On Pre-Novikov Algebras and Derived Zinbiel Variety

For a non-associative algebra $A$ with a derivation $d$, its derived algebra $A^{(d)}$ is the same space equipped with new operations $a\succ b = d(a)b$, $a\prec b = ad(b)$, $a,b\in A$. Given a variety ${\rm Var}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in ${\rm Var}$ and for all derivations $d$ of $A$. The same terminology is applied to binary operads governing varieties of non-associative algebras. For example, the operad of Novikov algebras is the derived one for the operad of (associative) commutative algebras. We state a sufficient condition for every algebra from a derived variety to be embeddable into an appropriate differential algebra of the corresponding variety. We also find that for ${\rm Var} = {\rm Zinb}$, the variety of Zinbiel algebras, there exist algebras from the derived variety (which coincides with the class of pre-Novikov algebras) that cannot be embedded into a Zinbiel algebra with a derivation.


Introduction
The class of nonassociative algebras with one binary operation satisfying the identities of left symmetry is known as the variety of Novikov algebras.Relations (1.1) and (1.2) emerged in [2,10] as a tool for expressing certain conditions on the components of a tensor of rank 3 considered as a collection of structure constants of a finite-dimensional algebra with one bilinear operation.In [10], it was a sufficient condition for a differential operator to be Hamiltonian; in [2] it was a condition to guarantee the Jacobi identity for a generalized Poisson bracket in the framework of Hamiltonian formalism for partial differential equations of hydrodynamic type.Novikov algebras may be obtained from (associative) commutative algebras with a derivation by means of the following operation-transforming functor (see [10]).Assume A is a commutative algebra with a multiplication * , and let d be a derivation of A, i.
In [9], it was proved that the free Novikov algebra Nov⟨X⟩ generated by a set X is a subalgebra of the free differential commutative algebra with respect to the operation (1.3).Moreover, it was shown in [4] that every Novikov algebra can be embedded into an appropriate commutative algebra with a derivation.
One may generalize the relation between commutative algebras with a derivation and Novikov algebras as follows.Let Var be a class of all algebras over a field k with one or more bilinear operations which is closed under direct products, subalgebras, and homomorphic images (HSPclass or variety).By the classical Birkhoff's theorem, a variety consists of all algebras that satisfy some family of identities.We will assume that Var is defined by multi-linear identities (this is not a restriction if char k = 0).For every A ∈ Var, let End(A) stand for the set of all linear operators on the space A. Then a linear operator d ∈ End(A) is called a derivation if the analogue of the Leibniz rule holds for every binary operation in A. The set of all derivations of a given algebra A forms a subspace (even a Lie subalgebra) Der(A) of End(A).The class of all pairs (A, d), A ∈ Var, d ∈ Der(A), is also a variety denoted Var Der defined by multi-linear identities.Given (A, d) ∈ Var Der, denote by A (d) the same space A equipped with two bilinear operations ≺, ≻ for each operation on A: The class of all systems A (d) , (A, d) ∈ Var Der, is closed under direct products, so all homomorphic images of all their subalgebras form a variety denoted DVar, called the derived variety of Var (see [19]).Alternatively, DVar consists of all algebras with duplicated family of operations that satisfy all those identities that hold on all A (d) for all (A, d) ∈ Var Der.For example, if Com is the variety of commutative (and associative) algebras then DCom = Nov since x ≺ y = y ≻ x.In general, the description of DVar may be obtained in the language of operads and their Manin products [19].Namely, if we identify the notations for a variety and its governing operad [11] then the operad DVar coincides with the Manin white product of operads Var and Nov.
As a corollary, the free algebra DVar⟨X⟩ in the variety DVar generated by a set X is isomorphic to the subalgebra in the free algebra in the variety Var Der generated by X.However, it is not clear that the following embedding statement holds in general: every DVar-algebra can be embedded into an appropriate differential Var-algebra (or Var Der-algebra).In other words, the problem is to determine whether the class of all subalgebras of all algebras A (d) , (A, d) ∈ Var Der, is closed under homomorphic images.Positive answers were obtained for Var = Com [4], Var = Lie [19], Var = Perm [18], Var = As [21].
