aa r X i v : . [ m a t h . R A ] J un GR ¨OBNER—SHIRSHOV BASES FOR REPLICATED ALGEBRAS
P.S. KOLESNIKOV
Abstract.
We establish a universal approach to solution of the word prob-lem in the varieties of di- and tri-algebras. This approach, for example, allowsto apply Gr¨obner—Shirshov bases method for Lie algebras to solve the idealmembership problem in free Leibniz algebras (Lie di-algebras). As another ap-plication, we prove an analogue of the Poincar´e—Birkhoff—Witt Theorem foruniversal enveloping associative tri-algebra of a Lie tri-algebra (CTD ! -algebra). Introduction
Gr¨obner bases theory is known as an effective computational technique in com-mutative algebra and related areas. Various questions in mathematics may bereduced to the ideal membership problem (or word problem ): given a set S of (com-mutative) polynomials, whether a given polynomial f belongs to the ideal generatedby S . In non-commutative (or even non-associative) settings, the same problem ap-pears mainly in theoretical studies rather than in computational context, see thereview [6]. The classical example is given by the Poincar´e—Birkhoff—Witt Theo-rem for Lie algebras and its analogues for other classes of algebras (see [18]).It worths mentioning that Gr¨obner bases in commutative algebras [7] appearedsimultaneously with standard bases (now called Gr¨obner—Shirshov bases, GSB) inLie algebras [19]. In the last years, GSB theories have been established for variousclasses of non-associative algebras, including associative di-algebras [5, 21]. Thelatter were introduced in [16] as “envelopes” of Leibniz algebras.Let us sketch what is a GSB theory for a given variety Var of (linear) algebrasover a field k . It usually includes the following components. • Description of the free algebra Var h X i generated by a set X . Elements ofVar h X i are called polynomials , they are linear combinations of monomials (normal words) that form a linear basis of Var h X i . • Linear order on monomials compatible with algebraic operations. • Elimination procedure of the leading word ¯ f of a polynomial f in a mono-mial u . • Definition of compositions and the notion of triviality of a compositionmodulo a given set of polynomials. • Composition-Diamond Lemma.If S is a set of monic polynomials (principal word appears with identity coefficient)such that every composition of its polynomials is trivial modulo S then S is saidto be a GSB in Var h X i . Principal point of GSB theory, the Composition-DiamondLemma (CD-Lemma), usually has the following form: S is a GSB if and only if for Mathematics Subject Classification.
Key words and phrases. di-algebra, tri-algebra, Gr¨obner—Shirshov basis.The research is supported by RSF (project 14-21-00065). every element f in the ideal generated by S there exists g ∈ S such that ¯ f admitselimination of ¯ g . In particular, if S is a GSB in Var h X i then the images of S -reduced monomials (those that do not admit elimination of ¯ g , g ∈ S ) form a linear basis ofthe quotient algebra Var h X | S i generated by X with defining relations S .In this paper, we propose a general approach to the GSB theory for a class ofvarieties obtained by replication procedures . The latter, from the categorical pointof view, may be easily explained in terms of operads. If Var is the variety governedby an operad P (see [9]) then replicated varieties di- Var and tri- Var are governedby Hadamard the products Perm ⊗P and ComTriAs ⊗P , respectively. Here Permand ComTriAs are the operads corresponding to the varieties of Perm-algebras andcommutative tri-algebras introduced in [8] and [20], respectively (see [11] for moredetails).In particular, for the varieties As and Lie of associative and Lie algebras, di- Asand di- Lie coincide with the varieties of associative di-algebras and Leibniz algebras,respectively. Leibniz algebras are known as the most common “non-commutative”generalization of Lie algebras. Note that for associative di-algebras GSB theoryhas been constructed in [5] and [21]. However, there is no GSB theory for Leibnizalgebras as well as for tri-algebras (i.e., systems from tri- Var). This paper aims atfilling this gap.Suppose GSB theory is known for a variety Var. Then we explicitly constructa “Var-envelope” for free algebras di- Var h X i and tri- Var h X i such that solution ofthe ideal membership problem in this Var-envelope induces a solution of the sameproblem in di- Var h X i or tri- Var h X i . This approach differs from the usual onedescribed above, but it is easier in applications. We will apply the technique devel-oped to study universal enveloping Lie tri-algebras of Lie algebras and associativeenveloping tri-algebras of Lie tri-algebras.2. Replication of varieties
