Existentially closed Leibniz algebras and an embedding theorem
aa r X i v : . [ m a t h . R A ] N ov EXISTENTIALLY CLOSED LEIBNIZ ALGEBRAS AND ANEMBEDDING THEOREM
CHIA ZARGEH
Abstract.
In this paper we introduce the notion of existentially closed Leib-niz algebras. Then we use HNN-extensions of Leibniz algebras in order toprove an embedding theorem.
IntroductionThe notion of algebraically closed was originally introduced for groups in a shortpaper of W. R. Scott [9]. A group G is said to be algebraically closed if everyfinite set of equations and inequations which is consistent with G already has asolution in G . Scott applied the concept of algebraically closed in order to provideimportant embedding theorems stating that every countable group G can be em-bedded in a countable algebraically closed group H . There exists a rich literatureon the properties of existentially closed groups and their applications, an interestedreader can see [4] and [8]. Having considered the properties of closure of algebraicsystems in both existentially and algebraically senses, we can claim that they areequivalent concepts for groups and Lie algebras. We recall that an algebraic system A is existentially closed, if every consistent finite set of existential sentences withparameters from A , is satisfiable in A . Shahryari in [10] used the concept of exis-tentially closed groups and Lie algebras to prove some embedding theorems. Forinstance, Shahryari showed that any Lie algebra L can be embedded in a simpleLie algebra in such a way for any non-zero elements a and b , there is x such that[ x, a ] = b .In this work, we introduce the concept of existentially closed for Leibniz algebraswhich are a non-antisymmetric generalization of Lie algebras introduced by Bloh [1]and Loday [7]. We follow Shahryari’s approach to provide an embedding theoremanalogous to Lie algebras. Our main tool in this work is the HNN-extension ofLeibniz algebras which has been constructed in[6].The paper is organized as follows. Section 1 is devoted to preliminary tools andthe concept of existentially closed for the case of Leibniz algebras. In Section 2, werecall the concept of HNN-extensions of Leibniz algebras with more details. In Sec-tion 3, we provide a theorem on embeddability of any Leibniz algebra in a solvableLeibniz algebra. Date : 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Existentially closed, Leibniz algebras, HNN-extension. Existentially closed Leibniz algebras
A right Leibniz algebra is defined as a vector space with a bilinear multiplicationsuch that the right multiplication is a derivation. Indeed, a Leibniz algebra is avector space L over a field K with some bilinear product [ − , − ] which satisfies theLeibniz identity [[ x, y ] , z ] = [[ x, z ] , y ] + [ x, [ y, z ]] . Let I be a subspace of a Leibniz algebra L . Then I is a subalgebra if [ I, I ] ⊂ I , aleft (resp. right) ideal if [ L, I ] ⊂ I (resp. [ I, L ] ⊂ I ). I is an ideal of L if it is botha left ideal and a right ideal. The Leibniz algebra Leib ( X ) is called free Leibnizalgebra with a set of generators X if, for any Leibniz algebra L , an arbitrary map X → L can be extended to an algebra homomorphism Leib ( X ) → L . Then X iscalled the set of free generators of Leib ( X ).One way of obtaining Leibniz algebras is to use a dialgebra D . This is a vectorspace equipped with two bilinear associative products ⊣ and ⊢ , and the laws x ⊣ y ⊣ z = x ⊣ ( y ⊢ z )( x ⊢ y ) ⊣ z = x ⊢ ( y ⊣ z )( x ⊣ y ) ⊢ z = x ⊢ y ⊢ z. If we define [ x, y ] = x ⊣ y − y ⊢ x , then ( D, [ − , − ]) becomes a Leibniz algebra. Definition 1.1.
Leibniz algebra L (with [ L, L ] = I ) is said to be simple if the onlyideals of L are { } , I , L . Definition 1.2.
A Leibniz algebra L is called solvable if there exists n ∈ N suchthat L [ n ] = 0, where L [1] = L , L [ s +1] = [ L [ s ] , L [ s ] ] for s ≥ Leib be the class of Leibniz algebras over field k . For A, B ∈ Leib , the notation A ∗ B stands for free product of A and B in Leib . If L ∈ Leib , then Φ ∈ L ∗ Leib ( X )can be considered as an L -valued function on L . An equation of the form Φ( x ) = 0is solvable over L if there exists an extension ¯ L of L such that the equation has asolution in ¯ L . In the case of finding such a solution in L itself then Φ( x ) = 0 is saidto be solvable in L . Definition 1.3.
