p L, α L q . The (Higgins)commutator of p H, α L | q and p K, α L | q , denoted by pr H, K s , α L | q , is the subalgebra of p L, α L q spanned by the elements r h, k s , h P H , k P K . Note that it is not necessarilyan ideal, so Huq and Higgins commutators do not always coincide. The idea behindthis claim is that the Hom-Jacobi identity may not help to break “ x, r h, k s ‰ into abracket of elements from H and K . A Hom-Lie algebra p L, α L q is called perfect if L “ r L, L s . Note that r L, L s is always an ideal. The quotient ´ L r L,L s , α L ¯ is anabelian object in HomLie and it is called the abelianisation of p L, α L q which we willdenote by ` L ab , α L ab ˘ . Definition 2.5 ([8]) . The centre of a Hom-Lie algebra p L, α L q is the ideal Z p L q “ t x P L | r α k p x q , y s “ for all y P L, k P N u . Remark . When α L : L Ñ L is a surjective endomorphism, then we have that Z p L q “ t x P L | r x, y s “ u . Definition 2.7 ([9]) . A short exact sequence of Hom-Lie algebras Ñ p
M, α M q i Ñ p
K, α K q π Ñ p
L, α L q Ñ is said to be central if r M, K s “ . Equivalently, M Ď Z p K q .Following [42], the homology with trivial coefficients of a Hom-Lie algebra p L, α L q is the homology of the complex p C αn p L q , d n q , n ě , where C αn p L q “ Λ n L and d n : C αn p L q ÝÑ C αn ´ p L q is given by d n p x ^ ¨ ¨ ¨ ^ x n q“ ÿ ď i ă j ď n r x i , x j s ^ α L p x q ^ ¨ ¨ ¨ ^ { α L p x i q ^ ¨ ¨ ¨ ^ { α L p x j q ^ ¨ ¨ ¨ ^ α L p x n q A routine check shows that H α p L, α L q “ K and H α p L, α L q “ L r L,L s .2.8. Crossed modules.Definition 2.9 ([10]) . Let p L, α L q and p M, α M q be Hom-Lie algebras. A Hom-action of p L, α L q on p M, α M q is a linear map L b M Ñ M, x b m ÞÑ x m , satisfyingthe following properties:a) r x,y s α M p m q “ α L p x q p y m q ´ α L p y q p x m q ,b) α L p x q r m, m s “ r x m, α M p m qs ` r α M p m q , x m s ,c) α M p x m q “ α L p x q α M p m q for all x, y P L and m, m P M .A Hom-action is called trivial if x m “ for all x P L and m P M . Remark . If p M, α M q is an abelian Hom-Lie algebra enriched with a Hom-actionof p L, α L q , then p M, α M q is a Hom-module over p L, α L q [41]. Example . a) Let p H, α H q be a subalgebra and p K, α K q an ideal of p L, α L q . Then thereexists a Hom-action of p H, α H q on p K, α K q given by the product in L . Inparticular, there is a Hom-action of p H, α H q on itself given by the productin H .b) Let L and M be Lie algebras. Any Lie action of L on M (see e.g. [18])defines a Hom-action of p L, id L q on p M, id M q . J.M. CASAS AND X. GARCÍA-MARTÍNEZ c) Let L be a Lie algebra and α : L Ñ L be an endomorphism. Let M bean L -module satisfying the condition α p x q m “ x m , for all x P L , m P M .Then p M, id M q is a Hom-module over the Hom-Lie algebra p L, α q consideredin Example 2.3 c). As an explicit example of this, we can consider L tobe the -dimensional vector space with basis t e , e u , together with theproduct r e , e s “ ´r e , e s “ e and zero elsewhere, α to be representedby the matrix ˆ ˙ , and M to be the ideal of L generated by t e u .d) Any homomorphism of Hom-Lie algebras f : p L, α L q Ñ p M, α M q inducesa Hom-action of p L, α L q on p M, α M q by x m “ r f p x q , m s , for x P L and m P M .e) Let , p M, α M q i , p K, α K q π , p L, α L q , , σ l r be a split ex-tension of Hom-Lie algebras. Then there is a Hom-action of p L, α L q on p M, α M q defined in the following way: x m “ i ´ r σ p x q , i p m qs , for all x P L , m P M . In fact, there is an equivalence between split extensions and ac-tions [8]. Definition 2.12 ([10]) . Let p M, α M q and p N, α N q be Hom-Lie algebras with Hom-actions on each other. The Hom-actions are said to be compatible if p m n q m “ r m , n m s and p n m q n “ r n , m n s for all m, m P M and n, n P N . Example . If p H, α H q and p H , α H q both are ideals of a Hom-Lie algebra p L, α L q , then the Hom-actions of p H, α H q and p H , α H q on each other, consideredin Example 2.11 a), are compatible.Crossed modules of Hom-Lie algebras were introduced in [40] in order to provethe existence of a one-to-one correspondence between strict Hom-Lie 2-algebras andcrossed modules of Hom-Lie algebras. Definition 2.14. A precrossed module of Hom-Lie algebras is a triple of the form ` p M, α M q , p L, α L q , µ ˘ , where p M, α M q and p L, α L q are Hom-Lie algebras togetherwith a Hom-action from p L, α L q over p M, α M q and a Hom-Lie algebra homomor-phism µ : p M, α M q Ñ p L, α L q such that the following identity hold:a) µ p x m q “ r x, µ p m qs ,for all m P M, x P L .A precrossed module ` p M, α M q , p L, α L q , µ ˘ is said to be a crossed module whenthe following identity is satisfied:b) µ p m q m “ r m, m s ,for m, m P M . Remark . For a crossed module ` p M, α M q , p L, α L q , µ ˘ , the subalgebra Im p µ q isan ideal of p L, α L q and Ker p µ q is contained in the centre of p M, α M q . Moreover p Ker p µ q , α M | q is a Hom- Coker p µ q -module.Since HomLie is a strongly protomodular category (by being a variety of distribu-tive Ω -groups [35]) it satisfies the “Smith is Huq” condition [36], and therefore thisway of introducing crossed modules corresponds to internal crossed modules in thesense of Janelidze [30]. Example . a) Let p H, α H q be a Hom-ideal of a Hom-Lie algebra p L, α L q . Then the triple ` p H, α H q , p L, α L q , inc ˘ is a crossed module, where the action of p L, α L q on p H, α H q is given in Example 2.11 a). There are two particular cases N THE CAPABLILITY OF HOM-LIE ALGEBRAS 5 which allow us to think a Hom-Lie algebra as a crossed module, namely p H, α H q “ p L, α L q and p H, α H q “ p , q . So ` p L, α L q , p L, α L q , id ˘ and ` p , q , p L, α L q , ˘ are crossed modules.b) Let p L, α L q be a Hom-Lie algebra and p M, α M q be a Hom-L-module. Then ` p M, α M q , p L, α L q , ˘ is a crossed module.3. The non-abelian tensor and exterior products of Hom Liealgebras
Non-abelian tensor product of Hom-Lie algebras.
