On Malcev algebras nilpotent by Lie center and corresponding analytic Moufang loops
Alexander Grishkov, Marina Rasskazova, Liudmila Sabinina, Mohamed Salim
aa r X i v : . [ m a t h . R A ] M a y On Malcev algebras nilpotent by Lie center andcorresponding analytic Moufang loops.
Alexander Grishkov, Marina Rasskazova,Liudmila Sabinina, Mohamed SalimMay, 2020
Abstract
In this note we describe the structure of finite dimensional Malcev al-gebras over a field of real numbers R , which are nilpotent module its Liecenter. It is proved that the corresponding analytic global Moufang loopsare nilpotent module their nucleus. Key words:
Malcev algebras, Moufang loops, Global Moufang loops . The theory of analytic loops started with the work of A.I. Malcev in [Ma1]. Inthis article, the correspondence between the analytic local diassociative loopsand the binary Lie algebras was established. A loop is a set M , endowed witha binary operation M × M → M , with the neutral element e ∈ M and thecondition that the equations ax = b, ya = b for all a, b ∈ M have a uniquesolution. A loop is called diassociative , if every two elements of this loop generatea subgroup. Malcev showed that for an analytic loop, with the Moufang identity:( xy )( zx ) = x ( yz ) x the corresponding tangent algebra satisfies the followingidentities: x = [ J ( x, y, z ) , x ] − J ( x, y, [ x, z ]) = 0 , where J ( x, y, z ) = [[ x, y ] , z ] + [[ x, z ] , x ] + [[ z, x ] , y ] (See [1]). Algebras with thesedefining identities are currently called Malcev algebras . The more difficult ques-tion, if every finite dimensional Malcev R -algebra is the tangent algebra ofsome local analytic Moufang loop was solved positively by E.Kuzmin in 1969 in[Kuzm2].Let us consider a pair ( a , L ), where a is some subvariety of binary Lie algebras(in particular Malcev algebras) and L is a subvariety of diassociative loops (inparticular, Moufang loops).A pair ( a , L ), will be called locally dual , if it satisfies the following conditions:1 every tangent algebra of a local analytic loop from the variety L belongsto the variety a , • every finite-dimensional R -algebra from the variety a is a tangent algebraof some local analytic loop from the variety L The description of all locally dual pairs is a meaningful and difficult problem.If a locally dual pair ( a , L ) satisfies a stronger condition, namely, if for everyglobal analytic loop S such that its local loop belongs to L , is in L too, we willcall such a pair ( a , L ) globally dual .Kerdman [Ker] showed that a pair ( m , M ) is globally dual if m is the va-riety of all Malcev algebras, and M is a variety of all Moufang loops. Inthis note we study the duality of a pair formed by two varieties: the firstone, N L k , is the subvariety of the variety m , which is defined by the identity J ([ x , x , ..., x k ] , y, z ) = 0 , where in the product [ x , x , ..., x k ] a distributionof parentheses is arbitrary and the second one, N G k , is the subvariety of thevariety M , defined by the identity ([ x , x , ..., x k ] , y, z ) = 1 . Here [ x , x , ..., x k ]is a commutator of length k with an arbitrary distribution of parentheses and( a, b, c ) = (( ab ) c )( a ( bc )) − is an associator.For a Malcev algebra M we define M = M , M n = i + j = n P i,j> [ M i , M j ]. Let F bea free Malcev algebra, then N L k is a variety of Malcev algebras defined by allidentities of the type J ( w, x, y ) = 0, where w ∈ F k , k ≥ . Let M be a finite dimensional Malcev algebra over a field C , and let G be asolvable radical of M. Then there exists a semisimple subalgebra (Levi factor) S, such that M = S ⊕ G ( [Gri1],[Kuzm1],[Car]).We will use the results and the terminology from [Gri2]. Let g ∈ M , theelement g is said to be splitting element if g = t + n , where n is a nilpotentelement and t is a semisimple element , i.e., the right multiplication operator R t is diagonalizable and the operators R t and R n commute. A Malcev algebra M issaid to be splitting if all elements of M are splitting. If M is a finite-dimensionalsplitting Malcev algebra over a field of characteristic 0, then M = S ⊕ T ⊕ N ,where S is a semisimple Levi factor, T is an abelian subalgebra of M such thateach element of T is semisimple ( toroidal subalgebra), and N is the nilpotentradical of M. Additionally [
S, T ] = 0, N = X α ∈ ∆ M N α , where ∆ ⊂ T ∗ = Hom k ( T, k ) and N α = { x ∈ N | [ x, t ] = α ( t ) x, ∀ t ∈ T } . (1)2oreover, [ N α , N β ] ⊆ N α + β , if α = β, and [ N α , N α ] ⊆ N α + N − α . Since [
T, S ] = 0, one has that N α is an S -module and hence N = N ⊕ N ,[ S, N ] = [ S, N ] = N and [ S, N ] = 0. Set M = S ⊕ X α ∈ ∆ \ M N α ⊕ N and let us denote by M the subalgebra generated by T ⊕ M . Notice that ingeneral, M = T ⊕ M , and [ N , N ] ⊆ N , [ M , N ] ⊆ M . Hence M is an ideal. Every finite-dimensional Malcev algebra M over a field of charac-teristic 0 is contained in some splitting Malcev algebra ˆ M .
