HNN-extension of involutive multiplicative Hom-Lie algebras
aa r X i v : . [ m a t h . R A ] J a n HNN-extension of involutive multiplicative Hom-Liealgebras
Sergei Silvestrov , Chia Zargeh Division of Mathematics and Physics, School of Education, Culture and Communication,Mälardalen University, Box 883, 72123 Västeras, Sweden.e-mail: [email protected] Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil.e-mail: [email protected]
Abstract
The construction of HNN-extensions of involutive Hom-associative algebras andinvolutive Hom-Lie algebras is described. Then, as an application of HNN-extension,by using the validity of Poincaré-Birkhoff-Witt theorem for involutive Hom-Lie algebras,we provide an embedding theorem.
One of the most important constructions in combinatorial group theory is Higman-Neumann-Neumann extension (or HNN-extension, for short), which states that if A and A are isomorphic subgroups of a group G , then it is possible to find a group H containing G such that A and A are conjugate to each other in H and G is embed-dable in H (see [45]). The HNN-extension of a group has a topological interpretationdescribed in [28] and [59], which is used as a motivation for its study. Spreading clas-sical techniques in combinatorial group theory to other algebraic structures has shownoutstanding capacities for solving problems in affine algebraic geometry, the theory ofLie algebras and mathematical physics. In this regard, HNN-extension of Lie algebraswas constructed by Lichtman and Shirvani [58] and Wasserman [79] through differentapproaches. They used HNN-extension in order to give a new proof for Shirshov’stheorem [75], namely, a Lie algebra of finite or countable dimension can be embeddedinto a -generator Lie algebra. Moreover, the idea of HNN-extension has been recentlyspread to Leibniz algebras in [51] and Lie superalgebras in [50], which are respectively,non-antisymmetric and natural generalization of Lie algebras.In this paper we intend to introduce HNN-extension for the Hom-generalizationof Lie algebras. Hom-Lie algebras and more general quasi-Hom-Lie algebras were in-troduced first by Hartwig, Larsson and Silvestrov in [43], where the general quasi-deformations and discretizations of Lie algebras of vector fields using more general Mathematics subject classification : 17D30, 17B61
Keywords : HNN-extension, Hom-Lie algebra, Hom-associative algebra -derivations (twisted derivations) and a general method for construction of defor-mations of Witt and Virasoro type algebras based on twisted derivations have beendeveloped, initially motivated by the q -deformed Jacobi identities observed for the q -deformed algebras in physics, q -deformed versions of homological algebra and discretemodifications of differential calculi. Hom-Lie superalgebras, Hom-Lie color algebrasand more general quasi-Lie algebras and color quasi-Lie algebras where introducedfirst in [54, 55, 76]. Quasi-Lie algebras and color quasi-Lie algebras encompass withinthe same algebraic framework the quasi-deformations and discretizations of Lie alge-bras of vector fields by σ -derivations obeying twisted Leibniz rule, and color Lie al-gebras, the well-known natural generalizations of Lie algebras and Lie superalgebras.In quasi-Lie algebras, the skew-symmetry and the Jacobi identity are twisted by de-forming twisting linear maps, with the Jacobi identity in quasi-Lie and quasi-Hom-Liealgebras in general containing six twisted triple bracket terms. In Hom-Lie algebras,the bilinear product satisfies the non-twisted skew-symmetry property as in Lie alge-bras, and the Hom-Lie algebras Jacobi identity has three terms twisted by a singlelinear map, reducing to the Lie algebras Jacobi identity when the twisting linear mapis the identity map. Hom-Lie admissible algebras have been considered first in [62],where in particular the Hom-associative algebras have been introduced and shown tobe Hom-Lie admissible, leading to Hom-Lie algebras using commutator map as newproduct, and thus constituting a natural generalization of associative algebras as Lieadmissible