Let us recall the non-abelian tensor product of Hom-Lie algebras introduced in [10] as a generalisationof the non-abelian tensor product of Lie algebras [17].Let p M, α M q and p N, α N q be Hom-Lie algebras acting on each other compatibly.Consider the Hom-vector space p M b N, α M b N q given by the tensor product M b N of the underlying vector spaces and the linear map α M b N : M b N Ñ M b N , α M b N p m b n q “ α M p m q b α N p n q . Denote by D p M, N q the subspace of M b N generated by all elements of the forma) r m, m s b α N p n q ´ α M p m q b m n ` α M p m q b m n ,b) α M p m q b r n, n s ´ n m b α N p n q ` n m b α N p n q ,c) n m b m n ,d) n m b m n ` n m b m n ,e) r n m, n m s b α N p m n q ` r n m , n m s b α N p m n q ` r n m , n m s b α N p m n q ,for m, m , m P M and n, n , n P N .The quotient vector space p M b N q{ D p M, N q with the product r m b n, m b n s “ ´ n m b m n (1)and together with the endomorphism p M b N q{ D p M, N q Ñ p M b N q{ D p M, N q induced by α M b N , is a Hom-Lie algebra, which is called the non-abelian tensorproduct of Hom-Lie algebras p M, α M q and p N, α N q (or Hom-Lie tensor product forshort). It will be denoted by p M ‹ N, α M ‹ N q and the equivalence class of m b n will be denoted by m ‹ n . Lemma 3.2 ([10]) . Let p M, α M q and p N, α N q be Hom-Lie algebras with compatibleactions on each other. Then the following statements hold: a) There are homomorphisms of Hom-Lie algebras ψ M : p M ‹ N, α M ‹ N q Ñ p M, α M q , ψ M p m ‹ n q “ ´ n m,ψ N : p M ‹ N, α M ‹ N q Ñ p N, α N q , ψ N p m ‹ n q “ m n. b) There are Hom-actions of p M, α M q and p N, α N q on the Hom-Lie tensorproduct ( M ‹ N, α M ‹ N ) given, for all m, m P M , n, n P N , by m p m ‹ n q “ r m , m s ‹ α N p n q ` α M p m q ‹ m n n p m ‹ n q “ n m ‹ α N p n q ` α M p m q ‹ r n , n s c) Ker p ψ M q and Ker p ψ N q are contained in the centre of p M ‹ N, α M ‹ N q . d) The induced Hom-action of Im p ψ q on Ker p ψ q and the induced Hom-actionaction of Im p ψ q on Ker p ψ q are trivial. e) The homomorphisms ψ M and ψ N satisfy the following properties for all m, m P M , n, n P N : ψ M p m p m ‹ n qq “ r α M p m q , ψ M p m ‹ n qs ,ψ N p n p m ‹ n qq “ r α N p n q , ψ N p m ‹ n qs , ψ M p m ‹ n q p m ‹ n q “ r α M ‹ N p m ‹ n q , m ‹ n s “ ψ N p m ‹ n q p m ‹ n q . J.M. CASAS AND X. GARCÍA-MARTÍNEZ
Non-abelian exterior product of Hom-Lie algebras.
We introduce nowthe non-abelian exterior product, following [16, 18]. Let us consider two crossedmodules of Hom-Lie algebras η : p M, α M q Ñ p L, α L q and µ : p N, α N q Ñ p L, α L q .Then there are induced compatible Hom-actions of p M, α M q and p N, α N q on eachother via the Hom-action of p L, α L q (in fact there is an equivalence between pairsof crossed modules with the same domain and compatible actions [15]). There-fore, we can construct the non-abelian tensor product p M ‹ N, α M ‹ N q . We define p M l N, α M l N q as the Hom-vector subspace of p M ‹ N, α M ‹ N q , where M l N is thevector subspace spanned by the elements of the form m ‹ n, m P M, n P N , suchthat η p m q “ µ p n q , and α M l N is the restriction of α M ‹ N to M l N . Proposition 3.4.