If such ˆ M does notcontain intermediate splitting subalgebra, which contains M , then ˆ M is calleda splitting of M . Each automorphism of the algebra M extends uniquely to anautomorphism of a splitting of M. If ˆ M is a splitting of M , then ˆ M = M ,and any ideal of the algebra M is an ideal of the algebra ˆ M and vice verse anyideal of ˆ M , which is in M is also the ideal of M . This result is analogous to onefor Lie algebras due to A. I. Malcev [Ma2].In what follows in this article the splitting algebra of an algebra M we willdenote by ˆ M .
Recall that
Lie ( M ) = { x ∈ M | J ( x, M, M ) = 0 } is the Lie center of M . Lemma 1.
In this notation
Lie ( M ) ⊆ Lie ( ˆ M ) . Proof.
By the construction ˆ M = n S i =1 M ( i ) , where M (1) = M, M ( n ) = ˆ M , dim M ( i ) =dim M ( i −
1) + 1 , M ( i ) = M ( i −
1) + R t i , i ≥ , where t i is asemisimple element, i.e. R t i : M ( i − → M ( i −
1) is a diagonalizable operator.Moreover there exists x i ∈ M ( i − n i = x i − t i is a nil element, i.e R n i is a nilpotent operator and R t i ◦ R n i = R n i ◦ R t i , [ n i , t i ] = 0 . Now consider l ∈ Lie ( M ( i − , x ∈ M ( i − . It is sufficient to show that J ( l, x, t i ) = 0 . Suppose that M ( i −
1) = Σ α ∈ ∆ ⊕ M ( i − α (2)is a Cartan decomposition with respect to the operator R t i or R x i . Without loss of generality one can assert that l ∈ Lie ( M ( i − ∩ M ( i − α , x ∈ M ( i − β . If α = β, then J ( l, x, t i ) = 0 due to the fact that any Malcev algebra is a binary-Lie algebra. 3onsider the case α = β = 0 . The equality J ( l, x, t i ) = 0 is equivalent to [ x, l ] ∈ M ( i − α . Indeed, suppose that [ x, l ] / ∈ M ( i − α . Then [ x, l ] = [ x, l ] α +[ x, l ] − α , where[ x, l ] α ∈ M ( i − α and 0 = [ x, l ] − α ∈ M ( i − − α . Recall that x i − n i = t i . J ( l, x, x i ) = [[ x, l ] α , ( t i + n i )]+[[ x, l ] − α , ( t i + n i )]+[[ x, ( t i + n i )] , l ] − [[ l, ( t i + n i )] , x ]We get2 α [ l, x ] α − α [ l, x ] − α +[[ x, l ] α +[ x, l ] − α , n i ]+[[ x, t i ] , l ]+[[ x, n i ] , l ] − [[ l, t i ] , x ] − [[ l, n i ] , x ]= 2 α [ l, x ] α − α [ l, x ] − α +[[( x, l ] α +[ x, l ] − α ) , n i ]+[[ x, n i ] , l ]+2 α ([ x, l ] α +[ x, l ] − α ) − [[ l, n i ] , x ] = − α [ x, l ] α + α [ x, l ] − α +2 α ([ x, l ] α +[ x, l ] − α )+[([ x, l ] α +[ x, l ] − α ) , n i ]+[[ x, n i ] , l ] α + [[ x, n i ] , l ] − α − [[ l, n i ] , x ] α − [[ l, n i ] , x ] − α = 0Finally we get3 α [ x, l ] − α + [[ x, l ] − α , n i ] + [[ x, n i ] , l ] − α − [[ l, n i ] , x ] − α = 0 (3)For every element 0 = a ∈ M ( i − α one can define the number | a | = min { m | R mn i a = 0 } . Then the equality [ x, l ] − α = 0 one can show byinduction on the sum | x | + | l | . If | x | + | l | = 2 one has [[ x, n i ] , l ] − α = [[ l, n i ] , x ] − α =0 . Now suppose that 2 < | x | + | l | = m. Then one has | [ x, n i ] | + | l | < m and | [ l, n i ] | + | x | < m and by induction hypothesis [[ x, n i ] , l ] − α = [[ l, n i ] , x ] − α = 0 . Therefore by (3) we get 3 α [ x, l ] − α + [[ x, l ] − α , n i ] = 0 (4)Now if | [ x, l ] − α | = s we can apply R s − n i to the equality (4) and get [ x, l ] − α = 0 . In the case α = β = 0 we have J ( l, x, t i ) = 0 , since [ N , N ] ⊆ N and [ N , t i ] = 0Thus one has Lie ( M ) ⊆ ... ⊆ Lie ( M ( i )) ⊆ ... ⊆ Lie ( ˆ M )In the theory of Lie algebras there exists the following construction of de-composable extension . Let L be a Lie algebra and let N be a subalgebra of theLie algebra Der L (the algebra of all derivations of L ). Then the direct sum N ⊕ L has a structure of Lie algebra with the multiplication:( a, l ) · ( b, r ) = ([ a, b ] , l b − r a + [ l, r ]) . (5)Notice that we are not assuming that N or L is abelian.4his construction has a generalization for Malcev algebras.Suppose that M is a Malcev algebra such that M = ˜ N + L, where L ⊆ Lie ( M ) , and M has an ideal I ∈ ˜ N such that J ( ˜ N ) ⊆ I, [ I, L ] = 0 , then˜ N/I ∼ = N is a Lie algebra. It means that N acts on L by derivations. In thiscase the formula (5) defines a Malcev algebra structure on ˜ M = ˜ N ⊕ L , where I acts trivially on the Lie algebra L by definition. This construction is calledthe decomposable extension of Malcev algebras . Notice that in this construction L is an ideal contained in the Lie center of M .In what follows the decomposable extension of Malcev algebra will be denotedby ˜ M .
The aim of this section is to show the following
Theorem 1.
Let M be a finite dimensional Malcev algebra from the variety N L k over a field of complex numbers C . Then1. M may be embedded into the splitting Malcev algebra ˆ M = S ⊕ T ⊕ N ∈ N L k , where S is a semi-simple Lie subalgebra, T is a toroidal subalgebra, N is a nilpotent ideal.2. N = N ⊕ [ S, N ] , where N is a Malcev subalgebra of N and the ideal M generated by S ⊕ T ⊕ (cid:16)P α ∈ ∆ \ L N α (cid:17) ⊕ [ S, N ] is contained in Lie ( ˆ M ) .3. There exists a Malcev algebra ˜ M = N ⊕ M of the variety N L k , whichis a decomposable extension of a nilpotent Malcev algebra N and a Liesubalgebra M , such that there exists an epimorphism π : ˜ M −→ ˆ M . In order to prove this Theorem we need to collect some intermediate resultswhich we will present in the following lemmata.Put ∞ T n =1 M i by M ω . Following introduced notation one has:
Lemma 2.
Let M be a splitting Malcev algebra from the variety N L k , k ≥ .Then ideal M , constructed above, contains M ω and is contained in the Liecenter Lie ( M ) . Proof.
By definition of the variety
N L k we get that M ω ∈ Lie ( M ) the Lie centerof M . By construction M ⊆ M ω ⊆ Lie ( M ), hence M ′ ⊆ Lie ( M ) where M ′ isa subalgebra of M generated by M . It is clear that M = T ⊕ M ′ , [ T, M ′ ] ⊆ M ′ = S ⊕ V with V ⊆ N . Hence for proving the lemma it is enough to provethat J ( x, y, z ) = 0, where x, y and z are elements of T ∪ ( ∪ α ∈ ∆ N α ). If x, y ∈ T ,then z ∈ N α and J ( x, y, z ) = [[ z, x ] , y ] + [[( y, z )] , x ] = α ( x ) α ( y ) z − α ( y ) α ( x ) z = 0 . If x ∈ T , y ∈ N α , z ∈ N β and α = 0 or β = 0, then y ∈ M ⊆ Lie ( M )or z ∈ Lie ( M ), hence J ( x, y, z ) = 0. At last, in the case α = β = 0 we get J ( x, y, z ) = 0 since [ N , N ] ⊆ N . 5 emma 3. Let M be a Malcev algebra from the variety N L k .Then [ J ( M ) , M ω ] = 0 . Proof.