In this paper, we derive a sufficient condition for a positive answer to the embedding statement for a given Var.Namely, if the Manin white product Nov • Var of the corresponding operads coincides with the Hadamard product Nov ⊗ Var then the embedding statement holds for Var.We also find an example of a variety Var governed by a binary quadratic operad such that the embedding statement fails for Var.It turns out that the variety of Zinbiel algebras Zinb (also known as commutative dendriform algebras, pre-commutative algebras, dual Leibniz algebras, half-shuffle algebras) introduced in [20] provides such an example: there exists a DZinb-algebra which cannot be embedded into a Zinbiel algebra with a derivation.To our knowledge, this is the first example of such a variety Var that an algebra from DVar does not embed into a differential Var-algebra.
The operad Zinb governing the variety of Zinbiel algebras is a particular case of a general construction called dendriform splitting of an operad [1].For every binary operad Var (not necessarily quadratic, see, e.g., [13]) there exists an operad preVar governing the class of systems with duplicated family of operations.The generic example of a preVar algebra may be obtained from a Var-algebra with a Rota-Baxter operator R of weight zero (see, e.g., [14]).If A ∈ Var and R : A → A is such an operator then (A, ⊢, ⊣), where , preLie is the classical variety of left-symmetric algebras (relative to ⊢), preAs is exactly the variety of dendriform algebras [20].
The theory of pre-algebras and relations between them is similar in many aspects to the theory of "ordinary" algebras.For example, every preAs-algebra with respect to the "dendriform commutator" a ⊢ b − b ⊣ a is a preLie-algebra.The properties of the corresponding left adjoint functor (universal envelope) are close to what we have for ordinary Lie algebras [6,12].The class of pre-Novikov algebras has been recently studied in [24]: it coincides with DZinb.Therefore, our results show that the embedding statement cannot be transferred from ordinary algebras to pre-algebras.

Derived algebras and Manin white product of binary operads
The details about (symmetric) operads may be found, for example, in [5].For an operad P denote P(n) the linear space (over a base field k) of degree n elements of P, the action of a permutation σ ∈ S n on an element f ∈ P(n) is denoted f σ , let id ∈ P(1) stand for the identity element, and the composition rule ..,mn .Recall that an operad P is said to be binary, if P(1) = kid and the entire P is generated (as an operad) by its degree 2 space P(2).
Let us fix a binary operad P. For every linear space A, consider an operad End A with End A (n) = End (A ⊗n , A).A morphism of operads P → End A defines an algebra structure on A with a set of binary operations corresponding to the generators of P from P(2).The class of all such algebras is a variety of P-algebras defined by multi-linear identities corresponding to the defining relations of the operad P.
Conversely, every variety of algebras with binary operations defined by multi-linear identities gives rise to an operad P constructed in such a way that P(n) is the space of multi-linear elements of degree n in the variables x 1 , . . ., x n of the free algebra in this variety generated by the countable set {x 1 , x 2 , . ..} (see, e.g., [17,Section 1.3.5]for details).In particular, id ∈ P(1) is presented by the element x 1 .Then the variety under consideration consists exactly of all P-algebras, in other words, it is governed by the operad P.
We will not distinguish notations for an operad P and for the corresponding variety of Palgebras.
Example 2.2.Suppose F 2 is the free binary operad with F 2 (2) ≃ kS 2 (as symmetric modules).Then the class of F 2 -algebras coincides with the variety of all nonassociative algebras with one non-symmetric binary operation, i.e., F 2 (n) may be identified with the linear span of all bracketed monomials x σ(1) If P is an operad governing a variety of algebras with one binary operation then P is a homomorphic image of F 2 .If the kernel of the morphism is generated (as an operadic ideal) by elements from F 2 (3) then the operad P is said to be quadratic.The same definition works for operads with more than one generator.
Example 2.3.Let Zinb stand for the variety of algebras with one multiplication satisfying the identity known as the Zinbiel identity [20].Then the space Zinb(n), n ≥ 1, is spanned by linearly independent monomials The product of two right-normed words in a Zinbiel algebra can be expressed via shuffle permutations on a Zinbiel algebra is associative and commutative (it is known as the shuffle product of words).
Suppose P 1 and P 2 are two binary operads.Then the Hadamard product of P 1 and P 2 is the operad denoted P = P 1 ⊗ P 2 such that P(n) = P 1 (n) ⊗ P 2 (n), n ≥ 1, the action of S n on P(n) and the composition rule are defined in the componentwize way.