The Hom-vector subspace p M l N, α M l N q is contained in thecentre of p M ‹ N, α M ‹ N q , so it is an ideal of p M ‹ N, α M ‹ N q .Proof. For any m ‹ n P M l N, m ‹ n P M ‹ N we have: r α kM ‹ N p m ‹ n q , m ‹ n s “ ´ α kN p n q α kM p m q ‹ m n “ ´ µ p α kN p n qq α kM p m q ‹ m n “ ´ α kL p µ p n qq α kM p m q ‹ m n “ ´ α kL p η p m qq α kM p m q ‹ m n “ ´ η p α kM p m qq α kM p m q ‹ m n “ ´r α kM p m q , α kM p m qs ‹ m n “ (cid:3) Definition 3.5.
Let η : p M, α M q Ñ p L, α L q and µ : p N, α N q Ñ p L, α L q be crossedmodules of Hom-Lie algebras. The non-abelian exterior product of the Hom-Liealgebras p M, α M q and p N, α N q is the quotient p M, α M q N p N, α N q “ ˆ M ‹ NM l N , α M ‹ N ˙ where α M ‹ N is the induced homomorphism by α M ‹ N on the quotient.The coset corresponding to m ‹ n is denoted by m N n, m P M, n P n . Definition 3.6.
Let η : p M, α M q Ñ p L, α L q and µ : p N, α N q Ñ p L, α L q be crossedmodules of Hom-Lie algebras. For any Hom-Lie algebra p P, α P q , the bilinear map h : M ˆ N Ñ P is said to be an exterior Hom-Lie pairing if the following propertiesare satisfied:a) h pr m, m s , α N p n qq “ h p α M p m q , m n q ´ h p α M p m q , m n q ,b) h p α M p m q , r n, n sq “ h p n m, α N p n qq ´ h p n m, α N p n qq ,c) h p n m, m n q “ ´r h p m, n q , h p m , n qs ,d) h p m, n q “ whenever η p m q “ µ p n q ,e) h ˝ p α M ˆ α N q “ α P ˝ h ,for all m, m P M , n, n P N .An exterior Hom-Lie pairing h : p M ˆ N, α M ˆ N q Ñ p P, α P q is said to be uni-versal if for any other exterior Hom-Lie pairing h : p M ˆ N, α M ˆ N q Ñ p Q, α Q q ,there is a unique homomorphism of Hom-Lie algebras θ : p P, α P q Ñ p Q, α Q q suchthat θ ˝ h “ h . Example . a) If α M “ id M , α N “ id N , α P “ id P , and α L “ id L , then the definition ofexterior Lie pairing in [18] is recovered. N THE CAPABLILITY OF HOM-LIE ALGEBRAS 7 b) Let p M, α M q , p N, α N q be Hom-ideals of p L, α L q and let η, µ be the inclusionmaps. Then the bilinear map h : M ˆ N Ñ M X N, h p m, n q “ r m, n s , is anexterior Hom-Lie pairing. Proposition 3.8.
Let η : p M, α M q Ñ p L, α L q and µ : p N, α N q Ñ p L, α L q be crossedmodules of Hom-Lie algebras. The map h : p M ˆ N, α M ˆ N q Ñ p M N N, α M N N q , h p m, n q “ m N n, is a universal exterior Hom-Lie pairing. Definition 3.9 ([10]) . It is said that a Hom-Lie algebra p L, α L q satisfies the α -identity condition if r L, Im p α L ´ id L qs “ which is equivalent to the condition r x, y s “ r α L p x q , y s for all x, y P L .Any Lie algebra included into HomLie satisfies the α -identity condition and moreexamples can be found in [10, 11]. Proposition 3.10.
Let η : p M, α M q Ñ p L, α L q and µ : p N, α N q Ñ p L, α L q becrossed modules of Hom-Lie algebras such that p L, α L q satisfies the α -identity con-dition. Then φ : p M N N, α M N N q Ñ p L, α L q , φ p m N n q “ ´ η p n m q “ µ p m n q , is aprecrossed module.Proof. The Hom-Lie action of p L, α L q on p M N N, α M N N q is given by l p m N n q “ l m N α N p n q ` α M p m q N l n First we need to check that φ is a homomorphism of Hom-Lie algebras: φ r m N n, m N n s “ φ p´ n m N m n q “ η ´ p m n q p n m q ¯ “ η ´ µ p m n q p n m q ¯ “ r µ p m n q , η p n m qs “ r µ p η p m q n q , η p µ p n q m qs“ “ r η p m q , µ p n qs , r µ p m q , η p n qs ‰ “ ´r η p´ n m q , η p´ n m qs“ r φ p m N n q , φ p m N n qs Obviously α L ˝ φ “ φ ˝ α M N N .Then, we check that it is satisfies the precrossed module condition: φ ` l p m N n q ˘ “ φ p l m N α N p n q ` α M p m q N l n q“ ´ η ´ α N p n q p l m q ¯ ´ η ´ p l n q α M p m q ¯ “ ´ η ´ µ p α N p n qq p l m q ¯ ´ η ´ µ p l n q α M p m q ¯ “ ´r α L p µ p n qq , η p l m qs ´ η ´ r l,µ p n qs α M p m q ¯ “ ´ “ α L p µ p n qq , r l, η p m qs ‰ ´ “ r l, µ p n qs , α L p η p m qq ‰ “ ´ “ α L p l q , r µ p n q , η p m qs ‰ “ ´ “ l, r µ p n q , η p m qs ‰ “ r l, η p´ n m qs “ r l, φ p m N n qs (cid:3) There is a surjective homomorphism of Hom-Lie algebras π : p M, α M q‹p N, α N q Ñp M, α M q N p N, α N q given by π p m ‹ n q “ m N n .Let p M, α M q , p N, α N q be ideals of a Hom-Lie algebra p L, α L q . According to Ex-ample 2.16 a), they can be seen as crossed modules through the inclusion in p L, α L q .Hence p M, α M q N p N, α N q “ ˆ M ‹ N t m ‹ m | m P M X N u , α M ‹ N ˙ (2) J.M. CASAS AND X. GARCÍA-MARTÍNEZ
Proposition 3.11.