algebras. Since the pioneering works [43, 53–56, 62], Hom-algebra structuresexpanded into a popular area with increasing number of publications in various di-rections. Hom-algebra structures of a given type include their classical counterpartsand open broad possibilities for deformations, Hom-algebra extensions of cohomologi-cal structures and representations, formal deformations of Hom-associative algebras andHom-Lie algebras, Hom-Lie admissible Hom-coalgebras, Hom-coalgebras, Hom-Hopf al-gebras, Hom-Lie algebras, Hom-Lie superalgebras, color Hom-Lie algebras, BiHom-Liealgebras, BiHom-associative algebras, BiHom-Frobenius algebras and n -ary generaliza-tions of Hom-algebra structures have been further investigated in various aspects forexample in [1–27, 35, 36, 38–41, 44, 46–49, 52, 57, 60–74, 76–78, 80–84, 86–88].Our approach for construction of the HNN-extension of Hom-generalization of Liealgebras is based on the corresponding construction for its envelope. Therefore, we con-centrate on the study of HNN-extensions for involutive Hom-Lie algebras in which theiruniversal enveloping algebras have been explicitly obtained in [42]. It is worth notingthat there exists another approach provided in [80] for obtaining the universal envelop-ing algebra of a Hom-Lie algebra as a suitable quotient of the free Hom-nonassociativealgebra through weighted trees, but the point of difficulty in the approach in [80] is thesize of the weighted trees. Involutive Hom-Lie algebras have been constructed in [85],and the classical theory of enveloping algebras of Lie algebras was extended to an ex-plicit construction of the free involutive Hom-associative algebra on a Hom-module inorder to obtain the universal enveloping algebra [42]. This construction leads to aPoincare-Birkhoff-Witt theorem for the enveloping associative algebra of an involutiveHom-Lie algebra. This approach has been extended to the enveloping algebras for colorHom-Lie algebras in [11, 12]. Extensions of Hom-Lie superalgebras and Hom-Lie coloralgebras have been considered in [9, 13]. Hom-associative Ore extensions have beenconsidered in [29–34]The paper is organized as follows. In Section 2, we recall the preliminary concepts elated to involutive Hom-associative algebras and involutive Hom-Lie algebras. InSection 3, we introduce the HNN-extension for involutive Hom-associative algebras. InSection 4, we construct the HNN-extension for involutive Hom-Lie algebras and providean embedding theorem. In this section we recall necessary concepts related to involutive Hom-associative andinvolutive Hom-Lie algebras.
Definition 2.1.
Let K be a field.(a) Hom-module is a pair ( V, α V ) consisting of a K -module V and a linear operator α V : V → V .(b) Hom-associative algebra is a triple ( A, ∗ A , α A ) consisting of a K -module A , a linearmap ∗ A : A ⊗ A → A , called the multiplication, and a linear operator α A : A → A satisfying the Hom-associativity α A ( x ) ∗ A ( y ∗ A z ) = ( x ∗ A y ) ∗ A α A ( z ) , for all x, y, z ∈ A. (c) Hom-associative algebra is said to be multiplicative if the linear map α is multi-plicative in the sense of satisfying α A ( x ∗ A y ) = α A ( x ) ∗ A α A ( y ) for all x, y ∈ A .(d) Hom-associative algebra ( A, ∗ A , α A ) (resp. Hom-module ( V, α V ) ) is said to be involutive if α A = id (resp. α V = id ) .(e) Let ( V, α V ) and ( W, α W ) be Hom-modules. A K -linear map f : V → W is calleda morphism of Hom-modules if f ( α V ( x )) = α W ( f ( x )) for all x ∈ V. (f) Let ( A, ∗ A , α A ) and ( B, ∗ B , α B ) be two Hom-associative algebras. A K -linear map f : A → B is a morphism of Hom-associative algebras if f ( x ∗ A y ) = f ( x ) ∗ B f ( y ) , and f ( α A ( x )) = α B ( f ( x )) , for all x, y ∈ A. (g) Let ( A, ∗ A , α A ) be a Hom-associative algebra. A submodule B ⊆ A is called aHom-associative subalgebra of A if B is closed under the multiplication ∗ A and α A ( B ) ⊆ B .(h) Let ( A, ∗ A , α A ) be a Hom-associative algebra. A submodule I ⊆ A is called aHom-ideal of A if x ∗ A y ∈ I , y ∗ A x ∈ I for all x ∈ I, y ∈ A , and α A ( I ) ⊆ I . Definition 2.2.