Let (Σ , ν ) be a language , i.e., a set of operations together with arity function ν : Σ → Z + . A Σ -algebra is a linear space A equipped with polylinear operations f : A ⊗ n → A , f ∈ Σ, n = ν ( f ). Denote by Alg = Alg Σ the class of all Σ-algebras,and let Alg h X i stand for the free Σ-algebra generated by a set X . (For example, ifΣ consists of one binary operation then Alg h X i is the magmatic algebra.)For a given language (Σ , ν ), define two replicated languages (Σ (2) , ν (2) ) and(Σ (3) , ν (3) ) as follows:Σ (2) = { f i | f ∈ Σ , i = 1 , . . . , ν ( f ) } , ν (2) ( f i ) = ν ( f );Σ (3) = { f H | f ∈ Σ , ∅ = H ⊆ { , . . . , ν ( f ) }} , ν (3) ( f H ) = ν ( f ) . Denote Alg (2) = Alg Σ (2) and Alg (3) = Alg Σ (3) .Note that Σ (2) may be considered as a subset of Σ (3) via f i = f { i } . This is whywe will later deal with Σ (3) assuming the same statements (in a more simple form)hold for Σ (2) .Suppose an element Φ ∈ Alg h X i is polylinear with respect to x , . . . , x n ∈ X .Then for every nonempty subset H ⊂ { , . . . , n } we may define Φ H ∈ Alg (3) h X i in the following way (see [11] for more details). Every monomial summand of Φmay be naturally considered as a rooted tree whose leaves are labelled by variables x , . . . , x n and nodes are labelled by symbols of operations from Σ. Every node R ¨OBNER—SHIRSHOV BASES FOR REPLICATED ALGEBRAS 3 labelled by f ∈ Σ has one “input” and ν ( f ) “outputs”, each output is attached toa subtree. Let us emphasize leaves x i for i ∈ H and change the labels of nodesby the following rule. For every node labelled by a symbol f ∈ Σ consider thesubset S of { , . . . , ν ( f ) } which consists of those output numbers that are attachedto subtrees containing emphasized leaves. If S is nonempty then replace f with f S ,if S is empty then replace f with f { } . Transforming every polylinear monomial ofΦ in this way, we obtain Φ H . (For Σ (2) , it is enough to consider | H | = 1.)Suppose Var is a variety of Σ-algebras defined by a collection of polylinear iden-tities S (Var) ⊂ Alg h X i , X = { x , x , . . . } . The following statement may be inter-preted as a definition of what is the variety tri- Var. Theorem 2.1 ([11]) . A Σ (3) -algebra belongs to the class tri- Var if and only if itsatisfies the following identities in Alg (3) h X i : f H ( x , . . . , x i − , g S ( x i , . . . , x i + m − ) , x i + m , . . . , x n + m − )= f H ( x , . . . , x i − , g Q ( x i , . . . , x i + m − ) , x i + m , . . . , x n + m − ) , (1) where f, g ∈ Σ , ν ( f ) = n , ν ( g ) = m , H ⊆ { , . . . , n } , S, Q ⊆ { , . . . , m } , i / ∈ H ,and Φ H ( x , . . . , x n ) = 0 , (2) where Φ ∈ S (Var) , deg Φ = n , H ⊆ { , . . . , n } . Example 2.1.
Let Σ consist of one binary product µ ( a, b ) = [ ab ]. Consider Var =Lie (char k = 2) with defining identities[ x x ] + [ x x ] = 0 , [[ x x ] x ] + [[ x x ] x ] + [[ x x ] x ] = 0 . Then Σ (3) consists of three binary operations µ { } ( a, b ) = [ a ⊣ b ] , µ { } ( a, b ) = [ a ⊢ b ] , µ { , } ( a, b ) = [ a ⊥ b ] . The family of defining identities of the variety tri- Lie of
Lie tri-agebras containsreplicated skew-symmetry[ x ⊢ x ] + [ x ⊣ x ] = 0 , [ x ⊥ x ] + [ x ⊥ x ] = 0 . This allows replace [ a ⊣ b ] with − [ b ⊢ a ] and express all defining relations in termsof [ · ⊢ · ] and [ · ⊥ · ]. It is easy to see that (1) turn into[[ x ⊢ x ] ⊢ x ] = − [[ x ⊢ x ] ⊢ x ] , (3)[[ x ⊥ x ] ⊢ x ] = [[ x ⊢ x ] ⊢ x ] , (4)and the replication of Jacobi identity leads (up to equivalence) to the followingthree relations:[[ x ⊢ x ] ⊢ x ] − [ x ⊢ [ x ⊢ x ]] + [ x ⊢ [ x ⊢ x ]] = 0 , (5)[[ x ⊢ x ] ⊥ x ] − [ x ⊢ [ x ⊥ x ]] − [[ x ⊢ x ] ⊥ x ] = 0 , (6)[[ x ⊥ x ] ⊥ x ] + [[ x ⊥ x ] ⊥ x ] + [[ x ⊥ x ] ⊥ x ] = 0 . (7)Note that (5) is the left Leibniz identity, (3) easily follows from (5). Therefore, aLie tri-algebra may be considered as a linear space with two operations [ · ⊢ · ] and[ · ⊥ · ] such that the first one satisfies the left Leibniz identity, the second one isLie, and (4), (6) hold. P. S. KOLESNIKOV
Remark 2.1.
