On Pre-Novikov Algebras and Derived Zinbiel Variety

On Pre-Novikov Algebras and Derived Zinbiel Variety

Pavel KOLESNIKOV a, Farukh MASHUROV b and Bauyrzhan SARTAYEV cd

P. Kolesnikov, F. Mashurov and B. Sartayev

a) Sobolev Institute of Mathematics, Novosibirsk, Russia<EMAIL_ADDRESS>
b) Shenzhen International Center for Mathematics (SICM),
b) Southern University of Science and Technology, Shenzhen, Guangdong, P.R.
China<EMAIL_ADDRESS>
c) Narxoz University, Almaty, Kazakhstan<EMAIL_ADDRESS>
d) United Arab Emirates University, Al Ain, United Arab Emirates

Received August 31, 2023, in final form February 05, 2024; Published online
February 28, 2024

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 $\mathrm{Var}$ of algebras, its
derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in
$\mathrm{Var}$ 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
$\mathrm{Var}=\mathrm{Zinb}$, 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.

Novikov algebra; derivation; dendriform algebra; Zinbiel algebra

17A36; 17A30; 18M60

##  1 Introduction

The class of nonassociative algebras with one binary operation satisfying the
identities of left symmetry

$(xy)z-x(yz)=(yx)z-y(xz)$ (1.1)

and right commutativity

$(xy)z=(xz)y$ (1.2)

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.e., a linear operator $d\colon A\to A$
satisfying the Leibniz rule

$d(a*b)=d(a)*b+a*d(b),\qquad a,b\in A.$

Then the new operation

$ab=a*d(b),\qquad a,b\in A,$ (1.3)

satisfies the identities (1.1) and (1.2).

In [9], it was proved that the free Novikov algebra $\mathrm{Nov}\langle
X\rangle$ 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 $\mathrm{Var}$ be a class of all algebras
over a field $\Bbbk$ with one or more bilinear operations which is closed
under direct products, subalgebras, and homomorphic images (HSP-class or
variety). By the classical Birkhoff’s theorem, a variety consists of all
algebras that satisfy some family of identities. We will assume that
$\mathrm{Var}$ is defined by multi-linear identities (this is not a
restriction if $\operatorname{char}\Bbbk=0$). For every $A\in\mathrm{Var}$,
let $\mathop{End}\nolimits(A)$ stand for the set of all linear operators on
the space $A$. Then a linear operator $d\in\mathop{End}\nolimits(A)$ 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) $\mathop{Der}\nolimits(A)$ of
$\mathop{End}\nolimits(A)$. The class of all pairs $(A,d)$,
$A\in\mathrm{Var}$, $d\in\mathop{Der}\nolimits(A)$, is also a variety denoted
$\mathrm{Var}\mathop{Der}\nolimits$ defined by multi-linear identities. Given
$(A,d)\in\mathrm{Var}\mathop{Der}\nolimits$, denote by $A^{(d)}$ the same
space $A$ equipped with two bilinear operations $\prec$, $\succ$ for each
operation on $A$:

$a\prec b=ad(b),\qquad a\succ b=d(a)b,\qquad a,b\in A.$

The class of all systems $A^{(d)}$,
$(A,d)\in\mathrm{Var}\mathop{Der}\nolimits$, is closed under direct products,
so all homomorphic images of all their subalgebras form a variety denoted
$D\mathrm{Var}$, called the derived variety of $\mathrm{Var}$ (see [19]).
Alternatively, $D\mathrm{Var}$ consists of all algebras with duplicated family
of operations that satisfy all those identities that hold on all $A^{(d)}$ for
all $(A,d)\in\mathrm{Var}\mathop{Der}\nolimits$.

For example, if $\mathrm{Com}$ is the variety of commutative (and associative)
algebras then $D\mathrm{Com}=\mathrm{Nov}$ since $x\prec y=y\succ x$. In
general, the description of $D\mathrm{Var}$ 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 $D\mathrm{Var}$
coincides with the Manin white product of operads $\mathrm{Var}$ and
$\mathrm{Nov}$.

As a corollary, the free algebra $D\mathrm{Var}\langle X\rangle$ in the
variety $D\mathrm{Var}$ generated by a set $X$ is isomorphic to the subalgebra
in the free algebra in the variety $\mathrm{Var}\mathop{Der}\nolimits$
generated by $X$. However, it is not clear that the following embedding
statement holds in general: every $D\mathrm{Var}$-algebra can be embedded into
an appropriate differential $\mathrm{Var}$-algebra (or
$\mathrm{Var}\mathop{Der}\nolimits$-algebra). In other words, the problem is
to determine whether the class of all subalgebras of all algebras $A^{(d)}$,
$(A,d)\in\mathrm{Var}\mathop{Der}\nolimits$, is closed under homomorphic
images. Positive answers were obtained for $\mathrm{Var}=\mathrm{Com}$ [4],
$\mathrm{Var}=\mathrm{Lie}$ [19], $\mathrm{Var}=\mathrm{Perm}$ [18],
$\mathrm{Var}=\mathrm{As}$ [21].

In this paper, we derive a sufficient condition for a positive answer to the
embedding statement for a given $\mathrm{Var}$. Namely, if the Manin white
product $\mathrm{Nov}\circ\mathrm{Var}$ of the corresponding operads coincides
with the Hadamard product $\mathrm{Nov}\otimes\mathrm{Var}$ then the embedding
statement holds for $\mathrm{Var}$.

We also find an example of a variety $\mathrm{Var}$ governed by a binary
quadratic operad such that the embedding statement fails for $\mathrm{Var}$.
It turns out that the variety of Zinbiel algebras $\mathrm{Zinb}$ (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 $D\mathrm{Zinb}$-algebra which cannot be embedded into a
Zinbiel algebra with a derivation. To our knowledge, this is the first example
of such a variety $\mathrm{Var}$ that an algebra from $D\mathrm{Var}$ does not
embed into a differential $\mathrm{Var}$-algebra.