A Leibniz algebra L is called existentially closed if every systemof equations which is solvable over L is solvable in L .2. HNN-extensions of Leibniz algebras
The Higman-Neumann-Neumann extensions (HNN-extensions) for groups wasalready introduced in [3]. If A is a subgroup of a group G and t ∈ G , then themapping a t − at is an isomorphism between the two subgroups A and t − At of G . The HNN construction tries to reverse the viewpoint. For a group G withan isomorphism φ between two of its subgroups A and B , H is an extension of G with an element t ∈ H such that t − at = φ ( a ) for every a ∈ A . The group H ispresented by H = h G, t | t − at = φ ( a ) , for all a ∈ A i and it implies that G is embedded in H . The HNN-extension of a group possessesan important position in algorithmic group theory which has been used for the proof XISTENTIALLY CLOSED LEIBNIZ ALGEBRAS AND AN EMBEDDING THEOREM 3 of the embedding theorem, namely, that every countable group is embeddable intoa group with two generators.Ladra et al. [6] studied the same construction for Leibniz algebras (as well astheir associative relatives, the so-called dialgebras) and proved that every Leibnizalgebra embeds into any of its HNN-extensions. The main difference between theconstruction of HNN-extension for groups and algebras is that the concepts of sub-groups and isomorphism are replaced by subalgebras and derivation, respectively.In other words, the derivation map defined on a subalgebra is used instead of iso-morphism between subgroups. In this section we recall the notion of HNN-extensionof Leibniz algebras. We note that the HNN-extension for Leibniz algebras has beenconstructed corresponding to both derivation and anti-derivation maps.
Definition 2.1.
A derivation of Leibniz algebras is defined in a similar way to thederivation of Lie algebras, that is, a linear map d : L → L satisfying d ([ x, y ] = [ d ( x ) , y ] + [ x, d ( y )] , for all x, y ∈ L . Definition 2.2. An anti-derivation of Leibniz algebras is defined as a linear map d ′ : L → L such that d ′ ([ x, y ]) = [ d ′ ( x ) , y ] − [ d ′ ( y ) , x ]for x, y ∈ L . HNN-extensions of Leibniz algebras.
Let L be a right Leibniz algebra and A be a subalgebra. We assume that the derivation d and anti-derivation d ′ aredefined on a subalgebra A instead of the whole L . The HNN-extensions of theLeibniz algebra L corresponding to the derivation d and the anti-derivation d ′ aredefined as follows, respectively:(2.1) L ∗ d := h L, t : d ( a ) = [ a, t ] , a ∈ A i , and(2.2) L ∗ d ′ := h L, t : d ′ ( a ) = [ t, a ] , a ∈ A i . Here t is a new symbol not belonging to L . By this, a new generating letter t isadded to any presentation of L . There are two special cases of HNN-extensions ofLeibniz algebras. • If A = L , then d is a derivation of L and L ∗ d is then the semidirect productof L with a one-dimensional Leibniz algebra which acts on L via d . • If A = 0, then L ∗ d is the free product of L with a one-dimensional Leibnizalgebra.If a Leibniz algebra L has a presentation h X | S i in the class of Leibniz algebras,then we have a presentation h X | S ( − ) i in the class of dialgebras, where S ( − ) is theset of polynomials obtained from S by changing the brackets as[ x, y ] = x ⊣ y − y ⊢ x. Indeed, for any Leibniz algebra L , there exists a unique universal enveloping di-algebra U ( L ). The next theorem can be considered as one of the applications ofHNN-extensions of Leibniz algebras. Theorem 2.3. [6]
Every Leibniz algebra embeds into its HNN-extension.
CHIA ZARGEH
The proof of the above theorem is based on the validity of Poincare-Birkhof-Witt theorem for Leibniz algebras which justifies the relation between constructionof HNN-extension for dialgebras and HNN-extensions for Leibniz algebras. For anextensive proof see [6]. 3.
Embedding theorem
In this section we provide an embedding theorem. To this end, we use thefollowing lemma and theorem proved by Shahryari in [10]. In fact, both lemma andtheorem can be considered for an arbitrary non-associative algebra.