The operad governing the variety tri- Lie is Koszul dual to the operadCTD governing the variety of commutative tridendriform algebras introduced in[15]. This is a particular case of a general relation between di- or tri- algebras andtheir dendriform counterparts [10]. In [22], tri- Lie is stated as CTD ! .In a similar way, one may construct the defining identities of the variety tri- As:the latter coincides with the variety of triassociative algebras introduced in [17]. Example 2.2.
Let C be the 2-dimensional space with a basis { e , e } equippedwith binary operations e i ⊥ e i = e i , e ⊢ e = e ⊣ e = e , e ⊢ e = e ⊣ e = e , other products are zero. It is easy to check that C ∈ tri- Com, where Com is thevariety of associative and commutative algebras.3. Construction of free di- and tri-algebras
In this section, we present simple construction of the free algebra tri- Var h X i generated by a given set X in the variety tri- Var. The same construction works fordi- Var after obvious simplifications. To make the results more readable, we restrictto the case when Σ consists of one binary product since this is the most commoncase in practice. However, there are no obstacles to the transfer of the followingconsiderations to an arbitrary language.Given a set X , denote by ˙ X the copy of X :˙ X = { ˙ x | x ∈ X } . Denote by F the free algebra Var h X ∪ ˙ X i in the variety Var. There exists uniquehomomorphism ϕ : F → Var h X i ⊂ F determined by x x , ˙ x x , x ∈ X .Let us define three binary operations on the space F as follows: f ⊢ g = ϕ ( f ) g, f ⊣ g = f ϕ ( g ) , f ⊥ g = f g, (8) f, g ∈ F . Denote the system obtained by F (3) . Lemma 3.1.
Algebra F (3) belongs to tri- Var .Proof. Obviously, ϕ = ϕ , so ϕ ( ϕ ( f ) g ) = ϕ ( f ϕ ( g )) = ϕ ( f ) ϕ ( g ) = ϕ ( f g ) , f, g ∈ F. (9)Relation (9) means that ϕ is a homomorphic averaging operator on F . It is wellknown (see [11, Theorem 2.13]) that in this case an algebra from Var relative tooperations (8) belongs to tri- Var. (cid:3) For a monomial u in F , denote by deg ˙ X u the degree of u with respect to variablesfrom ˙ X . Lemma 3.2.
The subalgebra V of F (3) generated by ˙ X coincides with the subspace W of F spanned by all monomials u such that deg ˙ X u > .Proof. It is clear that V ⊆ W . Indeed, consider u ⊢ v , u ⊣ v , u ⊥ v for u, v ∈ V .Inductive arguments allow to assume u, v ∈ W and thus all three products alsobelong to W by the definition of F (3) . The converse embedding W ⊆ V is provedanalogously. (cid:3) It is easy to see that the space V from Lemma 3.2 coincides with the ideal of F generated by ˙ X . R ¨OBNER—SHIRSHOV BASES FOR REPLICATED ALGEBRAS 5
Theorem 3.1.
The subalgebra V of F (3) is isomorphic to the free algebra in thevariety tri- Var generated by X .Proof. It is enough to prove universal property of V in the class tri- Var. Suppose A is an arbitrary tri-algebra in tri- Var, and let α : X → A be an arbitrary map.Our aim is to construct a homomorphism of tri-algebras χ : V → A such that χ ( ˙ x ) = α ( x ) for all x ∈ X .Recall the following construction (proposed in [10]). The subspace A = span { a ⊢ b − a ⊣ b, a ⊢ b − a ⊥ b | a, b ∈ A } is an ideal of A . The quotient ¯ A = A/A carries a natural structure of an algebrafrom Var given by ¯ a ¯ b = a ⊥ b . Consider the formal direct sum ˆ A = ¯ A ⊕ A equippedwith one well-defined product¯ ab = a ⊢ b, a ¯ b = a ⊣ b, ¯ a ¯ b = a ⊥ b, ab = a ⊥ b, for ¯ a, ¯ b ∈ ¯ A , where ¯ c = c + A ∈ ¯ A , c ∈ A . Then ˆ A ∈ Var.Another important fact on Var and tri- Var was established in [11]. For everycommutative tri-algebra C ∈ tri- Com and for every algebra