The operad P 1 ⊗P 2 may not be a binary one (in this paper, we will deal with such an example below).The sub-operad of P 1 ⊗ P 2 generated by P 1 (2) ⊗ P 2 (2) is called the Manin white product of P 1 and P 2 denoted by P 1 • P 2 .(For some operads P 1 , it may happen that P 1 • P 2 = P 1 ⊗ P 2 for all P 2 ; in [22], all such operads were described.) If P 1 and P 2 are quadratic binary operads then so is P 1 • P 2 .The defining relations of the last operad can be found as follows [11].Suppose R i ⊆ P i (3), i = 1, 2, are the spaces of defining relations of P i presented in the form of multi-linear identities in x 1 , x 2 , x 3 .Consider the space E(3) spanned by all possible compositions of degree 3 of operations from P 1 (2) ⊗ P 2 (2).Then the defining relations of Example 2.4.Let P 1 = Nov, P 2 = Zinb.Then P 1 (2) ⊗ P 2 (2) is spanned by four elements In order to find E(3), calculate all monomials of degree 3 in x 1 , x 2 , x 3 with operations ≻, ≺.For example, In the same way, the expressions for all 48 monomials may be calculated in Nov(3) ⊗ Zinb(3).In order to get defining relations of Nov • Zinb it is enough to find those linear combinations of these monomials that are zero in Nov(3) ⊗ Zinb(3).This is a routine problem of linear algebra.As a result, we obtain the following identities: Remark 2.5.Novikov algebras are closely related to conformal algebras in the sense of [16].
is a Lie conformal algebra [23].The reason of this relation is that the expression for the generalised Poisson bracket in [2] has the same form as the expression for the commutator of local chiral fields in [16].
Remark 2.6.The variety governed by the operad Nov • Zinb is also related to conformal algebras.It is straightforward to check that if V is an algebra over a field k, char k = 0, with two operations ≺, ≻ satisfying the identities (2.2) then the free k is a left-symmetric conformal algebra [15].We will explain this relation in the last section.
Let Var be a binary operad, we use the same notation for the corresponding variety of algebras.Given an algebra A in Var, denote by Der(A) the set of all derivations of A. Recall that a derivation of A is a linear map d : for all operations µ from Var(2).For a derivation d of an algebra A, denote by A (d) the linear space A equipped with derived operations for all µ in Var(2).The variety generated by the class of systems A (d) for all A ∈ Var and d ∈ Der(A) is denoted by DVar, the derived variety of Var.Obviously, the class of all such A (d) is closed under direct products, so, in order to get DVar, one has to consider all homomorphic images of all subalgebras of these A (d) .In many cases, it is enough to consider just subalgebras.The problem is to decide whether homomorphic images are actually needed.
For example, all relations that hold on every Zinbiel algebra with a derivation relative to the operations a ≻ b = d(a)b, a ≺ b = ad(b) follow from the identities (2.2).
If Var = Com is the variety of associative and commutative algebras then DVar = Nov, as follows from the construction of the free Novikov algebra [9].Here we have to mention that commutativity implies a ≻ b = b ≺ a in every algebra from DCom.
If Var = Lie then the algebras from DVar form exactly the class of all F 2 -algebras with one binary operation a ≻ b = −b ≺ a [19].Note that dim Nov(n) = 2n−2 n−1 [7], and Lie(n , where C n is the nth Catalan number.The number n!C n−1 coincides with the nth dimension of the operad F 2 .Hence, Nov • Lie = Nov ⊗ Lie. Suppose Var is a binary operad, DVar = Nov • Var, DVar⟨X⟩ is the free DVar-algebra generated by a countable set X = {x 1 , x 2 , . . .}. Denote by F = Var Der⟨X, d⟩ the free differential Var-algebra generated by X with one derivation d.Then there exists a homomorphism τ : DVar⟨X⟩ → F (d) sending X to X identically.An element from ker τ is an identity that holds on all Var-algebras with a derivation relative to the derived operations (2.3).Hence, τ is injective, i.e., the free DVar-algebra can be embedded into the free differential Var-algebra.
The next question is whether every DVar-algebra can be embedded into an appropriate differential Var-algebra.This is the same as to decide if the class of all subalgebras of A (d) , (A, d) ∈ Var Der, is closed under homomorphisms.The answer is positive for Var = Com [4], Lie [19], Perm [18], and As [21].In the following sections we derive a sufficient condition for Var to guarantee a positive answer.This condition is not necessary, but we find an example when the answer is negative.