The operad $\mathrm{Zinb}$ 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 $\mathrm{Var}$ (not necessarily quadratic,
see, e.g., [13]) there exists an operad $\mathrm{pre}\mathrm{Var}$ governing
the class of systems with duplicated family of operations. The generic example
of a $\mathrm{pre}\mathrm{Var}$ algebra may be obtained from a
$\mathrm{Var}$-algebra with a Rota–Baxter operator $R$ of weight zero (see,
e.g., [14]). If $A\in\mathrm{Var}$ and $R\colon A\to A$ is such an operator
then $(A,\vdash,\dashv)$, where

$a\vdash b=R(a)b,\qquad a\dashv b=aR(b),\qquad a,b\in A,$

is a $\mathrm{pre}\mathrm{Var}$-algebra. In this context,
$\mathrm{pre}\mathrm{Com}=\mathrm{Zinb}$ (since $a\vdash b=b\dashv a$),
$\mathrm{pre}\mathrm{Lie}$ is the classical variety of left-symmetric algebras
(relative to $\vdash$), $\mathrm{pre}\mathrm{As}$ 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
$\mathrm{pre}\mathrm{As}$-algebra with respect to the “dendriform commutator”
$a\vdash b-b\dashv a$ is a $\mathrm{pre}\mathrm{Lie}$-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
$D\mathrm{Zinb}$. Therefore, our results show that the embedding statement
cannot be transferred from ordinary algebras to pre-algebras.

##  2 Derived algebras and Manin white product of binary operads

The details about (symmetric) operads may be found, for example, in [5]. For
an operad $\mathcal{P}$ denote $\mathcal{P}(n)$ the linear space (over a base
field $\Bbbk$) of degree $n$ elements of $\mathcal{P}$, the action of a
permutation $\sigma\in S_{n}$ on an element $f\in\mathcal{P}(n)$ is denoted
$f^{\sigma}$, let $\mathop{id}\nolimits\in\mathcal{P}(1)$ stand for the
identity element, and the composition rule

$\mathcal{P}(n)\otimes\mathcal{P}(m_{1})\otimes\dots\otimes\mathcal{P}(m_{n})\to\mathcal{P}(m_{1}+\dots+m_{n})$

is denoted $\gamma^{m_{1}+\dots+m_{n}}_{m_{1},\dots,m_{n}}$.

Recall that an operad $\mathcal{P}$ is said to be binary, if
$\mathcal{P}(1)=\Bbbk\mathrm{id}$ and the entire $\mathcal{P}$ is generated
(as an operad) by its degree 2 space $\mathcal{P}(2)$.

Let us fix a binary operad $\mathcal{P}$. For every linear space $A$, consider
an operad $\mathop{End}\nolimits_{A}$ with
$\mathop{End}\nolimits_{A}(n)=\mathrm{End}\,(A^{\otimes n},A)$. A morphism of
operads $\mathcal{P}\to\mathop{End}\nolimits_{A}$ defines an algebra structure
on $A$ with a set of binary operations corresponding to the generators of
$\mathcal{P}$ from $\mathcal{P}(2)$. The class of all such algebras is a
variety of $\mathcal{P}$-algebras defined by multi-linear identities
corresponding to the defining relations of the operad $\mathcal{P}$.

Conversely, every variety of algebras with binary operations defined by multi-
linear identities gives rise to an operad $\mathcal{P}$ constructed in such a
way that $\mathcal{P}(n)$ is the space of multi-linear elements of degree $n$
in the variables $x_{1},\ldots,x_{n}$ of the free algebra in this variety
generated by the countable set $\\{x_{1},x_{2},\ldots\\}$ (see, e.g., [17,
Section 1.3.5] for details). In particular,
$\mathop{id}\nolimits\in\mathcal{P}(1)$ is presented by the element $x_{1}$.
Then the variety under consideration consists exactly of all
$\mathcal{P}$-algebras, in other words, it is governed by the operad
$\mathcal{P}$.

######  Example 2.1.

Let $\mu,\nu\in\mathcal{P}(2)$. Suppose we have identified $\mu$ with
$x_{1}x_{2}$ and $\nu$ with $x_{1}*x_{2}$. Then

$\displaystyle\begin{split}&\gamma^{3}_{1,2}(\mu,\mathop{id}\nolimits,\nu)=\gamma^{3}_{1,2}(x_{1}x_{2},x_{1},x_{1}*x_{2})=x_{1}(x_{2}*x_{3}),\\\
&\gamma^{4}_{2,2}\bigl{(}\nu^{(12)},\mu,\nu\bigr{)}=\gamma^{4}_{2,2}(x_{2}*x_{1},x_{1}x_{2},x_{1}*x_{2})=(x_{3}*x_{4})*(x_{1}x_{2}),\end{split}$

and so on.

We will not distinguish notations for an operad $\mathcal{P}$ and for the
corresponding variety of $\mathcal{P}$-algebras.

######  Example 2.2.

Suppose $\mathcal{F}_{2}$ is the free binary operad with
$\mathcal{F}_{2}(2)\simeq\Bbbk S_{2}$ (as symmetric modules). Then the class
of $\mathcal{F}_{2}$-algebras coincides with the variety of all nonassociative
algebras with one non-symmetric binary operation, i.e., $\mathcal{F}_{2}(n)$
may be identified with the linear span of all bracketed monomials
$\bigl{(}x_{\sigma(1)}\cdots x_{\sigma(n)}\bigr{)}$, $\sigma\in S_{n}$, so
$\dim\mathcal{F}_{2}(n)=n!C_{n-1}$, where $C_{k}$ is the $k$th Catalan number.

If $\mathcal{P}$ is an operad governing a variety of algebras with one binary
operation then $\mathcal{P}$ is a homomorphic image of $\mathcal{F}_{2}$. If
the kernel of the morphism is generated (as an operadic ideal) by elements
from $\mathcal{F}_{2}(3)$ then the operad $\mathcal{P}$ is said to be
quadratic. The same definition works for operads with more than one generator.