Let p M, α M q and p N, α N q be ideals of a Hom-Lie algebra p L, α L q .There is a homomorphism of Hom-Lie algebras θ M,N : p M, α M q N p N, α N q Ñ p M, α M q X p N, α N q given by θ M,N p m N n q “ r m, n s , for all m P M, n P N . Moreover θ M,N is a crossedmodule of Hom-Lie algebras when that p L, α L q satisfies the α -identity condition.Proof. The Hom-action of x P p
M, α M q X p N, α N q over m N n P p
M, α M q N p N, α N q is given by x p m N n q “ r x, m s N α N p n q ` α M p m q N r x, n s ,θ M,N p x p m N n qq “ “ r x, m s , α N p n q ‰ ` “ α M p m q , r x, n s ‰ “ ´ “ α L | p x q , r n, m s ‰ “ “ x, r m, n s ‰ “ x θ M,N p m N n q , θ M,N p m N n q p m N n q “ “ r m, n s , m ‰ N α N p n q ` α M p m q N “ r m, n s , m ‰ “ α L | r m, n s N α L | r m , n s “ r m, n s N r m , n s“ r m N n, m N n s . (cid:3) Proposition 3.12.
Let p L, α L q be a perfect Hom-Lie algebra. Then, p L, α L q ‹ p L, α L q “ p L, α L q N p L, α L q and the homomorphism θ L,L : p L, α L q N p L, α L q Ñ p L, α L q is the universal centralextension of p L, α L q . Moreover Ker p θ L,L q – H α p L, α L q .Proof. By Equation (1), when p L, α L q is perfect the ideal L l L is zero and therefore p L N L, α L N L q “ p L ‹ L, α L ‹ L q . Then, the second part is an immediate consequenceof Theorem 4.3 and Theorem 4.4 in [10]. (cid:3) Lemma 3.13. If p N, α N q is an ideal of a Hom-Lie algebra p L, α L q , then the fol-lowing induced sequence of Hom-Lie algebras p N N L, α N N L q , p L N L, α L N L q π N π , p LN N LN , α L N L q , is exact.Proof. It is a special case of the exact sequence of [10, Proposition 3.12]. Aftertaking quotients, we can use the relation x ^ y “ ´ y ^ x , which holds in L ^ L toerase one of the factors of the semi-direct product. (cid:3) Free Hom-Lie algebras were constructed in [8], where the following adjoint func-tors were obtained:
HomSet F r , K HomLie U l r where HomSet is the category of sets with a chosen endomorphism, F r is the functorthat assigns to a Hom-set p X, α X q the free Hom-Lie algebra F r p X, α X q and U isthe functor that assigns to a Hom-Lie algebra p B, α B q the Hom-set obtained byforgetting the operations. Note that since HomLie is a variety, it is monadic over
Set and therefore it has enough projectives.On the other hand, since
HomLie is semi-abelian the quotient p R, α R q X rp F, α F q , p F, α F qsrp F, α F q , p R, α R qs (3)is a Baer invariant [20, Theorem 6.9], i.e., it doesn’t depend on the chosen freepresentation Ñ p
R, α R q Ñ p F, α F q ρ Ñ p
G, α G q Ñ . This is the Schur multiplier of the Hom-Lie algebra p G, α G q and we denote it as M p G, α G q . N THE CAPABLILITY OF HOM-LIE ALGEBRAS 9
Lemma 3.14 ([12]) . Let p L, α L q be a Hom-Lie algebra. For any central extension Ñ p
N, α N q Ñ p G, α G q π Ñ p
L, α L q Ñ the extension Ñ p
N, α N q X rp G, α G q , p G, α G qs Ñ rp G, α G q , p G, α G qs Ñ pr L, L s , α L | q Ñ is also central. Moreover, if p L, α L q is perfect the commutator rp G, α G q , p G, α G qs isalso perfect. Let p L, α L q be a Hom-Lie algebra and let Ñ p
S, α S q Ñ p F, α F q τ Ñ p
L, α L q Ñ be a free presentation. Then pr F, S s , α F | q is an ideal of p F, α F q , so there exists asurjective homomorphism τ : p F, α F qpr F, S s , α F | q ։ p L, α L q , τ p f ` r F, S sq “ τ p f q . Moreover, Ñ ˆ S r F, S s , r α S ˙ Ñ ˆ F r F, S s , r α F ˙ τ Ñ p
L, α L q Ñ (4)is a central extension. By Lemma 3.14, the following extension Ñ p
S, α S qpr F, S s , α F | q č ˆ r F, F sr F, S s , α F | ˙ Ñ ˆ r F, F sr F, S s , r α F | ˙ r τ Ñ pr
L, L s , α L | q Ñ is also central. If in particular p L, α L q is perfect, the commutator ´ r F,F sr F,S s , α F | ¯ isalso perfect and the central extension is universal. Indeed, for any other centralextension Ñ p
A, α A q Ñ p K, α K q σ Ñ p
L, α L q Ñ of p L, α L q , there is a homomor-phism of Hom-Lie algebras such that the following diagram commutes , p S, α S q , p F, α F q τ , h (cid:12) (cid:18) ✤✤✤ p L, α L q , , p A, α A q , p K, α K q σ , p L, α L q , In fact, h induces a homomorphism h : ´ F r F,S s , α F ¯ Ñ p
K, α K q such that σ ˝ h “ τ .The restriction of h to ´ r F,F sr F,S s , α F | ¯ provides the required homomorphism, which isunique due to [9, Lemma 4.7].Since the universal central extension of a perfect Hom-Lie algebra is unique upto isomorphism, then Proposition 3.12 implies that p L N L, α L N L q – ´ r F,F sr F,S s , α F | ¯ and H α p L, α L q – ´ S Xr F,F sr F,S s , α F | ¯ for any perfect Hom-Lie algebra p L, α L q . Theorem 3.15.