By the result of Filippov (see [Fi],page 236) one has [ J ( M ) , Lie ( M )] = 0.On the other hand, M ω ⊆ Lie ( M ) since M ∈ N L k . Since M ⊆ M ω by Lemma 3 we have [ J ( M ) , M ] = 0 . Then the subalgebra N acts on the ideal M by derivations, hence it is possible to construct, asabove, a Malcev algebra ˜ M = N ⊕ M with a product given by (5).It is easy to see that the morphism ϕ : ˜ M −→ M, ϕ ( n, m ) = n + m is anepimorphism of Malcev algebras. Lemma 4.
The Malcev algebra ˜ M = N ⊕ M with the product given by (5)is a Malcev algebra of the variety N L k .Proof. Since M ⊂ Lie ( ˜ M ) , one has J ( ˜ M ) = J ( N ). By construction,[ M , J ( N )] = 0 . In this case, ˜ M is a Malcev algebra of the variety N L k if andonly if N is a Malcev algebra of the variety N L k ; which is exactly our case. Remark.
In general, if we have a Malcev algebra P = P + P , where P isa nilpotent subalgebra, P ⊆ Lie ( P ) is an ideal contained in the Lie center of P and P/P is a Malcev algebra of the variety N L k , then P is not necessarilya Malcev algebra of the the variety N L k . It is possible that P ∈ N L k +1 \ N L k . Example.
Set P = R { t, a, b, c | [ a, t ] = a, [ b, t ] = − b, [ a, b ] = c, [ c, t ] =[ a, c ] = [ b, c ] = 0 } and let it be a splitting Lie algebra. Choose any nilpotentMalcev algebra P which is not a Malcev algebra from the variety N L k , but P /Z is. Here Z = R z is some central ideal of P . It is easy to construct analgebra with those properties. Let us consider ˜ P = P ⊕ P and P = ˜ P /I , where I = R ( c − z ) is a central ideal. It is clear that P is not a Malcev algebra from thevariety N L k . But P = π ( P ) + π ( P ) , where π : ˜ P −→ P is a canonical homo-morphism. Notice that π ( P ) ∼ = P , π ( P ) ∼ = P ⊆ N ( P ) and P/π ( P ) = P /Z is a Malcev algebra from the variety N L k . Proof of the Theorem.
Consider ˆ M = S ⊕ T ⊕ N as splitting algebra of M = S ⊕ G. We will showthat ˆ M ∈ N L k if M ∈ N L k . Due to the construction of ˆ M ([Gri2]) any ideal I ✁ M is also the ideal of ˆ M and therefore M k = ˆ M k , k ≥
2. Now since
Lie ( M ) ⊆ Lie ( ˆ M ) and M k ⊆ Lie ( M ) one has M k ⊆ Lie ( M ) ⊆ Lie ( M k ) . Thismeans M k ∈ N L k . As it was noticed above N is a semisimple S ⊕ T - module,therefore N = Ann N ( S ⊕ T ) is the nilpotent subalgebra of ˆ M .
Moreover M ⊆ ˆ M ω ⊆ Lie ( ˆ M )Since [ T, N ] = 0 one has N ⊆ N . In other hand in general case N ∩ M = 0 . Recall that M is the ideal generated by T ⊕ M . Finally one gets ˜ M =6 ⊕ M . A variety M of Malcev algebras will be called (locally) smooth , if there exists avariety of Moufang loops L such that the pair ( M , L ) is (locally) globally dual.Analogously, a variety of Moufang loops L is (locally) smooth if there exists avariety of Malcev algebras M , such that the pair ( M , L ) is (locally) globallydual. A dual pair ( M, L ) will be called global if for any local analytic loop G of the variety L , there exists a global analytic loop ˜ G from the variety L whichis locally isomorphic to G . It is clear, that not all varieties of Moufang loopsare smooth. For example, the variety B n of Moufang loops of exponent n isnot smooth, since every analytic Moufang loop of a positive dimension is notperiodic. Nevertheless we have the following Conjecture: Conjecture 1.