######  Example 2.3.

Let $\mathrm{Zinb}$ stand for the variety of algebras with one multiplication
satisfying the identity

$(x_{1}\cdot x_{2})\cdot x_{3}=x_{1}\cdot(x_{2}\cdot x_{3}+x_{3}\cdot x_{2}),$
(2.1)

known as the Zinbiel identity [20]. Then the space $\mathrm{Zinb}(n)$, $n\geq
1$, is spanned by linearly independent monomials

$\bigl{(}x_{\sigma^{-1}(1)}\cdot\bigl{(}x_{\sigma^{-1}(2)}\cdot\bigl{(}\cdots\bigl{(}x_{\sigma^{-1}(n-1)}\cdot
x_{\sigma^{-1}(n)}\bigr{)}\cdots\bigr{)}\bigr{)}\bigr{)}=[x_{1}x_{2}\cdots
x_{n-1}x_{n}]^{\sigma},\qquad\sigma\in S_{n},$

so $\dim\mathrm{Zinb}(n)=n!$.

The product of two right-normed words in a Zinbiel algebra can be expressed
via shuffle permutations $S_{n,m}\subset S_{n+m}$: if $u=x_{1}\cdots x_{n}$,
$v=x_{n+1}\cdots x_{n+m}$ then

$[u]\cdot[v]=\sum\limits_{\sigma\in S_{n,m}\colon\sigma(1)=1}[uv]^{\sigma}.$

The new product $[u]\cdot[v]+[v]\cdot[u]$ on a Zinbiel algebra is associative
and commutative (it is known as the shuffle product of words).

Suppose $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are two binary operads. Then
the Hadamard product of $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ is the operad
denoted $\mathcal{P}=\mathcal{P}_{1}\otimes\mathcal{P}_{2}$ such that
$\mathcal{P}(n)=\mathcal{P}_{1}(n)\otimes\mathcal{P}_{2}(n)$, $n\geq 1$, the
action of $S_{n}$ on $\mathcal{P}(n)$ and the composition rule are defined in
the componentwize way.

The operad $\mathcal{P}_{1}\otimes\mathcal{P}_{2}$ may not be a binary one (in
this paper, we will deal with such an example below). The sub-operad of
$\mathcal{P}_{1}\otimes\mathcal{P}_{2}$ generated by
$\mathcal{P}_{1}(2)\otimes\mathcal{P}_{2}(2)$ is called the Manin white
product of $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ denoted by
$\mathcal{P}_{1}\circ\mathcal{P}_{2}$. (For some operads $\mathcal{P}_{1}$, it
may happen that
$\mathcal{P}_{1}\circ\mathcal{P}_{2}=\mathcal{P}_{1}\otimes\mathcal{P}_{2}$
for all $\mathcal{P}_{2}$; in [22], all such operads were described.)

If $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are quadratic binary operads then
so is $\mathcal{P}_{1}\circ\mathcal{P}_{2}$. The defining relations of the
last operad can be found as follows [11]. Suppose
$R_{i}\subseteq\mathcal{P}_{i}(3)$, $i=1,2$, are the spaces of defining
relations of $\mathcal{P}_{i}$ 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
$\mathcal{P}_{1}(2)\otimes\mathcal{P}_{2}(2)$. Then the defining relations of
$\mathcal{P}_{1}\circ\mathcal{P}_{2}$ form the space
$E(3)\cap(\mathcal{P}_{1}(3)\otimes R_{2}+R_{1}\otimes\mathcal{P}_{2}(3))$.

######  Example 2.4.

Let $\mathcal{P}_{1}=\mathrm{Nov}$, $\mathcal{P}_{2}=\mathrm{Zinb}$. Then
$\mathcal{P}_{1}(2)\otimes\mathcal{P}_{2}(2)$ is spanned by four elements

$\displaystyle x_{1}\prec x_{2}=x_{1}x_{2}\otimes x_{1}x_{2},\qquad$
$\displaystyle x_{1}\succ x_{2}=x_{2}x_{1}\otimes x_{1}x_{2},$ $\displaystyle
x_{2}\prec x_{1}=x_{2}x_{1}\otimes x_{2}x_{1},\qquad$ $\displaystyle
x_{2}\succ x_{1}=x_{1}x_{2}\otimes x_{2}x_{1}.$

In order to find $E(3)$, calculate all monomials of degree 3 in $x_{1}$,
$x_{2}$, $x_{3}$ with operations $\succ$, $\prec$. For example,

$\displaystyle(x_{1}\succ x_{3})\prec x_{2}$
$\displaystyle=\gamma^{3}_{1,2}(x_{2}\prec
x_{1},\mathop{id}\nolimits,x_{1}\succ x_{2})^{(12)}$
$\displaystyle=\gamma^{3}_{1,2}(x_{2}x_{1},\mathop{id}\nolimits,x_{2}x_{1})^{(12)}\otimes\gamma^{3}_{1,2}(x_{2}x_{1},\mathop{id}\nolimits,x_{1}x_{2})^{(12)}$
$\displaystyle=(x_{3}x_{2})x_{1}^{(12)}\otimes(x_{2}x_{3})x_{1}^{(12)}=(x_{3}x_{1})x_{2}\otimes(x_{2}x_{3})x_{1}\in\mathrm{Nov}(3)\otimes\mathrm{Zinb}(3).$

In the same way, the expressions for all 48 monomials may be calculated in
$\mathrm{Nov}(3)\otimes\mathrm{Zinb}(3)$. In order to get defining relations
of $\mathrm{Nov}\circ\mathrm{Zinb}$ it is enough to find those linear
combinations of these monomials that are zero in
$\mathrm{Nov}(3)\otimes\mathrm{Zinb}(3)$. This is a routine problem of linear
algebra. As a result, we obtain the following identities:

$\displaystyle(x_{1}\prec x_{2})\prec x_{3}=(x_{1}\prec x_{3})\prec x_{2},$
$\displaystyle x_{1}\succ(x_{2}\succ x_{3})=(x_{1}\succ x_{3})\prec
x_{2}-x_{1}\succ(x_{3}\prec x_{2}),$ $\displaystyle(x_{1}\succ x_{2})\succ
x_{3}=(x_{1}\succ x_{3})\succ x_{2}+(x_{1}\succ x_{3})\prec x_{2}-(x_{1}\succ
x_{2})\prec x_{3},$ (2.2) $\displaystyle(x_{1}\prec x_{2})\succ
x_{3}=x_{1}\prec(x_{2}\succ x_{3})+x_{1}\prec(x_{3}\prec x_{2})+(x_{1}\succ
x_{3})\prec x_{2}-(x_{1}\prec x_{3})\prec x_{2}.$

######  Remark 2.5.

Novikov algebras are closely related to conformal algebras in the sense of
[16]. Namely, if $V$ is a Novikov algebra then the free
$\Bbbk[\partial]$-module $C=\Bbbk[\partial]\otimes V$ equipped with a sesqui-
linear $\lambda$-product

$[u{}_{(\lambda)}v]=\partial(vu)+\lambda(vu+vu),\qquad u,v\in V,$

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 $\mathrm{Nov}\circ\mathrm{Zinb}$ is also
related to conformal algebras. It is straightforward to check that if $V$ is
an algebra over a field $\Bbbk$, $\operatorname{char}\Bbbk=0$, with two
operations $\prec$, $\succ$ satisfying the identities (2.2) then the free
$\Bbbk[\partial]$-module $C=\Bbbk[\partial]\otimes V$ equipped with a sesqui-
linear $\lambda$-product

$\bigl{(}u{}_{(\lambda)}v\bigr{)}=\partial(v\prec u)+\lambda(v\succ u+v\prec
u),\qquad u,v\in V,$

is a left-symmetric conformal algebra [15]. We will explain this relation in
the last section.

Let $\mathrm{Var}$ be a binary operad, we use the same notation for the
corresponding variety of algebras. Given an algebra $A$ in $\mathrm{Var}$,
denote by $\mathop{Der}\nolimits(A)$ the set of all derivations of $A$. Recall
that a derivation of $A$ is a linear map $d\colon A\to A$ such that

$d(\mu(a,b))=\mu(d(a),b)+\mu(a,d(b)),\qquad a,b\in A,$

for all operations $\mu$ from $\mathrm{Var}(2)$. For a derivation $d$ of an
algebra $A$, denote by $A^{(d)}$ the linear space $A$ equipped with derived
operations

$\mu^{\succ}(a,b)=\mu(d(a),b),\quad\mu^{\prec}(a,b)=\mu(a,d(b)),\qquad a,b\in
A,$ (2.3)

for all $\mu$ in $\mathrm{Var}(2)$. The variety generated by the class of
systems $A^{(d)}$ for all $A\in\mathrm{Var}$ and
$d\in\mathop{Der}\nolimits(A)$ is denoted by $D\mathrm{Var}$, the derived
variety of $\mathrm{Var}$. Obviously, the class of all such $A^{(d)}$ is
closed under direct products, so, in order to get $D\mathrm{Var}$, 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.

######  Theorem 2.7 ([19]).

For a binary operad $\mathrm{Var}$, the variety $D\mathrm{Var}$ is governed by
the operad ${\mathrm{Nov}\circ\mathrm{Var}}$.

For example, all relations that hold on every Zinbiel algebra with a
derivation relative to the operations $a\succ b=d(a)b$, $a\prec b=ad(b)$
follow from the identities (2.2).

If $\mathrm{Var}=\mathrm{Com}$ is the variety of associative and commutative
algebras then $D\mathrm{Var}=\mathrm{Nov}$, as follows from the construction
of the free Novikov algebra [9]. Here we have to mention that commutativity
implies $a\succ b=b\prec a$ in every algebra from $D\mathrm{Com}$.

If $\mathrm{Var}=\mathrm{Lie}$ then the algebras from $D\mathrm{Var}$ form
exactly the class of all $\mathcal{F}_{2}$-algebras with one binary operation
$a\succ b=-b\prec a$ [19]. Note that $\dim\mathrm{Nov}(n)=\binom{2n-2}{n-1}$
[7], and $\mathrm{Lie}(n)=(n-1)!$. Hence,
$\dim(\mathrm{Nov}\otimes\mathrm{Lie})(n)=\frac{(2n-2)!}{(n-1)!}$ which is
equal to $n!C_{n-1}$, where $C_{n}$ is the $n$th Catalan number. The number
$n!C_{n-1}$ coincides with the $n$th dimension of the operad
$\mathcal{F}_{2}$. Hence,
$\mathrm{Nov}\circ\mathrm{Lie}=\mathrm{Nov}\otimes\mathrm{Lie}$.

Suppose $\mathrm{Var}$ is a binary operad,
$D\mathrm{Var}=\mathrm{Nov}\circ\mathrm{Var}$, $D\mathrm{Var}\langle X\rangle$
is the free $D\mathrm{Var}$-algebra generated by a countable set
$X=\\{x_{1},x_{2},\dots\\}$. Denote by
$F=\mathrm{Var}\mathop{Der}\nolimits\langle X,d\rangle$ the free differential
$\mathrm{Var}$-algebra generated by $X$ with one derivation $d$. Then there
exists a homomorphism $\tau\colon D\mathrm{Var}\langle X\rangle\to F^{(d)}$
sending $X$ to $X$ identically. An element from $\ker\tau$ is an identity that
holds on all $\mathrm{Var}$-algebras with a derivation relative to the derived
operations (2.3). Hence, $\tau$ is injective, i.e., the free
$D\mathrm{Var}$-algebra can be embedded into the free differential
$\mathrm{Var}$-algebra.