Let p L, α L q be a perfect Hom-Lie algebra and let Ñ p
S, α S q Ñ p F, α F q τ Ñ p
L, α L q Ñ be a free presentation of p L, α L q . Then, H α p L, α L q – ˆ S X r
F, F sr F, S s , α F | ˙ Lemma 3.16. If F is a free Hom-Lie algebra, then F N F – r F, F s .Proof. It is done following the same strategy as in [17]. (cid:3) Stallings-Stammbach exact sequence
Let Ñ p
N, α N q Ñ p G, α G q π Ñ p
L, α L q Ñ be a short exact sequence ofHom-Lie algebras, and let Ñ p
R, α R q Ñ p F, α F q ρ Ñ p
G, α G q Ñ be a projec-tive presentation of p G, α G q , we can construct the following diagram of projectivepresentations: (cid:12) (cid:18) u ~ sssssssssss p R, α R q (cid:12) (cid:18) u ~ ssssssssss , p S, α S q , (cid:12) (cid:18) p F, α F q ρ (cid:12) (cid:18) τ “ π ˝ ρ (cid:31) ) ❏❏❏❏❏❏❏❏❏ , p N, α N q , (cid:12) (cid:18) p G, α G q π , (cid:12) (cid:18) p L, α L q , (cid:29) ' ❋❋❋❋❋❋❋❋❋
00 0 0 (5)Based on the ˆ -lemma, we can write the commutative diagram of Figure 1to obtain the following result: Theorem 4.1.
Let Ñ p
N, α N q Ñ p G, α G q π Ñ p
L, α L q Ñ be a short exactsequence of Hom-Lie algebras. There exists the following natural exact sequence: M p G, α G q Ñ M p L, α L q Ñ ˆ N r G, N s , r α N ˙ Ñ H α p G, α G q Ñ H α p L, α L q Ñ (6) Remark . If α G “ id G and α L “ id L , sequence (6) is the Stallings-Stammbachexact sequence associated to a short exact sequence of Lie algebras [29].If Ñ p
N, α N q Ñ p G, α G q π Ñ p
L, α L q Ñ is a central extension of Hom-Liealgebras, then sequence (6) gives rise to the following natural exact sequence M p G, α G q Ñ M p L, α L q Ñ p N, α N q Ñ H α p G, α G q Ñ H α p L, α L q Ñ . Moreover the kernel of M p G, α G q Ñ M p L, α L q is ´ r F,S sr F,R s , r α S | ¯ , since r F, S s Ď R under the centrality condition.Since p N, α N q and ´ G r G,G s , α G | ¯ are abelian Hom-Lie algebras, we can constructthe Hom-vector space ´ N b G r G,G s , α b ¯ , where α b p n b g q “ α N p n q b α G p g q . There-fore there is a well-defined surjective homomorphism ϕ : ˆ N b G r G, G s , α b ˙ Ñ ˆ r F, S sr F, R s , r α S | ˙ , ϕ p n b g q “ r f, s s ` r F, R s , where ρ p s q “ n, ρ p f q “ g, s P S, f P F . The composition G : ˆ N b G r G, G s , α b ˙ ϕ ։ ˆ r F, S sr F, R s , r α S | ˙ ã Ñ M p G, α G q N THE CAPABLILITY OF HOM-LIE ALGEBRAS 11 pr F, R s , α R | q (cid:12) (cid:18) (cid:12) (cid:18) ' . ' . ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ pr F, S s X
R, α R | q (cid:15) (cid:15) (cid:15) (cid:15) , , } (cid:7) } (cid:7) ✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝ pr F, F s X
R, α R | q (cid:15) (cid:15) (cid:15) (cid:15) * * * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ } (cid:7) } (cid:7) ✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝✝ ´ r F,S sX R r F,R s , α R | ¯ , , (cid:12) (cid:18) (cid:12) (cid:18) ´ r F,F sX R r F,R s , α R | ¯ / / / / (cid:12) (cid:18) (cid:12) (cid:18) ´ r F,F sX R r F,S sX R , α R | ¯ (cid:12) (cid:18) (cid:12) (cid:18) u ~ u ~ sssssssss pr F, S s , α S | q , , (cid:15) (cid:15) (cid:15) (cid:15) pr F, F s X
S, α S | q / / / / (cid:9) (cid:9) (cid:9) (cid:9) ✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒ ´ r F,F sX S r F,S s , r α S | ¯ (cid:5) (cid:5) (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ ‚ (cid:15) (cid:15) (cid:15) (cid:15) , , ´ r F,F sr F,R s , r α F | ¯ (cid:15) (cid:15) (cid:15) (cid:15) / / / / ‚ (cid:15) (cid:15) (cid:15) (cid:15) ` r N, G s , α N | ˘ – ´ r F,S sr F,S sX R , r α R | ¯ (cid:31) ) (cid:31) ) ❏❏❏❏❏❏❏❏❏❏❏ (cid:20) (cid:28) (cid:20) (cid:28) ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ pr Q, Q s , α Q | q – p N X r
G, G s , α N | q , , (cid:12) (cid:18) (cid:12) (cid:18) % % % % ❑❑❑❑❑❑❑❑❑❑ pr G, G s , α G | q / / / / (cid:12) (cid:18) (cid:12) (cid:18) – ´ r G,G s N Xr G,G s , r α G | ¯ (cid:12) (cid:18) (cid:12) (cid:18) ´ N Xr G,G sr G,N s , α N | ¯ (cid:12) (cid:18) (cid:12) (cid:18) p N, α N q , , (cid:15) (cid:15) (cid:15) (cid:15) & & & & ▲▲▲▲▲▲▲▲▲▲ p G, α G q / / / / (cid:15) (cid:15) (cid:15) (cid:15) p Q, α Q q (cid:15) (cid:15) (cid:15) (cid:15) ´ N r G,N s , r α N ¯ y y y y sssssssss ´ NN Xr G,G s , α N | ¯ , , ´ G r G,G s , α G | ¯ / / / / ´ Q r Q,Q s , α Q | ¯ Figure 1. gives rise to the following six-term exact sequence: ˆ N b G r G,G s , α b ˙ M p G, α G q M p L, α L qp N, α N q H α p G, α G q H α p L, α L q Capable Hom-Lie algebras
Definition 5.1.