The weight criterion and special derived algebras
Let Var be a binary operad.An algebra V with two binary operations ≺, ≻ from the variety DVar is called special if it can be embedded into a Var-algebra A with a derivation d such that u ≺ v = ud(v) and u ≻ v = d(u)v in A for all u, v ∈ V .The class of all Var-algebras with a derivation is a variety since it is defined by identities.The free differential Var-algebra Var Der⟨X, d⟩ generated by a set X is isomorphic as a Var-algebra to the free Var-algebra Var X (ω) generated by the set For a nonassociative monomial u ∈ X (ω) define its weight wt(u) ∈ Z as follows.For a single letter u = x (n) , set wt(u) = n − 1.If u = u 1 u 2 then wt(u) = wt(u 1 ) + wt(u 2 ).Since the defining relations of Var X (ω) are weight-homogeneous, we may define the weight function on Var Der⟨X, d⟩.Note that if f ∈ Var X (ω) is a weight-homogeneous polynomial then wt d(f ) = wt(f ) + 1. Lemma 3.1.Let Var be a binary operad such that Nov • Var = Nov ⊗ Var.Then for every set X an element f ∈ Var X (ω) belongs to DVar⟨X⟩ if and only if wt(f ) = −1.
Proof .The "only if" part of the statement does not depend on the particular operad Var.Indeed, every formal expression in the variables X relative to binary operations ≺ and ≻ turns into a weight-homogeneous polynomial of weight −1 in Var X (ω) .
For the "if" part, assume u is a monomial of weight −1 in the variables X (ω) .In the generic form, with some bracketing.Here Here the first tensor factor is a differential commutative monomial of degree n and of weight −1 which belongs to Nov(n) by [9].In the second factor, we put the nonassociative multi-linear word obtained from u by removing all derivatives and consecutive re-numeration of variables (the bracketing remains the same as in u).
By the assumption, [u] belongs to (Nov • Var)(n), i.e., can be obtained from x 1 ≺ x 2 and x 1 ≻ x 2 by compositions and symmetric groups actions.Hence, the monomial x (s 1 ) 1 . . .x (sn) n may be expressed in terms of operations ≻ and ≺ on the variables x 1 , . . ., x n ∈ X in the differential algebra Var X (ω) .It remains to make the substitution x j → x i j to get the desired expression for u in DVar⟨X⟩.
, where the monomials of degree 3 represent compositions of for a binary operad Var implies Nov • Var = Nov ⊗ Var.Indeed, the one-to-one correspondence between (Nov ⊗ Var)(n) and DVar(n) preserves compositions and symmetric groups actions.
Hence, if x Proof .In [7], a linear basis of the free Novikov algebra generated by an ordered set was described in terms of partitions and Young diagrams.To prove the assertion, we will use this basis, which consists of non-associative monomials constructed from Young diagrams with cells properly filled with generators, see [8,Section 4] for details.
Suppose h is a non-associative monomial of degree n in Nov X (ω) of weight −1.The problem is to show that h ∈ DNov⟨X⟩.Let us proceed by induction both on the degree of h number of letters from X (ω) and on the number of "naked" letters x = x (0) , x ∈ X, that appear in h.For brevity, letters of the form x (n) for n > 0 are called "derived".
If deg h = 1 then h = x ∈ X ⊂ DNov⟨X⟩.If deg h > 1 but h contains only one "naked" letter x ∈ X then all other letters are of the form y ′ i ∈ X ′ since wt(h) = −1.Then the identities of left symmetry (1.1) and right commutativity (1.2) allow us to rewrite h as a linear combination of nonassociative monomials with subwords of the form xy ′ or y ′ x.The latter may be processed in a way described below: e.g., xy ′ may be replaced with a new letter (x ≺ y) so that we get words of smaller degree in the extended alphabet.
Case 1.If the monomial h has a subword a (k) b or ab (k) for some a, b ∈ X and k ≥ 1, then we may transform h to an expression in the extended alphabet (adding a new letter a ≻ b to X) as or, similarly, for ab (k) .The expression in the right-hand side contains monomials either of smaller degree or with a smaller number of "naked" letters.Hence, h ∈ DNov⟨X⟩.Case 2. For the general case, we need to recall the description of a linear basis of the free Novikov algebra (see [7,8,9]).Suppose X (ω) is linearly ordered in such a way that every "naked" letter is smaller than every derived one.