Lemma 3.1. [10] . Let V be an inductive class of algebras over a field K . Suppose V is closed under subalgebra and L ∈ V . Then there exists an algebra H ∈ V containing L such that its dimension is at most max {ℵ , dimL, | K |} . Further, for any system S of equations and in-equations over L , there exists theeither of the following assertions: S has a solution in H For any extension H ⊂ E ∈ V , the system S has no solution in E .Proof. It is assumed that X is a countable set of variables and η = max {ℵ , dimL, | K |} . Any equation over L consists of finitely many elements of L and X . The number ofsystem of equations and in-equations over L is | L ∪ K | and denoted by κ . Note that | L | = max { dimL, | K |} , hence κ = | L | + ℵ = η . Let us consider a well-ordering inthe set of all systems as { S α } α , using ordinals 0 ≤ α ≤ κ . Suppose L = L .For any 0 ≤ γ ≤ α , the algebra L γ ∈ V is defined in such a way that | L γ | ≤ κ and β ≤ γ ⇒ L β ⊂ L γ . We put E α = [ γ ≤ α L γ , so E α ∈ V and, further, | E α | ≤ α | L γ | ≤ κ = κ . Suppose that S α has no solutionin any extension of E α . Then we set L α = E α . If there exists an extension E α ⊂ E ∈ V such that S α has a solution ( u , . . . , u n ) in E , then we set L α = h E α , u , . . . , u n i ⊂ E . Since V is closed under subalgebra, it follows that L α ∈ V and we have | L α | = | E α | ≤ κ. Now, we define H = [ ≤ α ≤ κ L α which is an element of V . We have | H | ≤ κ = κ , and hence max { dimH, | K |} ≤ max {ℵ , dimL, | K |} , therefore, we have dimH ≤ max {ℵ , dimL, | K |} . (cid:3) Theorem 3.2. [10]
Let V be an inductive class of algebras over field K . Suppose V is closed under subalgebra and L ∈ V . Then there exists an algebra L ∗ ∈ V withthe following properties, XISTENTIALLY CLOSED LEIBNIZ ALGEBRAS AND AN EMBEDDING THEOREM 5 L is a subalgebra of L ∗ . L ∗ is existentially closed in the class V . dimL ∗ ≤ {ℵ , dimL, | K |} . Proof.
Let H = L and H = H be an algebra satisfying requirements of theprevious lemma. Suppose H m is defined and let H m +1 be an algebra obtained bythe lemma from H m . Then dimH m +1 ≤ max {ℵ , dimH m , | K |} = max {ℵ , dimL, | K |} . Now, put L ∗ = [ m H m . Therefore, L ∗ is an algebra which has the properties (1)-(3). (cid:3) We recall the notion of biderivation of Leibniz algebras which has already beenintroduced in [2].
Definition 3.3.
Let L be a Leibniz algebra. A biderivation of L is a pair ( d, D )of K -linear maps d, D : L → L such that(3.1) d ([ l, l ′ ]) = [ d ( l ) , l ′ ] + [ l, d ( l ′ )] , (3.2) D ([ l, l ′ ]) = [ D ( l ) , l ′ ] − [ D ( l ) ′ , l ] , (3.3) [ l, d ( l ′ )] = [ l, D ( l ′ )]for all l, l ′ ∈ L. The set of all biderivations of L is denoted by Bider ( L ) which is a Leibniz algebrawith the Leibniz bracket given by[( d , D ) , ( d , D )] = ( d d − d d , D d − d D ) . As a quick example, let l ∈ L , then the pair ( ad ( l ) , Ad ( l )) with ad ( l )( l ′ ) = − [ l ′ , l ]and Ad ( l )( l ′ ) = [ l, l ′ ] for all l ′ ∈ L , is a biderivation and ( ad ( l ) , Ad ( l )) is called inner biderivation of L . We use this concept during the proof of the next theorem.On the basis of the properties of HNN-extensions of Leibniz algebras, we can providethe following embedding theorem. The proof of the theorem is similar to the caseof Lie algebras. Theorem 3.4.
Let L be a Leibniz algebra over field K . Then there exists a Leibnizalgebra L ∗ having the following properties: L is a subalgebra of L ∗ . For any nonzero a, b, b ′ ∈ L ∗ , there exists x, y ∈ L ∗ such that [ x, a ] = b and [ a, y ] = b ′ , and so L ∗ is solvable. dimL ∗ ≤ max {ℵ , dimL, | K |} . L ∗ is not finitely generated. Every finite-dimensional simple Leibniz algebra over field K embeds in L ∗ . If K is finite and A is finite-dimensional Leibniz algebra over K , then wehave Bider ( A ) ∼ = N L ∗ ( A ) C L ∗ ( A ) CHIA ZARGEH Proof.