For any non-negative integer k , a linear map D : A → A is called an α kA -derivation of involutive Hom-associative algebra ( A, ∗ A , α A ) , if D ◦ α kA = α kA ◦ D,D ◦ ( x ∗ A y ) = D ( x ) ∗ A α kA ( y ) + α kA ( x ) ∗ A D ( y ) . efinition 2.3. Let ( V, α V ) be an involutive Hom-module. A free involutive Hom-associative algebra on V is an involutive Hom-associative algebra ( F IHA ( V ) , ∗ F , α F ) together with a morphism of Hom-modules j V : ( V, α V ) → ( F IHA ( V ) , α F ) such that,for any involutive Hom-associative algebra ( A, ∗ A , α A ) together with a morphism ofHom-modules f : ( V, α V ) → ( A, α A ) , there is a unique morphism of Hom-associativealgebras f : ( F IHA ( V ) , ∗ F , α F ) → ( A, ∗ A , α A ) such that f = f ◦ j V . Definition 2.4.
A Hom-Lie algebra is a triple ( g , [ · , · ] g , β ) consisting of a vector space g , a skew-symmetric bilinear map (bracket) [ · , · ] g : g × g → g and a linear map β : g → g satisfying the following Hom-Jacobi identity: [ β ( u ) , [ v, w ] g ] g + [ β ( v ) , [ w, u ] g ] g + [ β ( w ) , [ u, v ] g ] g = 0 . (1)Hom-Lie algebra is called a multiplicative Hom-Lie algebra if β satisfies β ([ u, v ] g ) = [ β ( u ) , β ( v )] g . (2)A Hom-Lie algebra ( g , [ · , · ] g , β ) is called involutive if β = id g . Note that the classicalLie algebra can be recovered when β = id g , with the identity (1) becoming the Jacobiidentity for Lie algebras. Definition 2.5.
A morphism of Hom-Lie algebras f : ( g , [ · , · ] g , β g ) → ( h , [ · , · ] h , β h ) is a k -linear map f : g → h such that f ([ x, y ] g ) = [ f ( x ) , f ( y )] h and f ( β g ( x )) = β h ( f ( x )) for all x ∈ g . Hom-associative algebras were introduced in [62], and shown to be Hom-Lie admissi-ble, i.e. any Hom-associative algebra ( A, ∗ A , α A ) yields a Hom-Lie algebra ( A, [ · , · ] A , β A ) with β A = α A and [ x, y ] A = x ∗ A y − y ∗ A x for x, y ∈ A .For simplicity, we will restrict our considerations to multiplicative Hom-Lie algebrasand multiplicative Hom-associative algebras, meaning that the twisting map is notonly linear, but also an endomorphism of the Hom-Lie algebra or Hom-associativealgebra respectively. An interesting important problem is to understand completelythe role of the multiplicatives restriction and extend the results and constructions frommultiplicative to general, not necessarily multiplicative, Hom-Lie algebras and Hom-associative algebras. Definition 2.6 ( [42]) . Let ( g , [ · , · ] g , β ) be a Hom-Lie algebra. A universal envelopingHom-associative algebra of g is a Hom-associative algebra U g = ( U g , ∗ g , α U ) , togetherwith a morphism φ g : ( g , [ · , · ] g , β ) → ( U g , [ · , · ] U g , β Ug ) of Hom-Lie algebras, that satisfiesthe universal property.The following lemma describes the universal property in the involutive case. Lemma 2.7 ( [42]) . Let ( g , [ · , · ] g , β g ) be an involutive multiplicative Hom-Lie algebra. (a) Let ( A, ∗ A , α A ) be a multiplicative Hom-associative algebra, f : ( g , [ · , · ] g , β g ) → ( A, [ · , · ] A , β A ) be a morphism of Hom-Lie algebras, and B be the multiplicative Hom-associativesubalgbera of A generated by f ( g ) . Then B is involutive. b) The universal enveloping multiplicative Hom-associative algebra ( U g , φ g ) of ( g , [ · , · ] g , β g ) is involutive. (c) In order to verify the universal property of ( U g , φ g ) , we only need to considerinvolutive multiplicative Hom-associative algebras A := ( A, ∗ A , α A ) . Definition 2.8.
A linear subspace s ⊆ g is called a Hom-Lie subalgebra of a Hom-Liealgebras ( g , [ · , · ] g , β ) if β ( s ) ⊆ s and s is closed under the bracket operation [ · , · ] g : ∀ s , s ∈ s : [ s , s ] g ∈ s . Let ( g , [ · , · ] g , β ) be a multiplicative Hom-Lie algebra. For any nonnegative integer k ,denote by β k the k -times composition of β , i.e. β k = β . . . β ( k -times ) . In particular, β = Id and β = β . Definition 2.9.