A Hom-Lie algebra p L, α L q is said to be capable if there exists aHom-Lie algebra p H, α H q such that L – H { Z p H q .When α L “ id L and α H “ id H the above definition recovers the notion of capableLie algebra in [38]. It is well-known that capability of groups (respectively, Liealgebras) is closely related with the group of the inner automorphisms (respectively,the Lie algebra of the inner derivations). Let us recall some notions concerningderivations from [39]. We denote by α k the composition of α with itself k times. Definition 5.2. An α k -derivation of a Hom-Lie algebra p L, α L q is a linear map d : L Ñ L such thata) d ˝ α L “ α L ˝ d ,b) d r x, y s “ r d p x q , α k p y qs ` r α k p x q , d p y qs , for all x, y P L .We denote by Der α k p L q the set of all α k -derivations of p L, α L q . The algebra Der p L q “ à k ě Der α k p L q is a Hom-Lie algebra with respect to the usual bracket operation r d, d s “ d ˝ d ´ d ˝ d and the endomorphism r α : Der p L q Ñ Der p L q given by r α p d q “ α ˝ d .For any Hom-Lie algebra p L, α L q satisfying the α -identity condition (Defini-tion 3.9) and x P L , we define d k p x q : L Ñ L by d k p x qp y q “ r α k p x q , y s . Then d k p x q P Der α k ` p L q , which is called an inner α k ` -derivation. We denote by InnDer α k p L q the set of all inner α k -derivations, and InnDer p L q “ à k ě InnDer α k p L q is an ideal of Der p L q .There is a homomorphism of Hom-Lie algebras ϕ : L Ñ Der p L q , ϕ p x q “ p d p x q , d p x q , . . . , d k p x q , . . . q such that Im p ϕ q “ InnDer p L q and Ker p ϕ q “ Z p L q . This homomorphism shows thatif a Hom-Lie algebra p L, α L q satisfying the α -identity condition (Definition 3.9) isisomorphic to inner derivations of some Hom-Lie algebra p H, α H q that satisfies the α -identity condition, then p L, α L q is capable, i.e., we can obtain the following exactsequence Ñ p Z p H q , α H | q Ñ p H, α H q ϕ Ñ p
InnDer p H q , r α H q – p L, α L q Ñ . Definition 5.3.
Let p L, α L q be a Hom-Lie algebra. The tensor centre of p L, α L q is the set: Z ‹ α p L q “ t l P L | α k p l q ‹ x “ , for all x P L, k P N u . The exterior centre of p L, α L q is the set: Z N α p L q “ t l P L | α k p l q N x “ , for all x P L, k P N u . Lemma 5.4.
For any Hom-Lie algebra p L, α L q both Z ‹ α p L q and Z N α p L q are idealsof p L, α L q contained in Z p L q .Proof. Let l P Z N α p L q and x P L . We need to prove that r l, x s P Z N α p L q , i.e., α kL pr l, x sq N y “ for all k P N and y P L . But this is true since we already havethat α kL pr l, x sq “ r α kL p l q , α kL p x qs “ θ L,L ` α kL p l q N α kL p x q ˘ “ . The same argumentworks for Z ‹ α p L q with the tensor adapted version of θ L,L . (cid:3) It is obvious that Z ‹ α p L q Ď Z N α p L q . The equality follows whenever p L, α L q isperfect by Proposition 3.12. N THE CAPABLILITY OF HOM-LIE ALGEBRAS 13
Proposition 5.5.