Every element of Nov X (ω) may be presented as a linear combination of non-associative words of the form where ω) .The letters are ordered in such a way that the following conditions hold: In particular, if at least one of the words W = W l contains both "naked" and derived letters then there are two options: (i) the last letter a l,1 is a derived one; (ii) a l,1 is "naked".In the first case, the final subword (a l,2 a l,1 ) of W is of the form considered in Case 1 since a l,2 has to be "naked" due to the choice of order on X (ω) .In the second case, we may find a suffix of W which is of the following form: An easy induction on s ≥ 1 shows that the suffix may be transformed (by means of left symmetry) to a sum of monomials considered in Case 1.Hence, it remains to consider the case when each W l contains either only "naked" letters or only derived ones.Due to the ordering of letters in X (ω) , the word h of the form (3.1) has the following property: there exists 1 ≤ l < k such that all W i for i ≤ l consist of only derived letters and for i > l all W i are nonassociative words in "naked" letters.Then use right-commutativity to transform h to the form h Here W 1 = y (n) u, n > 0, u consists of derived letters, W l+1 = xv, x ∈ X, v consists of "naked" letters.The subword W 1 W l+1 may be transformed to y (n) (xv) u, n > 0, by right commutativity, and its prefix y (n) (xv) transforms (by induction on deg v) to a form described in Case 1 by left symmetry: If Var is a binary operad such that Var • Nov = Var ⊗ Nov then every DVaralgebra is special.
Proof .Suppose V is a DVar-algebra.Then V may be presented as a quotient of a free algebra DVar⟨X⟩ modulo an ideal I. Consider the embedding DVar⟨X⟩ ⊂ Var X (ω) and denote by J the differential ideal of Var X (ω) generated by I. Then U = Var X (ω) /J is the universal enveloping differential Var-algebra of V .It remains to prove that J ∩ DVar⟨X⟩ = I, namely, the "⊆" part is not a trivial one.Assume f ∈ J. Then there exists a family of (differential) polynomials Φ i ∈ Var (X ∪{t}) (ω) such that f = i Φ i | t=g i , for some g i ∈ I. If, in addition, f ∈ DVar⟨X⟩ then wt(f ) = −1.Since wt g i = −1, we should have wt Φ i = −1 for all i.By Lemma 3.1, every polynomial Φ i may be represented as an element of DVar⟨X ∪ {t}⟩, so Φ i | t=g i ∈ I for all i, and thus f ∈ I. ■ Corollary 3.6.Every DNov-algebra V with operations ≻ and ≺ can be embedded into a commutative algebra with two commuting derivations d and ∂ so that x ≻ y = ∂(x)d(y), x ≺ y = x∂d(y) for all x, y ∈ V .
Proof .For a free DNov-algebra generated by a set X, we have the following chain of inclusions given by Proposition 3.4: Here The elements of DNov⟨X⟩ are exactly those polynomials in Com X (ω,ω) that can be presented as linear combinations of monomials The same arguments as in the proof of Theorem 3.5 imply the claim.■ Apart from the operads Com and Lie considered above, the operads Pois and As governing the varieties of Poisson and associative algebras, respectively, also satisfy the conditions of Theorem 3.5 [19,21].However, even if Nov • Var ̸ = Nov ⊗ Var then it is still possible that every DVar-algebra is special.For example, if Var = Jord is the variety of Jordan algebras then the corresponding operad is not quadratic and, in particular, the element 3) does not belong to (Nov • Jord)(3).The operad Nov • Jord is generated by a single operation x 1 ≻ x 2 = x 2 ≺ x 1 due to commutativity of Jord.Hence, Nov • Jord is a homomorphic image of the free operad F 2 .On the other hand, we have Proposition 3.7.For every non-associative algebra V with a multiplication ν : V ⊗ V → V there exists an associative algebra (A, •) with a derivation d such that In other words, we are going to show that every non-associative algebra V embeds into the derived anti-commutator algebra A (+) (d) for an appropriate associative differential algebra (A, d).
Proof .Let us choose a linear basis B of V equipped with an arbitrary total order ≤ such that (B, ≤) is a well-ordered set.Then define F to be the free associative algebra generated by B (ω) .Induce the order ≤ on B (ω) by the following rule: and expand it to the words in B (ω) by the deg-lex rule (first by length, then lexicographically).