For any nonnegative integer k , a linear map d : g → g is called a β k -derivation of the involutive Hom-Lie algebra ( g , [ · , · ] g , β ) , if [ d, β ] = 0 , that is, d ◦ β k = β k ◦ d, (3) ∀ u, v ∈ g : d [ u, v ] g = [ d ( u ) , β k ( v )] g + [ β k ( u ) , d ( v )] g . (4) Example 2.10.
Let ( g , [ · , · ] g , α ) be an involutive multiplicative Hom-Lie algebra. For x ∈ g , let consider α ( x ) = x , then ad x : g → g defined by ad x ( y ) = [ x, y ] g for all y ∈ g is an α -derivation of ( g , [ · , · ] g , α ) . Let ( A, ∗ A , α A ) be an involutive Hom-associative algebra over ring of integers. Let ( B i , ∗ A , α A | Bi ) ( i ∈ I ) be a family of Hom-associative subalgebras of A as definedin Definition 1 (g), with injective morphisms θ i : B i → A , and for each i ∈ I , a θ i -derivation δ i : B i → A such that α A commutes with θ i and δ i . The associatedHNN-extension is presented as H = h A, B i , t i , δ i , θ i : i ∈ I i , which is an involutive Hom-associative algebra H := ( A ∪ { t i } , ∗ H , α H ) in such a waythat x ∗ H y = α H ( x ∗ A y ) , where α H ( t i ) = t i and α H ( a ) = α A ( a ) along with a homo-morphism φ : ( A, ∗ A , α A ) → ( H, ∗ H , α H ) with the following conditions:(i) t i ∗ H ( φ ( b )) − φ ( θ i ( b )) ∗ H t i = φ ( δ i ( b )) for all b ∈ B i and all i ∈ I .(ii) Given any involutive Hom-associative algebra ( S, ∗ S , α S ) with elements σ i ∈ S satisfying α S ( σ i ) = σ i , a morphism f : ( A, α A ) → ( S, α S ) such that σ i ∗ S α S ( f ( b )) − α S ( f ( b )) ∗ S σ i = f ( δ i ( b )) for all b ∈ B i and i ∈ I , there exists a uniquemorphism θ : ( H, ∗ H , α H ) → ( S, ∗ A , α A ) such that θ ( t i ) = σ i and θ ( φ ( a )) = f ( a ) for all a ∈ A . ssume a single letter t in the condition (i) of construction of HNN-extension of invo-lutive multiplicatve Hom-associative algebra. Since δ is an α A -derivation, δ ( α A ( b )) = t ∗ H α A ( b ) − α A ( b ) ∗ H t = α H ( t ∗ A α A ( b )) − α H ( α A ( b ) ∗ A t ) (by definition of ∗ H ) = α H ( t ) ∗ A α A ( b ) − α A ( b ) ∗ A α H ( t ) (by Def. 2.1 (c), (d)) = t ∗ A b − b ∗ A t = α A ( δ ( b )) , which implies that in the construction of HNN-extension for the case of involutiveHom-associative algebras, it is essential to consider the multiplicative property. Itis worth pointing out that the second property of α -derivations in Definition 2.2 isstraightforward by Hom-associativity.A left Hom- B i -module A/B i is a Hom-module ( A/B i , α A/B i ) that comes equippedwith a left B i -action, B i ⊗ A/B i → A/B i , with b ∗ A/B i ( a + B i ) = ( b ∗ A a ) + B i and α A/B i : A/B i → A/B i with α A/B i ( a + B i ) = α A ( a ) + B i , for all b ∈ B i . Let X i be afree basis of free left Hom- B i -module A/B i . We define a normal sequence as ( t i ∗ A α A ( x )) ∗ A ( t i ∗ A α A ( x )) ∗ A · · · ∗ A ( t i r ∗ A α A ( x r )) , with i j ∈ I and x α ∈ X i j for ≤ α ≤ r . The set of all normal sequences is denoted by V . Theorem 3.1 concerns the embeddability of involutive Hom-associative algebra intoits HNN-extension. We follow the Lichtman and Shirvani’s approach [58] in order toprove that. Theorem 3.1.