B ∈ Var the linearspace C ⊗ B equipped with( p ⊗ a ) ∗ ( q ⊗ b ) = ( p ∗ q ) ⊗ ab, ∗ ∈ {⊢ , ⊣ , ⊥} , a, b ∈ B, p, q ∈ C, is a tri-algebra in the variety tri- Var. There is a natural relation between A ∈ tri- Var and ˆ A ∈ Var. For the tri-algebra C from Example 2.2 we have an embed-ding of tri-algebras ι : A → C ⊗ ˆ A given by ι : a e ⊗ ¯ a + e ⊗ a, a ∈ A. Now, construct ˆ α : X ∪ ˙ X → ˆ A asˆ α ( x ) = αx ∈ ¯ A, ˆ α ( ˙ x ) = α ( x ) ∈ A, for x ∈ X . The map ˆ α induces a homomorphism of algebras ˆ ψ : F → ˆ A . Finally,define ψ : F → C ⊗ ˆ A by ψ ( f ) = e ⊗ ˆ ψ ( ϕ ( f )) + e ⊗ ˆ ψ ( f ) , f ∈ F, (10)Let us show that ψ is a homomorphism of tri-algebras. For every f, g ∈ F , ∗ ∈{⊢ , ⊣ , ⊥} ψ ( f ) ∗ ψ ( g ) = ( e ⊗ ˆ ψ ( ϕ ( f )) + e ⊗ ˆ ψ ( f )) ∗ ( e ⊗ ˆ ψ ( ϕ ( g )) + e ⊗ ˆ ψ ( g ))= ( e ∗ e ) ⊗ ˆ ψ ( ϕ ( f ) ϕ ( g ))+( e ∗ e ) ⊗ ˆ ψ ( ϕ ( f ) g )+( e ∗ e ) ⊗ ˆ ψ ( f ϕ ( g ))+( e ∗ e ) ⊗ ˆ ψ ( f g )= e ⊗ ˆ ψ ( ϕ ( f g )) + e ⊗ ˆ ψ ( ϕ ( f ) g ) , ∗ = ⊢ ,e ⊗ ˆ ψ ( ϕ ( f g )) + e ⊗ ˆ ψ ( f ϕ ( g )) , ∗ = ⊣ ,e ⊗ ˆ ψ ( ϕ ( f g )) + e ⊗ ˆ ψ ( f g ) , ∗ = ⊥ . (11)On the other hand, it is straightforward to compute ψ ( f ∗ g ). Since ϕ is a homo-morphic averaging operator, the results coincide with those in (11).It is easy to see from the definition that ψ ( ˙ x ) = ι ( α ( x )). Therefore, ψ ( V ) ⊆ ι ( A ) ⊆ C ⊗ ˆ A by Lemma 3.2. Finally, the desired homomorphism χ may beconstructed as χ = ι − ◦ ψ | V : V → A. P. S. KOLESNIKOV
In other words, the diagram˙ X ⊆ −−−−→ V ⊆ −−−−→ F (3) ϕ y y χ y ψ X α −−−−→ A ι −−−−→ C ⊗ ˆ A is commuting. (cid:3) Remark 3.1.
All constructions of this section make sense for di-algebras. It isenough to consider only operations ⊢ and ⊣ on F given by the same rules. The roleof V ⊆ F is played by the subspace of polynomials linear in ˙ X . Example 3.1.
Let Var = Lie, di- Lie is the variety of Leibniz algebras. Thendi- Lie h X i ≃ V ⊆ Lie h X ∪ ˙ X i . Suppose X is linearly ordered; let us extend the order to X ∪ ˙ X in the natural way: x > y ⇒ ˙ x > ˙ y, ˙ x > y for all x, y ∈ X . It is easy to see that all words of the form[ . . . [[ ˙ x x ] x ] . . . x n ](with left-justified bracketing) are linearly independent in F since so are their im-ages in U ( F ) ≃ As h X ∪ ˙ X i . These words correspond to[ . . . [[ x ⊣ x ] ⊣ x ] ⊣ · · · ⊣ x n ] ∈ di- Lie h X i , (12)where [ · ⊣ · ] satisfies (right) Leibniz identity[ x ⊣ [ y ⊣ z ]] = [[ x ⊣ y ] ⊣ z ] − [[ x ⊣ z ] ⊣ y ] . Obviously, every Leibniz polynomial may be rewritten as a linear combination ofmonomials (12), so the latter form a linear basis of di- Lie h X i [13, 14].4. Ideal membership problem
As above, let Var be a variety of algebras (with one binary product) defined bypolylinear identities, di- Var and tri- Var are the corresponding varieties of di- andtri-algebras.Suppose X is a nonempty set of generators. By Theorem 3.1 and Remark 3.1,free systems tri- Var h X i and di- Var h X i may be considered as subspaces of F =Var h X ∪ ˙ X i . As in Section 3, let us denote these subspaces by V . We will considerthe case of tri-algebras in details.For every S ⊆ V ⊆ F (3) denote by ( S ) (3) the ideal of V generated by S . Everytri-algebra A ∈ tri- Var may be presented by generators and defining relations as A ≃ tri- Var h X | S i ≃ V / ( S ) (3) for appropriate X and S . In order to understandthe structure of A we have to know how to decide whether a given f ∈ V belongsto ( S ) (3) . This kind of problems is the main target of the Gr¨obner—Shirshov basis(GSB) method. In order to translate GSB theory from the class Var to tri- Var(and to di- Var) we need the following Theorem 4.1.