The next question is whether every $D\mathrm{Var}$-algebra can be embedded
into an appropriate differential $\mathrm{Var}$-algebra. This is the same as
to decide if the class of all subalgebras of $A^{(d)}$,
$(A,d)\in\mathrm{Var}\mathop{Der}\nolimits$, is closed under homomorphisms.
The answer is positive for $\mathrm{Var}=\mathrm{Com}$ [4], $\mathrm{Lie}$
[19], $\mathrm{Perm}$ [18], and $\mathrm{As}$ [21]. In the following sections
we derive a sufficient condition for $\mathrm{Var}$ to guarantee a positive
answer. This condition is not necessary, but we find an example when the
answer is negative.

##  3 The weight criterion and special derived algebras

Let $\mathrm{Var}$ be a binary operad. An algebra $V$ with two binary
operations $\prec$, $\succ$ from the variety $D\mathrm{Var}$ is called special
if it can be embedded into a $\mathrm{Var}$-algebra $A$ with a derivation $d$
such that $u\prec v=ud(v)$ and $u\succ v=d(u)v$ in $A$ for all $u,v\in V$. The
class of all $\mathrm{Var}$-algebras with a derivation is a variety since it
is defined by identities. The free differential $\mathrm{Var}$-algebra
$\mathrm{Var}\mathop{Der}\nolimits\langle X,d\rangle$ generated by a set $X$
is isomorphic as a $\mathrm{Var}$-algebra to the free $\mathrm{Var}$-algebra
$\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$ generated by the set

$X^{(\omega)}=\bigl{\\{}x^{(n)}\mid x\in X,\,n\in\mathbb{Z}_{+}\bigr{\\}},$

with the derivation $d$ defined by $d\bigl{(}x^{(n)}\bigr{)}=x^{(n+1)}$, $x\in
X$, $n\in\mathbb{Z}_{+}$.

For a nonassociative monomial $u\in X^{(\omega)}$ define its weight
$\mathop{wt}\nolimits(u)\in\mathbb{Z}$ as follows. For a single letter
$u=x^{(n)}$, set $\mathop{wt}\nolimits(u)=n-1$. If $u=u_{1}u_{2}$ then
$\mathop{wt}\nolimits(u)=\mathop{wt}\nolimits(u_{1})+\mathop{wt}\nolimits(u_{2})$.
Since the defining relations of
$\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$ are weight-homogeneous,
we may define the weight function on $\mathrm{Var}\mathop{Der}\nolimits\langle
X,d\rangle$. Note that if
$f\in\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$ is a weight-
homogeneous polynomial then $\mathop{wt}\nolimits
d(f)=\mathop{wt}\nolimits(f)+1$.

######  Lemma 3.1.

Let $\mathrm{Var}$ be a binary operad such that
$\mathrm{Nov}\circ\mathrm{Var}=\mathrm{Nov}\otimes\mathrm{Var}$. Then for
every set $X$ an element
$f\in\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$ belongs to
$D\mathrm{Var}\langle X\rangle$ if and only if $\mathop{wt}\nolimits(f)=-1$.

###### Proof.

The “only if” part of the statement does not depend on the particular operad
$\mathrm{Var}$. Indeed, every formal expression in the variables $X$ relative
to binary operations $\prec$ and $\succ$ turns into a weight-homogeneous
polynomial of weight $-1$ in
$\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$.

For the “if” part, assume $u$ is a monomial of weight $-1$ in the variables
$X^{(\omega)}$. In the generic form,

$u=\bigl{(}x_{i_{1}}^{(s_{1})}\cdots x_{i_{n}}^{(s_{n})}\bigr{)},\qquad
x_{i_{j}}\in X,\quad s_{j}\geq 0,$

with some bracketing. Here $s_{1}+\dots+s_{n}=n-1$. Consider the element

$[u]=x_{1}^{(s_{1})}\cdots x_{n}^{(s_{n})}\otimes(x_{1}\cdots
x_{n})\in\mathrm{Nov}(n)\otimes\mathrm{Var}(n).$

Here the first tensor factor is a differential commutative monomial of degree
$n$ and of weight $-1$ which belongs to $\mathrm{Nov}(n)$ 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 $(\mathrm{Nov}\circ\mathrm{Var})(n)$,
i.e., can be obtained from $x_{1}\prec x_{2}$ and $x_{1}\succ x_{2}$ by
compositions and symmetric groups actions. Hence, the monomial
$\bigl{(}x_{1}^{(s_{1})}\dots x_{n}^{(s_{n})}\bigr{)}$ may be expressed in
terms of operations $\succ$ and $\prec$ on the variables $x_{1},\dots,x_{n}\in
X$ in the differential algebra
$\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$. It remains to make the
substitution $x_{j}\to x_{i_{j}}$ to get the desired expression for $u$ in
$D\mathrm{Var}\langle X\rangle$. ∎

######  Example 3.2.

For example, if
$u=(x_{1}(x_{1}^{\prime\prime}x_{2}))\in\mathrm{Lie}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$
then $[u]=x_{1}x_{2}^{\prime\prime}x_{3}\otimes(x_{1}(x_{2}x_{3}))$. It is
straightforward to check that $[u]=x_{1}\prec(x_{2}\succ
x_{3})-x_{2}\succ(x_{1}\prec x_{3})-(x_{1}\prec x_{2})\prec x_{3}$, where the
monomials of degree 3 represent compositions of $x_{1}\prec x_{2}$,
$x_{2}\prec x_{1}$, $x_{1}\succ x_{2}$, $x_{2}\succ x_{1}$, and
$\mathop{id}\nolimits=x_{1}\otimes x_{1}$ as in Example 2.4. Hence,
$u=x_{1}\prec(x_{1}\succ x_{2})-x_{1}\succ(x_{1}\prec x_{2})-(x_{1}\prec
x_{1})\prec x_{2}$.