Let V be the class of all Leibniz algebras. Theorem 3.4 implies that thereexists an existentially closed Leibniz algebra L ∗ containing L such that dimL ∗ ≤ max {ℵ , dimL, | K |} . Let 0 = a, b ∈ L ∗ . Let d and d ′ : h a i → L be a derivation and an anti-derivation,respectively, and d ( a ) = b and d ′ ( a ) = b ′ . Let consider HNN-extensions 2.1 and 2.2of Leibniz algbera L . Then the embeddability theorem 2.3 implies that L embedsin both HNN-extensions. Therefore, the equations [ x, a ] = b and [ a, x ] = b ′ havesolutions in L ∗ d and L ∗ d ′ , respectively, so 2 is proved.Let suppose x , . . . , x n be a finite set of elements of Leibniz algebra L ∗ and considerthe following systems of equations[ x, x i ] = 0 , [ x i , x ] = 0 , where 1 ≤ i ≤ n, x = 0 . These systems have solutions in the Leibniz algebra L ∗ × h x i , and so we have C L ∗ ( h x , . . . , x n i ) = 0. Therefore, L ∗ is not finitelygenerated.Suppose H is a finite-dimensional simple Leibniz algebra with basis u , . . . , u n with[ u i , u j ] = P r λ rij u r . Let consider the system[ x i , x j ] = X r λ rij x r for 1 ≤ i, j ≤ n , x i = 0 where 1 ≤ i ≤ n . This system has a solution in L ∗ × H andso there is a nonzero homomorphism H → L ∗ and H embeds in L ∗ .To prove 6, let K be finite and A be finite-dimensional subalgebra of L ∗ . Let d and d ′ be derivation and anti-derivation maps, respectively. Let consider HNN-extensions corresponding to both derivation and anti-derivation L ∗ d := h L, t : d ( a ) = [ a, t ] , a ∈ A i , and L ∗ d ′ := h L, t : d ′ ( a ) = [ t, a ] , a ∈ A i , in which the system [ a, x ] = d ( a ) and [ y, a ] = d ′ ( a ) have solutions. Therefore, thereare x, y ∈ L ∗ such that d ( a ) = [ a, x ] and d ′ ( a ) = [ y, a ] for all a ∈ A , so x is inthe left normalizer and y is in the right normalizer. Let N L ∗ ( A ) be the normalizerof A which is the intersection of left and right normalizer. Therefore, there is anepimorphism N L ∗ ( A ) → Bider ( L ) with the kernel C L ∗ ( A ) and we have Bider ( A ) ∼ = N L ∗ ( A ) C L ∗ ( A ) . (cid:3) References
1. A. Bloh, A generalization of the concept of a Lie algebra.
Sov. Math. Dokl.
Theory and Applications of Categories ,
33 (2) , (2018) 23–42.3. G. Higman, B. H. Neumann, H. Neumann, Embedding theorems for groups,
J. London. Math.Soc , , (1949), 247-254. MR: 11:322d.4. G. Higman, E. L. Scott, Existentially closed groups, Clarendon Press , (1988).5. P. S. Kolesnikov, L. G. Makar-Limanov, I. P. Shestakov, The Freiheitssatz for Generic PoissonAlgebras,
SIGMA (2014) 115–130. XISTENTIALLY CLOSED LEIBNIZ ALGEBRAS AND AN EMBEDDING THEOREM 7
6. M. Ladra, M. Shahryari, C. Zargeh, HNN-extensions of Leibniz algebras,
Journal of Algebra ,
532 (15) , (2019), 183–200.7. J.,-L, Loday, Une Version non commutative des algebras de Lie: les algebras de Leibniz,
Enseign. Math. , , (1993), 269-293.8. R. C. Lyndon, P. E. Combinatorial group theory, Springer-Verlag, (2001).9. W. R. Scott, Algeberaically closed groups,
Proc. of AMS, (1951) 118–121.10. M. Shahryari, Existentially closed structures and some embedding theorems, MathematicalNotes
6, (2017), 1023–1032.
Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, S˜ao Paulo, Brazil.
Email address ::