Let S ⊂ V ⊂ F (3) . Then ( S ) (3) = ( S ∪ ϕ ( S )) ∩ V, where ( P ) stands for the ideal of F generated by its subset P . R ¨OBNER—SHIRSHOV BASES FOR REPLICATED ALGEBRAS 7
Here ϕ is the endomorphism of F defined in Section 3. Proof.
Denote I = ( S ∪ ϕ ( S )). Obviously, I = [ s ≥ I s , I ⊆ I ⊆ . . . , where I = span ( S ∪ ϕ ( S )), I s +1 = I s + F I s + I s F, s ≥ . Similarly, for J = ( S ) (3) we have J = [ s ≥ J s , J ⊆ J ⊆ . . . , where J = span ( S ), J s +1 = J s + V ⊢ J s + J s ⊣ V + V ⊣ J s + J s ⊢ V + J s ⊥ V + V ⊥ J s , s ≥ . Since S ⊂ V , ϕ ( S ) ⊂ Var h X i , and Var h X i ∩ V = 0, we have I ∩ V = J .Moreover, I = J + I ′ , where I ′ = span ϕ ( S ) = ϕ ( J ) ⊆ Var h X i .Assume I s = J s + ϕ ( J s ) for some s ≥
0. Note that F = V + Var h X i andVar h X i = ϕ ( V ) by Lemma 3.2. Then I s +1 = I s + ( V + Var h X i ) I s + I s ( V + Var h X i )= J s + V J s + V ϕ ( J s ) + Var h X i J s + J s V + ϕ ( J s ) V + J s Var h X i + ϕ ( J s ) + ϕ ( J s ) Var h X i + Var h X i ϕ ( J s ) . It remains to note that
V J s = V ⊥ J s , V ϕ ( J s ) = V ⊣ J s , Var h X i J s = V ⊣ J s ,J s V = J s ⊥ V, ϕ ( J s ) V = J s ⊣ V, J s Var h X i = J s ⊣ V. Hence, I s +1 = J s +1 + ϕ ( J s ) + ϕ ( J s ) Var h X i + Var h X i ϕ ( J s ) , but the latter three summands obviously give ϕ ( J s +1 ).We have proved I s = J s + ϕ ( J s ) for all s ≥
0. Therefore, I s ∩ V = J s , and I ∩ V = J , as required. (cid:3) Remark 4.1.
For di-algebras, the statement of Theorem 4.1 holds true: the inter-section of V ≃ di- Var h X i with the ideal generated by S ∪ ϕ ( S ) in F is equal to theideal generated by S in the di-algebra V .Let S ⊂ tri- Var h X i . Then I = ( S ∪ ϕ ( S )) is a ϕ -invariant ideal of F , so onemay induce tri-algebra structure on F/I . Corollary 4.1. tri- Var h X | S i is isomorphic to the subalgebra of ( F/I ) (3) gener-ated by ˙ X . Similar statement holds for di-algebras. Theorem 4.1 and Remark 4.1 provide an easy approach to GSB theory for theclasses of tri- and di-algebras (tri- Var and di- Var, respectively) modulo the analo-gous theory for the variety Var. In order to find Gr¨obner—Shirshov basis of an idealgenerated by S ⊆ tri- Var h X i one should rewrite the relations from S as elementsof F = Var h X ∪ ˙ X i , and find the GSB ˆ S of the ideal in F generated by S ∪ ϕ ( S ).(In the case of di-algebras, it is enough to find a part of the latter GSB, namely, P. S. KOLESNIKOV those polynomials of degree ≤ X .) To find a linear basis of tri- Lie h X | S i oneshould just consider ˆ S -reduced monomials in F and choose those that belong to V .In order to present an example, let us recall the main features of the Gr¨obner—Shirshov bases theory for Lie algebras [19] (see also [3]).First, suppose X is a linearly ordered set of generators, X ∗ is the set of all asso-ciative words in X equipped with deg-lex order , i.e., two words are first comparedby their length and then lexicographically. The set of associative Lyndon—Shirshov words (LS-words) consists of all such words u that for every presentation u = vw , v, w ∈ X ∗ , we have u > wv . Every associative LS-word u has a unique standard bracketing [ u ] such that [ u ] = [[ v ][ w ]], where w is the longest proper LS-suffixof u . An associative LS-word with standard bracketing is called a non-associativeLS-word; linear order on such words is induced by the deg-lex order on X ∗ .The set of all non-associative LS-words in the alphabet X is a linear basis ofthe free Lie algebra Lie h X i . Given 0 = f ∈ Lie h X i , ¯ f denotes its principal non-associative LS-word.Next, recall the notion of a composition. For every associative LS-words w and v such that w = uvu ′ for some u and