Consider the set of defining relations All relations in the set S are obtained from a ′ b+ba ′ −ν(a, b) by formal derivation d : x (s) → x (s+1) , x (s) ∈ B (ω) .Hence, A = F/(S) is a differential associative algebra, and the map φ : The principal parts of f ∈ S relative to the order ≤ are a (n) b, a, b ∈ B, n ≥ 1.These words have no compositions of inclusion or intersection, hence, S is a Gröbner-Shirshov basis in F and the images of all variables from B are linearly independent in A since they are S-reduced (see, e.g., [3] for the definitions).Therefore, φ : V → A (+) is the desired embedding. ■ Consider the variety SJord generated by all special Jordan algebras (i.e., embeddable into associative ones with respect to the anti-commutator).In particular, for every associative algebra (A, •) with a derivation d the same space A equipped with a new operation ν is an algebra from DSJord.Proposition 3.7 implies that there are no identities that hold for the binary operation ν like that.Hence, the varieties DSJord = DJord coincide with the variety of all nonassociative algebras with one operation.However, again from Proposition 3.7 every DJord-algebra embeds into an appropriate Jordan algebra (even a special Jordan algebra) with a derivation.
In the next section, we find an example of a variety Var for which the embedding statement fails.
4 Dendriform splitting and a non-special pre-Novikov algebra Another example of a variety Var not satisfying the conditions of Theorem 3.5 is the class Zinb of Zinbiel (dual Leibniz or pre-commutative) algebras.This is a particular case of the dendriform splitting of a binary operad described in [1,13].Namely, if Var is a variety of algebras with (one or more) binary operation µ(x, y) = xy satisfying a family of multi-linear identities Σ then preVar is a variety of algebras with duplicated set of binary operations µ ⊢ (x, y) = x ⊢ y, µ ⊣ (x, y) = x ⊣ y satisfying a set of identities preΣ defined as follows.Assume f = f (x 1 , . . ., x n ) is a multi-linear polynomial of degree ≤ n, and let k ∈ {1, . . ., n}.Suppose u is a nonassociative monomial in the variables x 1 , . . ., x n such that each x i appears in u no more than once.Define a polynomial u [k]  in x 1 , . . ., x n relative to the operations µ ⊢ , µ ⊣ by induction on the degree.
Transforming each monomial u in the multi-linear polynomial f in this way, we get f [k] (x 1 , . . ., x n ).The collection of all such f [k] for f ∈ Σ, k = 1, . . ., deg f , forms the set of defining relations of a new variety denoted preVar.
If the initial operation was commutative or anti-commutative then the set of identities preΣ includes x 1 ⊢ x 2 = ±x 2 ⊣ x 1 , so the operations in preVar are actually expressed via µ ⊢ or µ ⊣ .For example, Var = Lie produces the variety preLie of left-or right-symmetric algebras (depending on the choice of ⊢ or ⊣).If Var = Com then, in terms of the operation x • y = x ⊣ y = y ⊢ x, all three identities (4.1) of pre-associative algebras are equivalent to (2.1).
In a similar way, one may derive the identities of a preNov-algebra by means of the dendriform splitting applied to (1.1) and (1.2).Routine simplification leads us to the following definition: a preNov-algebra is a linear space with two bilinear operations ⊢, ⊣ satisfying The formal change of operations x ⊣ y = x ≺ y, x ⊢ y = y ≻ x turns (4.2) exactly into (2.2).Hence, the operad preNov = preDCom defines the same class of algebras as DZinb = DpreCom.
Remark 4.1.This is not hard to compute that preDLie = DpreLie and preDAs = DpreAs.In general, for every binary operad Var there exists a morphism of operads preDVar → DpreVar (i.e., every DpreVar-algebra is a preDVar-algebra).We do not know an example when this morphism is not an isomorphism, i.e., when the operations pre and D applied to a binary operad do not commute.