######  Remark 3.3.

The condition

$D\mathrm{Var}\langle
X\rangle=\bigl{\\{}f\in\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}\mid\mathop{wt}\nolimits(f)=-1\bigr{\\}}$

for a binary operad $\mathrm{Var}$ implies
$\mathrm{Nov}\circ\mathrm{Var}=\mathrm{Nov}\otimes\mathrm{Var}$. Indeed, the
one-to-one correspondence

$x_{1}^{(s_{1})}\cdots x_{n}^{(s_{n})}\otimes(x_{i_{1}}\cdots
x_{i_{n}})\leftrightarrow\bigl{(}x_{i_{1}}^{(s_{i_{1}})}\cdots
x_{i_{n}}^{(s_{i_{n}})}\bigr{)},\qquad\sum_{i}s_{i}=n-1,$

between $(\mathrm{Nov}\otimes\mathrm{Var})(n)$ and $D\mathrm{Var}(n)$
preserves compositions and symmetric groups actions. Hence, if
$\bigl{(}x_{i_{1}}^{(s_{i_{1}})}\cdots x_{i_{n}}^{(s_{i_{n}})}\bigr{)}$ may be
expressed via $X$ in terms of operations $\mu^{\succ}$, $\mu^{\prec}$
($\mu\in\mathrm{Var}(2)$) then $x_{1}^{(s_{1})}\cdots
x_{n}^{(s_{n})}\otimes(x_{i_{1}}\cdots
x_{i_{n}})\in(\mathrm{Nov}\circ\mathrm{Var})(n)$.

######  Proposition 3.4.

The operad $\mathrm{Var}=\mathrm{Nov}$ satisfies the conditions of Lemma 3.1,
i.e., $\mathrm{Nov}\circ\mathrm{Nov}=\mathrm{Nov}\otimes\mathrm{Nov}$.

###### 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
$\mathrm{Nov}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$ of weight $-1$. The
problem is to show that $h\in D\mathrm{Nov}\langle X\rangle$. Let us proceed
by induction both on the degree of $h$ (number of letters from $X^{(\omega)}$)
and on the number of “naked” letters $x=x^{(0)}$, $x\in X$, that appear in
$h$. (For brevity, letters of the form $x^{(n)}$ for $n>0$ are called
“derived”.)

If $\deg h=1$ then $h=x\in X\subset D\mathrm{Nov}\langle X\rangle$. If $\deg
h>1$ but $h$ contains only one “naked” letter $x\in X$ then all other letters
are of the form $y_{i}^{\prime}\in X^{\prime}$ since
$\mathop{wt}\nolimits(h)=-1$. 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 $xy^{\prime}$ or
$y^{\prime}x$. The latter may be processed in a way described below: e.g.,
$xy^{\prime}$ may be replaced with a new letter $(x\prec y)$ 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
$a,b\in X$ and $k\geq 1$, then we may transform $h$ to an expression in the
extended alphabet (adding a new letter $a\succ b$ to $X$) as

$a^{(k)}b=(a\succ b)^{(k-1)}-\sum\limits_{s\geq
1}\binom{k-1}{s}a^{(k-s)}b^{(s)},$

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, $h\in D\mathrm{Nov}\langle X\rangle$.

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 $X^{(\omega)}$ is
linearly ordered in such a way that every “naked” letter is smaller than every
derived one.

Every element of $\mathrm{Nov}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$ may be
presented as a linear combination of non-associative words of the form

$h=(\cdots(W_{1}W_{2})\cdots W_{k-1})W_{k},$ (3.1)

where $W_{1}=a_{1,r_{1}+1}(a_{1,r_{1}}(\cdots(a_{1,2}a_{1,1})\cdots))$,
$W_{l}=a_{l,r_{l}}(a_{l,r_{l}-1}(\cdots(a_{l,2}a_{l,1})\cdots))$,
$l=2,\dots,k$, $r_{1}\geq r_{2}\geq\dots\geq r_{k}$, $a_{i,j}\in
X^{(\omega)}$. The letters are ordered in such a way that the following
conditions hold:

  * •

If $r_{i}=r_{i+1}$ then $a_{i,1}\geq a_{i+1,1}$ for $i=1,\dots,k-1$;

  * •

$a_{1,r_{1}+1}\geq a_{1,r_{1}}\geq\dots\geq a_{1,2}\geq
a_{2,r_{2}}\geq\dots\geq a_{2,2}\geq\dots\geq a_{k-1,2}\geq
a_{k,r_{k}}\geq\dots\geq a_{k,2}$.

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 $X^{(\omega)}$. In
the second case, we may find a suffix of $W$ which is of the following form:

$y^{(n)}(x_{1}(x_{2}\cdots(x_{s-1}x_{s})\cdots)),\qquad x_{i},y\in X,\quad
n>0.$

An easy induction on $s\geq 1$ 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
$X^{(\omega)}$, the word $h$ of the form (3.1) has the following property:
there exists $1\leq l<k$ such that all $W_{i}$ for $i\leq l$ 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
$h=(\cdots((\cdots((W_{1}W_{l+1})W_{2})\cdots W_{l})W_{l+2})\cdots W_{k})$.
Here $W_{1}=y^{(n)}u$, $n>0$, $u$ consists of derived letters, $W_{l+1}=xv$,
$x\in X$, $v$ consists of “naked” letters. The subword $W_{1}W_{l+1}$ may be
transformed to $\bigl{(}y^{(n)}(xv)\bigr{)}u$, $n>0$, 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:

$y^{(n)}(xv)=\bigl{(}y^{(n)}x\bigr{)}v-\bigl{(}xy^{(n)}\bigr{)}v+x\bigl{(}y^{(n)}v\bigr{)}.$

######  Theorem 3.5.

If $\mathrm{Var}$ is a binary operad such that
$\mathrm{Var}\circ\mathrm{Nov}=\mathrm{Var}\otimes\mathrm{Nov}$ then every
$D\mathrm{Var}$-algebra is special.