u ′ (they may be empty) there exists uniquebracketing { u ∗ u ′ } on the word u ∗ u ′ in the alphabet X ∪ {∗} such that { u [ v ] u ′ } = [ w ] ∈ Lie h X i . Suppose f and g are to monic elements from Lie h X i , ¯ f = [ w ], ¯ g = [ v ]. If w = uvu ′ as above, then we say that f and g have a composition of inclusion ( f, g ) w = f − { ugu ′ } . If w = uu ′ and v = u ′ u ′′ for some u, u ′ , u ′′ ∈ X ∗ then we say that f and g have a composition of intersection ( f, g ) uu ′ u ′′ = { f u ′′ } − { ug } . Finally, recall the definition of a Gr¨obner—Shirshov basis (GSB) for Lie algebras.A set of monic elements S ⊂ Lie h X i is said to be a GSB in Lie h X i if for every f, g ∈ S every their composition ( f, g ) w is trivial , i.e., may be presented as( f, g ) w = X i α i { u i s i u ′ i } , s i ∈ S, where { u i s i u ′ i } = [ u i ¯ s i u ′ i ] < [ w ].A non-associative LS-word [ w ] is said to be S -reduced if w may not be presentedas w = uvu ′ , where [ v ] = ¯ s for some s ∈ S . Theorem 4.2 (CD-Lemma, [3, 19]) . For a set of monic elements S ⊆ Lie h X i denote ( S ) the ideal generated by S . Then the following conditions are equivalent: (1) S is a GSB in Lie h X i ; (2) f ∈ ( S ) implies ¯ f is not S -reduced; (3) the images of S -reduced words form a linear basis of Lie h X i / ( S ) . For associative algebras, one may just “erase brackets” in all definitions andstatements [2, 4].
Example 4.1.
Let X = { x, y } . Consider the Leibniz algebra L ∈ di- Lie generatedby X with one defining relation f = [ x ⊣ y ] + [ y ⊣ x ] + y. R ¨OBNER—SHIRSHOV BASES FOR REPLICATED ALGEBRAS 9
Define the order on X ∪ ˙ X by ˙ x > ˙ y > x > y . According to the general scheme,denote F = Lie h X ∪ ˙ X i . Then f may be interpreted as [ ˙ xy ] + [ ˙ yx ] + ˙ y ∈ F , and ϕ ( f ) = y . Obviously, f and ϕ ( f ) have a composition of inclusion ( f, ϕ ( f )) [ ˙ xy ] =[ ˙ yx ] + ˙ y , and the latter has no more compositions with ϕ ( f ). Therefore, y, [ ˙ yx ] + ˙ y is a GSB in F . The linear basis of di- Lie h x, y | f i consists of all those non-associative Lyndon—Shirshov words in Lie h ˙ x, ˙ y, x, y i that are linear in ˙ X that donot contain y or ˙ yx as (associative) subwords. Obviously, these are˙ y, [ . . . [[ ˙ xx ] x ] . . . x ] . Hence, the linear basis of L consists of y, [ . . . [[ x ⊣ x ] ⊣ x ] ⊣ · · · ⊣ x ] . Applications
Gr¨obner bases in commutative algebra are known as an efficient tool for solvingcomputational problems. In non-commutative (and non-associative) settings, GSBtechnique is mainly used for solving theoretical problems. A wide family of suchproblems is related with the structure of universal envelopes.Namely, suppose Var and Var are two varieties of algebras, and let ω be afunctor from Var to Var which turns A ∈ Var into A ( ω ) ∈ Var , where A ( ω ) isthe same linear space as A equipped with algebraic operations expressed in termsof operations in Var . (In terms of operads, this exactly means that the functor ω is induced by a morphism of corresponding operads, see, e.g., [9]. For example, thewell-known functor − : As → Lie turns an associative algebra A into Lie algebra A ( − ) with new operation [ ab ] = ab − ba , a, b ∈ A .) Then for every B ∈ Var thereexists unique (up to isomorphism) universal enveloping algebra U ω ( B ) ∈ Var .The most natural way to construct U ω ( B ) is to consider a linear basis X of B andexpress the multiplication table of B as a system S of defining relations in Var h X i .Then U ω ( B ) ≃ Var h X | S i . In this section, we consider two such functors on thevarieties tri- Lie and tri- As, and determine the structure of corresponding universalenvelopes: • forgetful functor ⊥ : tri- Lie → Lie, [ ab ] = [ a ⊥ b ]; • tri-commutator functor − : tri- As → tri- Lie, where[ a ⊢ b ] = a ⊢ b − b ⊣ a, [ a ⊣ b ] = a ⊣ b − b ⊢ a, [ a ⊥ b ] = a ⊥ b − b ⊥ a. (13)Let L be a Lie algebra with linear basis X and multiplication table µ : X × X → k X , µ ( x, y ) is a linear form in X for every x, y ∈ X . Assume X is linearly ordered.Then U ⊥ ( L ) ≃ tri- Lie h X | S i , where S = { [ x ⊥ y ] − µ ( x, y ) | x, y ∈ X, x > y } .According to the scheme described in Section 4, we have to consider U = Lie h X ∪ ˙ X | [ ˙ x ˙ y ] − ˙ µ ( x, y ) , [ xy ] − µ ( x, y ) , x, y ∈ X, x > y i , where ˙ µ ( x, y ) = P i α i ˙ z i for µ ( x, y ) = P i α i z i .Note that polynomials from S = { [ ˙ x ˙ y ] − ˙ µ ( x, y ) | x, y ∈ X, x > y } do not havecompositions, so S is a GSB. The same applies to ϕ ( S ) = { [ xy ] − µ ( x, y ) | x, y ∈ X, x > y } . Moreover, polynomials from S and ϕ ( S ) have no compositions sincethey depend on different variables. Hence, S ∪ ϕ ( S ) is a GSB and U is isomorphicto the free product ˙ L ∗ L [19], where ˙ L is the isomorphic copy of L . Corollary 4.1implies Theorem 5.1.
The universal enveloping Lie tri-algebra U ⊥ ( L ) of a given Lie al-gebra L is isomorphic as a linear space to the ideal of ˙ L ∗ L generated by ˙ L . Corollary 5.1.
The pair of varieties (tri- Lie , Lie) is a PBW-pair in the sense of [18] . Indeed, L embeds into U ⊥ ( L ) and there exists a basis of U ⊥ ( L ) which does notdepend on the particular multiplication table of L .Now, let A be an associative tri-algebra with operations ⊢ , ⊣ , and ⊥ . Then A ( − ) with new operations (13) is known to be a Lie tri-algebra. Given L ∈ tri- Lie,denote its universal enveloping associative tri-algebra by U − ( L ). The structure of U − ( L ) was studied in [11]. Let us show how to apply GSB approach to get the sameresult. A similar computation for di-algebras was performed in [5] in the frameworkof GSB theory for associative di-algebras developed in that paper. Our aim is toshow that the approach proposed in Section 4 allows to solve such problems withshorter computations.Suppose L ∈ tri- Lie, L = span { [ a ⊢ b ] − [ a ⊣ b ] , [ a ⊢ b ] − [ a ⊥ b ] | a, b ∈ L } asin Section 3, and let X be a basis of L such that X = X ∪ X , X is a basis of L .Denote by µ ⊢ , µ ⊣ , and µ ⊥ the linear forms corresponding to the operations on L .Since µ ⊢ ( x, y ) = − µ ⊣ ( y, x ) and µ ⊥ ( x, y ) = − µ ⊥ ( y, x ), we have to consider U = As h X ∪ ˙ X | S ∪ ϕ ( S ) i , where S = S ⊣ ∪ S ⊥ , S ⊣ = { ˙ xy − y ˙ x − ˙ µ ⊣ ( x, y ) | x, y ∈ X } ,S ⊥ = { ˙ x ˙ y − ˙ y ˙ x − ˙ µ ⊥ ( x, y ) | x, y ∈ X, x > y } . Note that ϕ ( S ⊣ ) = { xy − yx − µ ⊣ ( x, y ) | x, y ∈ X } = ¯ S ⊣ ∪ ¯ S ⊢ ∪ ¯ S , where ¯ S ⊣ = { xy − yx − µ ⊣ ( x, y ) | x, y ∈ X, x > y } , ¯ S ⊢ = { xy − yx + µ ⊣ ( y, x ) | x, y ∈ X, x > y } , ¯ S = { µ ⊣ ( x, x ) | x ∈ X } , and ϕ ( S ⊥ ) = { xy − yx − µ ⊥ ( x, y ) | x, y ∈ X, x > y } . Hence, the elements of ϕ ( S ) have the following compositions of inclusion: µ ⊣ ( x, y ) + µ ⊣ ( y, x ) , µ ⊣ ( x, y ) − µ ⊥ ( x, y ) , µ ⊣ ( y, x ) + µ ⊥ ( x, y ) , µ ⊣ ( x, x ) , where x > y . Obviously, the linear space of these compositions coincides with L ,so we may add letters X to the defining relations. Since µ ⊣ ( X, X ) = 0 in L and µ ⊣ ( X , X ) ⊂ L , it is enough to consider the following defining relations inAs h X ∪ ˙ X i : x, x ∈ X ; (14) xy − yx − µ ⊣ ( x, y ) , x, y ∈ X , x > y ; (15)˙ xy − y ˙ x − ˙ µ ⊣ ( x, y ) , x ∈ X, y ∈ X ; (16)˙ x ˙ y − ˙ y ˙ x − ˙ µ ⊥ ( x, y ) , x, y ∈ X, x > y. (17)Let us use the same order on X ∪ ˙ X as in Example 3.1 along with deg-lex order on( X ∪ ˙ X ) ∗ . R ¨OBNER—SHIRSHOV BASES FOR REPLICATED ALGEBRAS 11
Theorem 5.2.