Let ( A, ∗ A , α A ) be an involutive Hom-associative algebra over ring ofintegers, B i a family of Hom-associative subalgberas, with injective homomorphisms θ i : B i → A , a θ i -derivations δ i : B i → A . Assume that A/B i is a free left Hom- B i -module for all i , and let ( H, φ ) be the corresponding HNN-extension as above. Then themap φ is an embedding of A into H .Proof. Let us consider the free left Hom- A -module on the set of normal sequences, V ,and denote it by Q = ( ⊕ u ∈ V Au, α Q ) , α Q ( u , . . . , u r ) = ( α H ( u ) , . . . , α H ( u r )) . Consider the morphism of ( A, α A ) into S = (End Z ( Q ) , α S ) mapping a ∈ A to leftmultiplication by a on every factor denoted by a ¯ a and α S = α A . In the sequel, weneed to define suitable σ i ∈ S for all i ∈ I . If q ∈ Q is written as q = X u ∈ V X x ∈ X i ( b x,u ∗ A/B x ) ∗ A u = X u ∈ V X x ∈ X i ( b x,u ∗ A α A ( x )) ∗ A u = X u ∈ V X x ∈ X i b x,u ∗ A ( α A ( x ) ∗ A u ) for b x,u ∈ B i , define σ i ( q ) = X u ∈ V X x ∈ X i ( θ i ( b x,u ) ∗ A (( t i ∗ A α A ( x )) ∗ A u ) + δ i ( b x,u ) ∗ A ( α A ( x ) ∗ A u )) . e have P x ∈ X i ( δ i ( b x,u ) ∗ A α A ( x )) ∈ A and every (( t i ∗ A α A ( x )) ∗ A u ) ∈ V . For anyelement b ∈ B i ( i ∈ I ) , we recall that the left multiplication by b is denoted by ¯ b , so wehave σ i (¯ b ( q )) = σ i ( X u ∈ V X x ∈ X i (( b ∗ B b x,u ) ∗ A ( α A ( x ) ∗ A u )))= X u,x ( θ i ( b ∗ B b x,u ) ∗ A (( t i ∗ A α A ( x )) ∗ A u ))+ X u,x ( δ i ( b ∗ B b x,u ) ∗ A ( α A ( x ) ∗ A u )) , and θ i ( b )( σ i ( q )) = X i ( θ i ( b )) ∗ A ( X u ∈ V X x ∈ X i ( θ i ( b x,u ) ∗ A (( t i ∗ A α A ( x )) ∗ A u )))+ X i ( θ i ( b )) ∗ A ( X u ∈ V X x ∈ X i ( δ i ( b x,u ) ∗ A ( α A ( x ) ∗ A u ))) . Hence, σ i (¯ b ( q )) − θ i ( b )( σ i ( q )) = X u,x (( δ i ( b ) ∗ A b x,u ) ∗ A ( α A ( x ) ∗ A u )) = δ i ( b )( q ) . Therefore, the property (2) implies that there exists θ : ( H, ∗ H , α H ) → ( S, ∗ S , α S ) suchthat θ ( t i ) = σ i and θ ( φ ( a )) = ¯ a for all a ∈ A . Let ( A, ∗ A , α A ) be an arbitrary Hom-associative algebra, and let ( A, [ · , · ] A , β A ) be theHom-Lie algebra defined by [ x, y ] A = x ∗ A y − y ∗ A x, and β A = α A , for x, y ∈ A . If ( g , [ · , · ] g , β g ) is an involoutive Hom-Lie algebra, then ( U g , φ g ) is called a universal enveloping Hom-associative algebra of g , if φ g : ( g , [ · , · ] g , β g ) → ( U g , [ · , · ] U g , β U g ) is a homomorphism of Hom-Lie algebras, φ g ([ x, y ] g ) = [ φ g ( x ) , φ g ( y )] U g , φ g ( β g ( x )) = β U g ( φ g ( x )) , satisfying the following universal property: for any involutive Hom-associative algebra A = ( A, ∗ A , α A ) and any Hom-Lie algebra morphism ε : ( g , [ · , · ] g , β g ) → ( A, [ · , · ] A , β A ) ,there exists a unique morphism η : U g → A of Hom-associative algebras such that ηφ g = ε . For any involutive Hom-Lie algebra there exists a universal enveloping Hom-associative algebra, which is involutive and Poincare-Birkhoff-Witt theorem is valid forit. This shows that the map φ g is injective, and we can say that every β g -derivation ofinvolutive Hom-Lie algebra ( g , [ · , · ] g , β g ) extends to β U g -derivation of U g . efinition 4.1. Let ( g , [ · , · ] g , β g ) be an involutive Hom-Lie