An equivalent way to define the variety preVar was proposed in [13].Let Perm stand for the variety of associative algebras that satisfy left commutativity An algebra V with two operations ⊣, ⊢ is a preVar-algebra if and only if for every P ∈ Perm the space P ⊗ V equipped with the single operation is a Var-algebra.The same statement holds in the case when the binary operad Var is generated by several operations.Remark 4.4.Let V be a DZinb-algebra.In terms of pre-Novikov operations ⊣ and ⊢, the conformal algebra structure mentioned in Remark 2.6 is expressed as This is indeed a left-symmetric conformal algebra which is easy to check via the conformal analogue of (4.3).By slight abuse of notations, for every Perm-algebra P the operation is exactly the quadratic Lie conformal algebra structure [23] on k[∂] ⊗ P ⊗ V corresponding to the Novikov algebra P ⊗ V : [x (λ) y] = ∂(yx) + λ(xy + yx) for x = p ⊗ u, y = q ⊗ v, and the product is given by (4.3).
Hence the construction of a preLie conformal algebra from a DZinb-algebra is a quite clear consequence of the commutativity of tensor product.
The final statement of this section shows a substantial difference between the properties of Novikov algebras and preNov-algebras.Although the defining identities of preNov are exactly those that hold on differential Zinbiel algebras with operations (2.3) (i.e., the dendriform analogue of [9,Theorem 7.8] holds), the general embedding statement (i.e., the dendriform analogue of [4, Theorem 3]) turns to be wrong.Theorem 4.5.If the characteristic of the base field k is not 2 or 3 then there exists a DZinbalgebra which cannot be embedded into a differential Zinbiel algebra.
Proof .Consider the free Zinbiel algebra F generated by {a, b} (ω) = a, b, a ′ , b ′ , . . ., a (n) , b (n) , . . . .This is the free differential Zinbiel algebra with two generators a, b, its derivation d maps x (n) to x (n+1) for x ∈ {a, b}.The product of two elements f, g ∈ F is denoted f • g.
For every f, g ∈ F , define f ≺ g, f ≻ g by the rule (2.3): Then (F, ≺, ≻) is a DZinb-algebra, and its subalgebra generated by a, b is isomorphic to the free algebra DZinb⟨a, b⟩.
F , and let J stand for the ideal in F generated by f and all its derivatives: In particular, Let us show that h ∈ DZinb⟨a, b⟩ ⊂ J. Indeed, As in Example 2.3, we denote by x n ] the following expression in a Zinbiel algebra: Recall [20] that all such expressions with x i from a set X form a linear basis of the free Zinbiel algebra generated by X (i.e., this is a normal form in preCom⟨X⟩).
Then, in general, there is no commutative algebra (A, * ) with a derivation d and a Rota-Baxter operator ρ such that W ⊆ A, uv = u * d(v), R(u) = ρ(u), for u, v ∈ W , and ρd = dρ.In other words, a Novikov Rota-Baxter algebra cannot be in general embedded into a commutative Rota-Baxter algebra with a derivation.Indeed, assume such a system (A, * , d, ρ) exists for every Novikov algebra with a Rota-Baxter operator.Every pre-Novikov algebra V with operations ⊢ and ⊣ can be embedded into a Novikov algebra W = V with a Rota-Baxter operator so that u ⊢ v = R(u)v, u ⊣ v = uR(v), u, v ∈ V (see, e.g., [13]).We may further embed this W into a differential commutative Rota-Baxter algebra for u, v ∈ V .Therefore, we would embed a DZinb-algebra into a differential Zinbiel algebra which is not the case.

(
xy)z − x(yz) = (yx)z − y(xz) e., a linear operator d : A → A satisfying the Leibniz rule d(a * b) = d(a) * b + a * d(b), a, b ∈ A. Then the new operation ab = a * d(b), a, b ∈ A,

a
≺ b = ad(b), a ≻ b = d(a)b, a, b ∈ A.

Remark 4 . 2 .
For an arbitrary binary operad, there is a morphism of operads ζ : preVar → Zinb • Var.Namely, for every A ∈ Var and for every Z ∈ Zinb the space Z ⊗ A equipped with two operations(z ⊗ a) ⊢ (w ⊗ b) = (w • z) ⊗ ab, (z ⊗ a) ⊣ (w ⊗ b) = (z • w) ⊗ ab, for z, w ∈ Z, a, b ∈ A, is a preVar-algebra.However, preVar and Zinb • Var are not necessarily isomorphic.For example, if Var is defined by the identity (x 1 • x 2 ) • x 3 = 0 then the kernel of ζ is nonzero.As a corollary, we obtain Proposition 4.3.The operad preNov is isomorphic to the Manin white product Zinb • Nov.