Relations (14) – (17) form a GSB in As h X ∪ ˙ X i .Proof. Relations (15) correspond to the multiplication table of the Lie algebra ¯ L = L/L , so all their compositions of intersection are trivial. The same holds for (17):it corresponds to the Lie algebra L ( ⊥ ) . It remains to compute two families ofcompositions ( f, g ) w :(1) f = yz − zy − µ ⊣ ( y, z ), g = ˙ xy − y ˙ x − ˙ µ ⊣ ( x, y ), w = ˙ xyz , where y, z ∈ Z , y > z , x ∈ X ;(2) ˙ x ˙ y − ˙ y ˙ x − ˙ µ ⊥ ( x, y ), g = ˙ yz − z ˙ y − ˙ µ ⊣ ( y, z ), w = ˙ x ˙ yz , x, y ∈ X , x > y , z ∈ X .Let us compute in details the second one:( f, g ) w = f z − ˙ xg = − ˙ y ˙ xz − ˙ µ ⊥ ( x, y ) z + ˙ xz ˙ y + ˙ x ˙ µ ⊣ ( y, z ) . For h, h ′ ∈ As h X ∪ ˙ X i , let us write h ≡ h ′ if h − h ′ = P i α i u i s i u ′ i , where s i belongto (14)–(17), α i ∈ k , and u i ¯ s i u ′ i < w = ˙ x ˙ yz . Then( f, g ) w ≡ − ˙ yz ˙ x − ˙ y ˙ µ ⊣ ( x, z ) − ˙ µ ⊥ ( x, y ) z + z ˙ x ˙ y + ˙ µ ⊣ ( x, z ) ˙ y + ˙ x ˙ µ ⊣ ( y, z ) ≡ − ˙ µ ⊣ ( y, z ) ˙ x − ˙ y ˙ µ ⊣ ( x, z ) − ˙ µ ⊥ ( x, y ) z + z ˙ µ ⊥ ( x, y ) + ˙ µ ⊣ ( x, z ) ˙ y + ˙ x ˙ µ ⊣ ( y, z )= ˙ x ˙ µ ⊣ ( y, z ) − ˙ µ ⊣ ( y, z ) ˙ x + ˙ µ ⊣ ( x, z ) ˙ y − ˙ y ˙ µ ⊣ ( x, z ) + z ˙ µ ⊥ ( x, y ) − ˙ µ ⊥ ( x, y ) z ≡ ˙ µ ⊥ ( x, µ ⊣ ( y, z )) + ˙ µ ⊥ ( µ ⊣ ( x, z ) , y ) − ˙ µ ⊣ ( µ ⊥ ( x, y ) , z ) (18)The right-hand side of (18) is equal to [ x ⊥ [ y ⊣ z ]]+[[ x ⊣ z ] ⊥ y ] − [[ x ⊥ y ] ⊣ z ] ∈ L which is zero in every Lie tri-algebra. (cid:3) Corollary 5.2 ([11]) . As a linear space, U − ( L ) is isomorphic to U ( ¯ L ) ⊗ U ( L ( ⊥ ) ) ,where U ( ¯ L ) is the usual universal enveloping associative algebra of the Lie algebra ¯ L = L/L and U ( L ( ⊥ ) ) is the augmentation ideal of U ( L ( ⊥ ) ) .Proof. Linear basis of U − ( L ) consists of those associative words of degree > X that are reduced relative to (14)–(17): x . . . x k ˙ y . . . ˙ y m , x i ∈ X , y j ∈ X, where x ≤ · · · ≤ x k , k ≥ y ≤ · · · ≤ y m , m ≥ (cid:3) Remark 5.1.
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