###### Proof.

Suppose $V$ is a $D\mathrm{Var}$-algebra. Then $V$ may be presented as a
quotient of a free algebra $D\mathrm{Var}\langle X\rangle$ modulo an ideal
$I$. Consider the embedding $D\mathrm{Var}\langle
X\rangle\subset\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$ and
denote by $J$ the differential ideal of
$\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}$ generated by $I$. Then
$U=\mathrm{Var}\bigl{\langle}X^{(\omega)}\bigr{\rangle}/J$ is the universal
enveloping differential $\mathrm{Var}$-algebra of $V$. It remains to prove
that $J\cap D\mathrm{Var}\langle X\rangle=I$, namely, the “$\subseteq$” part
is not a trivial one.

Assume $f\in J$. Then there exists a family of (differential) polynomials
$\Phi_{i}\in\mathrm{Var}\bigl{\langle}(X\cup\\{t\\})^{(\omega)}\bigr{\rangle}$
such that $f=\sum_{i}\Phi_{i}|_{t=g_{i}}$, for some $g_{i}\in I$. If, in
addition, $f\in D\mathrm{Var}\langle X\rangle$ then
$\mathop{wt}\nolimits(f)=-1$. Since $\mathop{wt}\nolimits g_{i}=-1$, we should
have $\mathop{wt}\nolimits\Phi_{i}=-1$ for all $i$. By Lemma 3.1, every
polynomial $\Phi_{i}$ may be represented as an element of
$D\mathrm{Var}\langle X\cup\\{t\\}\rangle$, so $\Phi_{i}|_{t=g_{i}}\in I$ for
all $i$, and thus $f\in I$. ∎

######  Corollary 3.6.

Every $D\mathrm{Nov}$-algebra $V$ with operations $\succ$ and $\prec$ can be
embedded into a commutative algebra with two commuting derivations $d$ and
$\partial$ so that $x\succ y=\partial(x)d(y)$, $x\prec y=x\partial d(y)$ for
all $x,y\in V$.

###### Proof.

For a free $D\mathrm{Nov}$-algebra generated by a set $X$, we have the
following chain of inclusions given by Proposition 3.4:

$D\mathrm{Nov}\langle X\rangle\subset\mathrm{Nov}\mathop{Der}\nolimits\langle
X,\partial\rangle=\mathrm{Nov}\bigl{\langle}X^{(\omega)}\bigr{\rangle}\subset\mathrm{Com}\mathop{Der}\nolimits\bigl{\langle}X^{(\omega)},d\bigr{\rangle}=\mathrm{Com}\bigl{\langle}X^{(\omega,\omega)}\bigr{\rangle}.$

Here
$X^{(\omega,\omega)}=\bigl{(}X^{(\omega)}\bigr{)}^{(\omega)}=\bigl{\\{}x^{(n,m)}\mid
x\in X,n,m\in\mathbb{Z}_{+}\bigr{\\}}$, a variable $x^{(n,m)}$ represents
$d^{n}\partial^{m}(x)$. The elements of $D\mathrm{Nov}\langle X\rangle$ are
exactly those polynomials in
$\mathrm{Com}\bigl{\langle}X^{(\omega,\omega)}\bigr{\rangle}$ that can be
presented as linear combinations of monomials

$x_{1}^{(n_{1},m_{1})}\cdots
x_{k}^{(n_{k},m_{k})},\qquad\sum_{i}n_{i}=\sum_{i}m_{i}=k+1.$

The same arguments as in the proof of Theorem 3.5 imply the claim. ∎

Apart from the operads $\mathrm{Com}$ and $\mathrm{Lie}$ considered above, the
operads $\mathrm{Pois}$ and $\mathrm{As}$ governing the varieties of Poisson
and associative algebras, respectively, also satisfy the conditions of Theorem
3.5 [19, 21]. However, even if
$\mathrm{Nov}\circ\mathrm{Var}\neq\mathrm{Nov}\otimes\mathrm{Var}$ then it is
still possible that every $D\mathrm{Var}$-algebra is special. For example, if
$\mathrm{Var}=\mathrm{Jord}$ is the variety of Jordan algebras then the
corresponding operad is not quadratic and, in particular, the element
$[u]=x_{1}x_{2}^{\prime}x_{3}^{\prime}\otimes
x_{1}(x_{2}x_{3})\in\mathrm{Nov}(3)\otimes\mathrm{Jord}(3)$ does not belong to
$(\mathrm{Nov}\circ\mathrm{Jord})(3)$. The operad
$\mathrm{Nov}\circ\mathrm{Jord}$ is generated by a single operation
$x_{1}\succ x_{2}=x_{2}\prec x_{1}$ due to commutativity of $\mathrm{Jord}$.
Hence, $\mathrm{Nov}\circ\mathrm{Jord}$ is a homomorphic image of the free
operad $\mathcal{F}_{2}$. On the other hand, we have

######  Proposition 3.7.

For every non-associative algebra $V$ with a multiplication $\nu\colon
V\otimes V\to V$ there exists an associative algebra $(A,\cdot)$ with a
derivation $d$ such that $V\subseteq A$ and $\nu(u,v)=d(u)\cdot v+v\cdot d(u)$
for all $u,v\in V$.

In other words, we are going to show that every non-associative algebra $V$
embeds into the derived anti-commutator algebra
$\bigl{(}A^{(+)}\bigr{)}^{(d)}$ for an appropriate associative differential
algebra $(A,d)$.

###### Proof.

Let us choose a linear basis $B$ of $V$ equipped with an arbitrary total order
$\leq$ such that $(B,\leq)$ is a well-ordered set. Then define $F$ to be the
free associative algebra generated by $B^{(\omega)}$. Induce the order $\leq$
on $B^{(\omega)}$ by the following rule:

$a^{(n)}\leq b^{(m)}\iff(n,a)\leq(m,b)\ \text{lexicographically},$

and expand it to the words in $B^{(\omega)}$ by the deg-lex rule (first by
length, then lexicographically).