Every dual pair ( M, L ) of Malcev algebras and their correspond-ing Moufang loops is global. Notice that if the pair ( M , L ) is locally dual and the variety M containsonly Lie algebras, then all finite dimensional Lie algebras from M are solvable.Indeed, if M ∈ M is not solvable finite dimensional then M = S ⊕ G, where S is semisimple Lie subalgebra. Hence S contains some simple 3 − dimensional Liesubalgebra L. But the corresponding Lie group G ( L ) contains free subgroup.Hence L is variety of all groups and M is the variety of all Lie algebras.We will prove the Conjecture 1 for the pairs ( N L k , G k ), where N L k is thevariety defined in the last section and G k is a variety of Moufang loops definedby all identities of the type ( w, x, y ) = 1, where w ∈ F k , k ≥ F is aninfinite free generated Moufang loop such that F = F , and F k is the normalsubloop generated by k − Π i =1 [ F i , F k − i ]. Proposition 1.
The pair ( N L k , G k ) is dual for any k ≥ .Proof. Let M be a Malcev R -algebra of dimension n of the variety N L k . Then M ∼ = R n . There exists a small ball M ǫ = { x ∈ M | | x | ≤ ǫ } , which is a localMoufang loop with the product given by the Campbell-Hausdorff formula x · y := CH( x, y ) = x + y + 12 [ x, y ] + · · · . (6)Notice that the element 0 of M is the unit of this local analytic loop. From(6) we have that for every subalgebra P of M the corresponding local subgroupis given by P ǫ = P ∩ M ǫ . The subgroup P ǫ is normal if and only if P is an idealof M. { x, y } = x − · y − · x · y = [ x, y ] + ∞ X s =3 a s ( x, y ) , (7)where a s ( x, y ) ∈ M s if x, y ∈ M . Hence every commutator w in the local Mo-ufang loop ( M, · ) of length k ≥ w = ∞ P i = k w i , with w i ∈ M i . Since M ∈ N L k , then M s ⊆ Lie ( M ) for s ≥ k . Hence the corresponding commutatorsubloop M kǫ of local Moufang loop M ǫ is contained in Lie ( M ) . E.Kuzmin proved [ ? ], that in a local Moufang loop ( M, · ) the associator canbe expressed as: ( x, y, z ) = 16 J ( x, y, z ) + ∞ X i =7 a i ( x, y, z ) , (8)where a i ( x, y, z ) is an element of degree i of the ideal J ( M ) ⊂ M. By (8) we get that
Lie ( M ) ∩ M ǫ ⊆ N uc ( M ǫ ) , where N uc ( M ǫ ) = { x ∈ M ǫ | ( x, a, b ) = 0 , ∀ a, b ∈ M ǫ } . Hence M kǫ ⊂ Lie ( M ) ∩ M ǫ ⊂ N uc ( M kǫ ) . It meansthat ( M ǫ , · ) ∈ G k . Now suppose that ( M ǫ , · ) ∈ G k . Following the previous notation, we have:
Lemma 5. M k ∩ M ǫ = ( M ǫ , · ) k , where ( M ǫ , · ) k is a commutator subloop of thelocal loop ( M ǫ , · ) of degree k. Proof.
From the construction of the local loop ( M ǫ , · ), for every ideal I ofthe Malcev algebra M there is a corresponding normal subloop I ǫ = I ∩ M ǫ of ( M ǫ , · ) . It is clear that for nilpotent of class k Malcev algebra
M/M k thecorresponding local Moufang loop is (( M/M k ) ǫ , · ), which is nilpotent of class k. Hence ( M ǫ , · ) k ⊆ M k .Suppose that ( M ǫ , · ) is a nilpotent local loop of class k. By induction weprove that the Malcev algebra M is nilpotent of the same class k . It is clearfor k = 1. If the Malcev algebra M is not nilpotent of degree k then for some x , ..., x k ∈ M we have w = [ x , ..., x k ] = 0 for some distribution of parentheses.By (7) we get in ( M ǫ , · ): u t = { tx , tx , ..., tx k } = t k w + P i>k t i w i , where w i is an element of M i and t ∈ R . Since ( M, · ) is nilpotent of degree k , u = 0 in( M, · ) . Then w = 0 in M which yields to a contradiction.With all considerations above Lemma 5 is proved.Now we can finish the proof of Proposition 1. Let w = [ x , ..., x k ] ∈ M k ,we have to prove that w ∈ Lie ( M ) . For some t ∈ R we have by Lemma 5that t k w = [ tx , ...., tx k ] ∈ M k ∩ M ǫ ⊂ ( M ǫ , · ) k ⊂ N uc ( M ǫ ) . Here we usedthat ( M ǫ , · ) ∈ G k . Hence ( t k w, x, y ) = 0 for all x, y ∈ M ǫ . By 8 we get that J ( t k w, x, y ) = 0 . It means that w ∈ Lie ( M ) . Proposition 1 is proved.8ow we are ready to prove the main result of this paper.