Let Ñ p
S, α S q Ñ p F, α F q τ Ñ p
L, α L q Ñ be a free pre-sentation of p L, α L q and let ψ : p C, α C q Ñ p L, α L q be the central extension (4) ´ i . e . p C, α C q “ ´ F r F,S s , α F | ¯¯ . Then, there is an isomorphism of Hom-Lie algebras r F,F sr F,S s “ r C, C s – L N L . Moreover, x P Z p C q if and only if ψ p x q P Z N α p L q .Proof. The first part is a consequence of Lemmas 3.13 and 3.16, where the isomor-phism p L N L, α L N L q – ´ r F,F sr F,S s , α F | ¯ is induced by the map l N l ÞÑ r x, y s ` r
F, S s ,such that τ p x q “ l , τ p y q “ l .Let x P Z p C q . Then, “ r α k p x q , y s ” l N l “ ψ ` α k p x q ˘ N ψ p y q “ α k p ψ p x qq N ψ p y q hence ψ p x q P Z N α p L q .Conversely, let ψ p x q P Z N α p L q . Since r α k p x q , y s ” α k p l q N l “ α k ` ψ p x q ˘ N ψ p y q “ , then x P Z p C q . (cid:3) Proposition 5.5 implies that ψ ` Z p C q ˘ Ď Z N α p L q . On the other hand, ψ ´ p x q Ď Z p C q for any x P Z N α p L q , therefore ψ ` Z p C q ˘ “ Z N α p L q . (8) Theorem 5.6.
A Hom-Lie algebra p L, α L q is capable if and only if Z N α p L q “ .Proof. Assume Z N α p L q “ . Let ψ : p C, α C q ։ p L, α L q be the central extension (4).It is enough to show that Ker p ψ q “ Z p C q . To see the non-trivial inclusion, considerany x P Z p C q , then by Proposition 5.5 we have that ψ p x q P Z N α p L q “ , hence x P Ker p ψ q .Assume now that p L, α L q is a capable Hom-Lie algebra, i.e., there exists a Hom-Lie algebra p G, α G q such that L – G { Z p G q . Then, there is a surjective homo-morphism of Hom-Lie algebras π : p G, α G q ։ p L, α L q such that Ker p π q “ Z p G q .Consider a diagram of free presentations as (5), then we have the following com-mutative diagram: , p S, α S q , (cid:12) (cid:18) ✤✤✤ p F, α F q τ , ρ (cid:15) (cid:15) (cid:15) (cid:15) p L, α L q , , Z p G q , p G, α G q π , p L, α L q , Moreover τ pr F, S sq “ , then there exists r τ : ´ F r F,S s , r α F ¯ Ñ p
G, α G q . Let p C, α C q “ ´ F r F,S s , r α F ¯ and ψ “ π ˝ r τ .Since r τ is a surjective homomorphism, then r τ ` Z p C q ˘ Ď Z p G q “ Ker p π q , hence ψ ` Z p C q ˘ “ π ˝ r τ ` Z p C q ˘ Ď π p Ker p π qq “ , i.e., Z p C q Ď Ker p ψ q , and conse-quently, Ker p ψ q “ Z p C q . Now applying identity (8) we have Z Np L q α “ ψ ` Z p C q ˘ “ ψ ` Ker p ψ q ˘ “ . (cid:3) Definition 5.7.
The epicentre of a Hom-Lie algebra p L, α L q is the subalgebra Z ˚ α p L q “ č f ` Z p G q ˘ for all central extension f : p G, α G q ։ p L, α L q . Remark . Note that f ` Z p G q ˘ is an ideal of p L, α L q , so Z ˚ α p L q is also an idealof p L, α L q . Lemma 5.9.
Given a free presentation Ñ p
S, α S q Ñ p F, α F q τ Ñ p
L, α L q Ñ ,consider the central extension (4) . Then, Z ˚ α p L q “ τ ˆ Z ˆ F r F, S s , α F ˙˙ Proof.
We only need to show that τ ´ Z ´ F r F,S s ¯¯ Ď ϕ ` Z p H q ˘ for any central exten-sion Ñ p
A, α A q Ñ p H, α H q ϕ Ñ p
L, α L q Ñ .Since p F, α F q is a free Hom-Lie algebra, then there exists a homomorphism δ : p F, α F q Ñ p H, α H q such that ϕ ˝ δ “ τ . Moreover, δ p S q Ď A and δ pr F, S sq Ďr
A, H s “ . Therefore, we have that δ induces a homomorphism δ : ´ F r F,S s , α F ¯ Ñp H, α H q .Let us see that δ ´ Z ´ F r F,S s ¯¯ Ď Z p H q . Indeed, for any f P Z ´ F r F,S s ¯ , we havethat r α kF p f q , f s P r F, S s , for all f P F . Since as a vector space H is the direct sumof A and L , any h P H can be written as h “ δ p f q ` a, f P F, a P A , then r δ ` α kF p f q ˘ , h s “ δ pr α kF p f q , f sq ` r δ ` α kF p f q ˘ , a s “ Finally, τ ´ Z ´ F r F,S s ¯¯ “ ϕ ˝ δ ´ Z ´ F r F,S s ¯¯ Ď ϕ ` Z p H q ˘ . (cid:3) Proposition 5.10.
A Hom-Lie algebra p L, α L q is capable if and only if Z ˚ α p L q “ .Proof. If p L, α L q is capable, then the central extension Ñ p Z p H q , α H | q Ñ p H, α H q f Ñ p
L, α L q Ñ implies that Z ˚ α p L q Ď f p Z p H qq “ .Conversely, if Z ˚ α p L q “ , by Lemma 5.9 any free presentation Ñ p
S, α S q Ñ p F, α F q τ Ñ p
L, α L q Ñ induces that τ ´ Z ´ F r F,S s ¯¯ “ , and therefore ˆ Z ˆ F r F, S s ˙˙ Ď S r F, S s “
Ker p τ q Moreover, since (4) is a central extension, then S r F,S s Ď Z ´ F r F,S s ¯ . Thus, Ñ Z ˆ F r F, S s , α F ˙ Ñ ˆ F r F, S s , α F ˙ τ Ñ p
L, α L q Ñ is a central extension as well. (cid:3) Theorem 5.11.