algebra and s be a subalge-bra. Assume that d : s → g is a β g -derivation. The associated HNN-extension is givenby the following presentation h := h g , t : d ( s ) = [ t, s ] h , s ∈ s i , which is an involutive Hom-Lie algebra ( h , [ · , · ] h , β h ) with β h ( t ) = t , β h ( g ) = β g ( g ) for g ∈ g . This means that the presentation of g is augmented by adding a new generatingsymbol t , and for each s ∈ s , the relation [ t, s ] h = d ( s ) is added. We note that [ g , g ] h = [ g , g ] g , for all g , g ∈ g .Let assume that in the Definition 4.1, s = g , therefore, d is a β g -derivation of g and h is then the semi-direct product of g with a one-dimensional involutive Hom-Liealgebra which acts on g via d . In order to make this special case more clear, we recallthe concepts of Hom-action and semidirect product of Hom-Lie algebras in the sequelin accordance with [37]. Definition 4.2.
Let ( l , α l ) and ( m , α m ) be Hom-Lie algebras. A Hom-action from ( l , α l ) on ( m , α m ) is expressed by a bilinear map σ : l ⊗ m → m , σ ( x ⊗ m ) = x m such that(a) [ x, y ] α m ( m ) = α l ( x ) ( y m ) − α l ( y ) ( x m ) ,(b) α l ( x ) [ m, m ′ ] = [ x m, α m ( m ′ )] + [ α m ( m ) , x m ′ ] ,(c) α m ( x m ) = α l ( x ) α m ( m ) ,for all x, y ∈ l and m, m ′ ∈ m . Definition 4.3 ( [37]) . Let ( l , α l ) and ( m , α m ) be Hom-Lie algebras with an actionfrom ( l , α l ) on ( m , α m ) . The semidirect product ( m ⋊ l , ˜ α ) is the Hom-Lie algebra withunderlying K -vector space m ⊕ l , with bracket [( m , x ) , ( m , x )] = ([ m , m ] + x m − x m , [ x , x ]) and endomorphism ˜ α : m ⊕ l → m ⊕ l , ˜ α ( m, x ) = ( α m ( m ) , α l ( x )) for all x, x , x ∈ l and m, m , m ∈ m .If in the Definition 4.1 of HNN-extension of involutive Hom-Lie algebras, β g -deri-vation map is defined on the whole involutive Hom-Lie algebra ( g , [ · , · ] g , β g ) , then asemidirect product of one-dimensional involutive Hom-Lie algebra with g with respectto β g -derivation map will be obtained. Theorem 4.4.
Any involutive Hom-Lie algebra embeds into its HNN-extension.Proof.
Let ( U g , φ g ) and ( U s , φ s ) be the universal enveloping Hom-associative algebrascorresponding to, respectively, the involutive Hom-Lie algebra g and its subalgebra s ,which are involutive with respect to Lemma 2.7. Let h = h g , t : d ( s ) = [ t, s ] h , s ∈ i be the HNN-extension of involutive Hom-Lie algebra ( g , [ · , · ] g , β g ) as above. Byextending d to a β U g -derivation of U g defined on U s we form the HNN-extension ofinvolutive Hom-associative algebra U g which is denoted by M = h U g , U s , t, δ i . Let ( R, ∗ R , α R ) be an arbitrary involutive Hom-associative algebra with a homomorphismof Hom-Lie algebras ( h , [ · , · ] h , β h ) → ( R, [ · , · ] R , β R ) . The restriction to g extends toa homomorphism U g → R , which extends to a homomorphism M → R , so we have U h ≃ M . As U g / U s is a free left Hom- U s -module, Theorem 3.1 implies that U g isembedded into M , and so g embeds into its HNN-extension. Acknowledgement
Chia Zargeh was supported by postdoctoral scholarship CNPq, Conselho Nacional deDesenvolvimento Científico e Tecnológico - Brasil (152453/2019-9).
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