Theorem 2.
The dual pair ( N L k , G k ) is global.Proof. Let G ∈ G k be a local analytic loop and let L ( G ) = M ∈ N L k be its corresponding Malcev algebra. Let ϕ : M ֒ → ˆ M be an embedding of M in a splitting Malcev algebra ˆ M = S ⊕ T ⊕ N , (see notation of Theorem1). By definition ˆ M is minimal with this property. Hence [ ˆ M , ˆ M ] = [ M, M ] . Since M ∈ N L k then ˆ M ∈ N L k by Theorem 1. By [Ker] there exists thecorresponding to ˆ M global analytic simply connected Moufang loop ˆ G. By construction of ˆ G in [Ker] we get that ˆ G = P · Q, where P is simply con-nected semisimple Lie group with corresponding Lie algebra S and Q = Q · Q is simply connected solvable Moufang loop with corresponding Malcev algebra T ⊕ N, and Q ≃ R t ≃ T is abelian vectorial group Lie, corresponding to Liesubalgebra T, Q is simply connected nilpotent normal subloop corresponding tothe nilpotent ideal N. By Theorem 1 we have that ˆ M = S ⊕ T ⊕ N = N + M , where N ⊆ N is a nilpotent subalgebra and M is an ideal of ˆ M that iscontained in Lie ( ˆ M ) . Since exponential map from N to Q is a bijection then Exp ( N ) = Q ⊂ Q is a nilpotent simply connected subloop of ˆ G. Sinceˆ M = N + M then ˆ G = Q .G , where G is simply connected normal groupLie corresponding to the ideal M ⊂ Lie ( ˆ M ) . Since ˆ M is a splitting by Theorem 1 there exists a Malcev algebra ˜ M = N ⊕ M ∈ N L k and an epimorphism π : ˜ M ˆ M .
Let ˜ G be simply connected analytic Moufang loop corresponding to Malcevalgebra ˜ M . Then ˜ G = Q × G , where Q and G are subloops of ˆ G. Noticethat multiplication in ˜ G may be given the following analogue of (3).( r , g ) · ( r , g ) = ( r r , g r g ) (9)Where g r = r − g r is natural action of Moufang loop Q on the Lie group G by automorphisms, since A ( Q ) = J ( N ) acts trivially on G . Here A ( Q ) isan associator of Q and we used that [ J ( ˆ M ) , M ] ⊆ [ J ( ˆ M ) , Lie ( ˆ M )] = 0.It is clear that G is contained in the nucleus of ˜ G and ˜ G ∈ G k if and onlyif Q ∈ G k . Lemma 6.
Let N be a nilpotent finite dimensional Malcev algebra and R be thecorresponding simply connected analytic Moufang loop.If the corresponding local analytic Moufang loop R ǫ satisfies some identity f ( x , . . . , x n ) = 1 , then the global analytic loop R satisfies the same identity.In particular, N ∈ N L k if and only if R ∈ G k . Proof.
It is possible to assume that R = N ≃ R m with multiplication (6). Let f be an identity of the local analytic loop R ǫ . Then f ( x , . . . , x n ) = P j f j , where f j = f j ( x , . . . , x n ) is a Lie word in x , . . . , x n . Let v , . . . , v m be a basisof N. Then f = P mj =1 g j v j , where g j = g j ( x , . . . , x n ) is a polynomial functionin x , . . . , x n , j = 1 , . . . , m. Since f is a local identity then g j = 0 if | x s | < ε,s = 1 , . . . , n and ε is small enough. But any polynomial function, which is equal9o zero in some neighborhood of ¯0 is equal to zero for all values of the variables.Hence f = 1 is an identity of the loop R .Returning to the proof of the Theorem 2 we have that ˜ G ∈ G k due to Lemma6. Hence ˆ G ∈ G k , since ˆ G is a homomorphic image of ˜ G. With this the proof of the Theorem 2 is done.
Acknowledgements
The first author was supported by grants CNPq 307824/2016-0 and FAPESP2018/ 23690-6. The second author was supported by grant FAPESP 2018/11292-6. The third author thanks for the support to FAPESP for grant 2019/24418-0and the University of Sao Paulo. The first and fourth authors was supportedby UAEU UPAR grant G00002599.
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