Let p A, α A q be a central ideal of a Hom-Lie algebra p L, α L q . Then p A, α A q Ď Z ˚ α p L, α L q if and only if the homomorphism in sequence (7) G : ´ A b G r G,G s , α b ¯ Ñ M p L, α L q associated to the central extension Ñ p
A, α A q Ñ p L, α L q π Ñ ` LA , α L ˘ Ñ is the zero map.Proof. With a a similar diagram to (5) we have the free presentation Ñ p
S, α S q Ñp F, α F q π ˝ ρ Ñ p LA , α L q Ñ .We know from the construction of the exact sequence (7) that there is an iso-morphism Im p G q – ´ r F,S sr F,R s , α F | ¯ . Then, by Lemma 5.9 we have the following N THE CAPABLILITY OF HOM-LIE ALGEBRAS 15 commutative diagram: Z ´ F r F,R s ¯ , (cid:127) _ (cid:12) (cid:18) Z ˚ α p L q (cid:127) _ (cid:12) (cid:18) , ´ R r F,R s , α R ¯ , = rrrrrrrrrr ´ F r F,R s , α F ¯ τ , ǫ (cid:15) (cid:15) (cid:15) (cid:15) p L, α L q , γ (cid:15) (cid:15) (cid:15) (cid:15) ‚ „ , ´ LZ ˚ α p L q , α L ¯ Then, G “ ô r F, S sr F, R s “ ô ˆ S r F, R s , α S ˙ Ď Z ˆ F r F, R s ˙ ô γ ˝ τ ˆ R r F, R s ˙ “ ǫ ˆ R r F, R s ˙ “ ô p A, α A q “ τ p S, α S q “ τ ˆ S r F, S s , α S ˙ Ď Ker p γ q “ Z ˚ α p L q (cid:3) Corollary 5.12.
For any Hom-Lie algebra p L, α L q , the following statements areequivalent: a) Any central extension f : p G, α G q ։ p L, α L q satisfies that f p Z p G q , α G | q “p Z p L q , α L | q . b) The map G : ´ Z p L q b G r G,G s , α b ¯ Ñ M p L, α L q in sequence (7) associatedto the central extension Ñ p Z p L q , α L | q Ñ p L, α L q Ñ ´ LZ p L q , α L ¯ Ñ isthe zero map. c) The canonical homomorphism M p L, α L q Ñ M ´ LZ p L q , α L ¯ is injective.Proof. It is a direct consequence of the exactness of sequence (7) associated to thecentral extension Ñ p Z p L q , α L | q Ñ p L, α L q Ñ ´ LZ p L q , α L ¯ Ñ . (cid:3) Remark . If p L, α L q satisfies any of the equivalent statements of Corollary 5.12,then sequence (7) associated to the central extension p L, α L q Ñ ´ LZ p L q , α L ¯ impliesthat M p L, α L q “ Ker ˆ M ˆ LZ p L q , α L ˙ Ñ Z p L, α L q ˙ . Lemma 5.14.
Let Ñ p
N, α N q Ñ p L, α L q π Ñ p LN , α L q Ñ be an exact sequenceof Hom-Lie algebras. If p L, α L q is perfect, then Ker ˆ H α p L, α L q Ñ H α ˆ LN , α L ˙˙ “ Ker p θ N,L : p N N L, α N N L q Ñ p N, α N qq Proof.
First observe that p LN , α L q is also a perfect Hom-Lie algebra. Then, considerthe following diagram of exact rows: p N N L, α N N L q , θ N,L (cid:12) (cid:18) p L N L, α L N L q π N π , θ L,L (cid:12) (cid:18) p LN N LN , α L N L q , θ L { N,L { N (cid:12) (cid:18) , p N, α N q , p L, α L q π , p LN , α L q , and apply the Snake lemma. (cid:3) Remark . Note that if p N, α N q is a central ideal in p L, α L q , then Ker ` θ N,L : p N N L, α N N L q Ñ p N, α N q ˘ “ p N N L, α N N L q . Theorem 5.16.
For any perfect Hom-Lie algebra p L, α L q , the following statementshold: a) Z N α p L q is the smallest central ideal of p L, α L q containing all the centralideals p N, α N q for which the canonical morphism M p L, α L q Ñ M ` LN , α L ˘ is a monomorphism (or, equivalently, for which the canonical surjectivehomomorphism π N π : p L N L, α L N L q Ñ p LN N LN , α L N L q is an isomorphism). b) Z N α p L q “ Z ˚ α p L q .Proof. The first part is obtained from the following commutative diagram con-structed for any central ideal p N, α N q and using Lemmas 3.13 and 5.14: p N N L, α N N L q , H α p L, α L q , (cid:12) (cid:18) (cid:12) (cid:18) H α ` LN , α L ˘ (cid:12) (cid:18) (cid:12) (cid:18) p N N L, α N N L q , p L N L, α L N L q , (cid:15) (cid:15) (cid:15) (cid:15) ` LN N LN , α L N L ˘ (cid:15) (cid:15) (cid:15) (cid:15) , p L, α L q π / / / / ` LN , α L ˘ The second part is a direct consequence of Corollary 5.12 and the previous dia-gram where the central ideal p N, α N q is the epicentre of p L, α L q . (cid:3) Acknowledgements
The authors would like to thank the referee for his helpful comments and sug-gestions that improved the manuscript.
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