Consider the set of defining relations

$\displaystyle S=\bigg{\\{}a^{(n)}b+\sum\limits_{s\geq
1}\binom{n-1}{s}\\!\bigl{(}a^{(n-s)}b^{(s)}+b^{(s)}a^{(n-s)}\bigr{)}+ba^{(n)}-\nu(a,b)^{(n-1)}\mid
a,b\in B,n\geq 1\bigg{\\}}.$

All relations in the set $S$ are obtained from
$a^{\prime}b+ba^{\prime}-\nu(a,b)$ by formal derivation $d\colon x^{(s)}\to
x^{(s+1)}$, $x^{(s)}\in B^{(\omega)}$. Hence, $A=F/(S)$ is a differential
associative algebra, and the map $\varphi\colon V\to A^{(+)}$,
$\varphi(v)=v+(S)$, $v\in V$, preserves the operation, i.e.,
$\varphi(\nu(u,v))=d(\varphi(u))\cdot\varphi(v)+\varphi(v)\cdot d(\varphi(u))$
for all $u,v\in V$.

The principal parts of $f\in S$ relative to the order $\leq$ are $a^{(n)}b$,
$a,b\in B$, $n\geq 1$. 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, $\varphi\colon
V\to A^{(+)}$ is the desired embedding. ∎

Consider the variety $\mathrm{SJord}$ generated by all special Jordan algebras
(i.e., embeddable into associative ones with respect to the anti-commutator).
In particular, for every associative algebra $(A,\cdot)$ with a derivation $d$
the same space $A$ equipped with a new operation $\nu(u,v)=d(u)\cdot v+v\cdot
d(u)$, $u,v\in A$, is an algebra from $D\mathrm{SJord}$. Proposition 3.7
implies that there are no identities that hold for the binary operation $\nu$
like that. Hence, the varieties $D\mathrm{SJord}=D\mathrm{Jord}$ coincide with
the variety of all nonassociative algebras with one operation. However, again
from Proposition 3.7 every $D\mathrm{Jord}$-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 $\mathrm{Var}$ for which
the embedding statement fails.

##  4 Dendriform splitting and a non-special pre-Novikov algebra

Another example of a variety $\mathrm{Var}$ not satisfying the conditions of
Theorem 3.5 is the class $\mathrm{Zinb}$ 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 $\mathrm{Var}$ is a
variety of algebras with (one or more) binary operation $\mu(x,y)=xy$
satisfying a family of multi-linear identities $\Sigma$ then
$\mathrm{pre}\mathrm{Var}$ is a variety of algebras with duplicated set of
binary operations $\mu_{\vdash}(x,y)=x\vdash y$, $\mu_{\dashv}(x,y)=x\dashv y$
satisfying a set of identities $\mathrm{pre}\Sigma$ defined as follows. Assume
$f=f(x_{1},\dots,x_{n})$ is a multi-linear polynomial of degree $\leq n$, and
let $k\in\\{1,\dots,n\\}$. Suppose $u$ is a nonassociative monomial in the
variables $x_{1},\dots,x_{n}$ such that each $x_{i}$ appears in $u$ no more
than once. Define a polynomial $u^{[k]}$ in $x_{1},\dots,x_{n}$ relative to
the operations $\mu_{\vdash}$, $\mu_{\dashv}$ by induction on the degree. If
$u=x_{i}$ then $u^{[k]}=x_{i}$; if $u=vw$ and $x_{k}$ appears in $v$ (or in
$w$) then $u^{[k]}=v^{[k]}\dashv w^{[k]}$ (or, respectively, $v^{[k]}\vdash
w^{[k]}$); if $x_{k}$ does not appear in $u$ then set $u^{[k]}=v^{[k]}\dashv
w^{[k]}+v^{[k]}\vdash w^{[k]}$. Transforming each monomial $u$ in the multi-
linear polynomial $f$ in this way, we get $f^{[k]}(x_{1},\dots,x_{n})$. The
collection of all such $f^{[k]}$ for $f\in\Sigma$, $k=1,\dots,\deg f$, forms
the set of defining relations of a new variety denoted
$\mathrm{pre}\mathrm{Var}$.

For example, for $f(x_{1},x_{2},x_{3})=(x_{1}x_{2})x_{3}-x_{1}(x_{2}x_{3})$
the polynomials $f^{[k]}$, $k=1,2,3$, are given by

$\displaystyle f^{[1]}=(x_{1}\dashv x_{2})\dashv x_{3}-x_{1}\dashv(x_{2}\dashv
x_{3}+x_{2}\vdash x_{3}),$ $\displaystyle f^{[2]}=(x_{1}\vdash x_{2})\dashv
x_{3}-x_{1}\vdash(x_{2}\dashv x_{3}),$ $\displaystyle f^{[3]}=(x_{1}\vdash
x_{2}+x_{1}\dashv x_{2})\vdash x_{3}-x_{1}\vdash(x_{2}\vdash x_{3}).$ (4.1)

These identities define the variety of pre-associative or dendriform algebras
[20].

If the initial operation was commutative or anti-commutative then the set of
identities $\mathrm{pre}\Sigma$ includes $x_{1}\vdash x_{2}=\pm x_{2}\dashv
x_{1}$, so the operations in $\mathrm{pre}\mathrm{Var}$ are actually expressed
via $\mu_{\vdash}$ or $\mu_{\dashv}$. For example, $\mathrm{Var}=\mathrm{Lie}$
produces the variety $\mathrm{pre}\mathrm{Lie}$ of left- or right-symmetric
algebras (depending on the choice of $\vdash$ or $\dashv$). If
$\mathrm{Var}=\mathrm{Com}$ then, in terms of the operation $x\cdot y=x\dashv
y=y\vdash x$, 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
$\mathrm{pre}\mathrm{Nov}$-algebra by means of the dendriform splitting
applied to (1.1) and (1.2). Routine simplification leads us to the following
definition: a $\mathrm{pre}\mathrm{Nov}$-algebra is a linear space with two
bilinear operations $\vdash$, $\dashv$ satisfying

$\displaystyle(x_{1}\dashv x_{2})\dashv x_{3}=(x_{1}\dashv x_{3})\dashv
x_{2},$ $\displaystyle(x_{1}\vdash x_{2})\dashv x_{3}=(x_{1}\vdash
x_{3})\vdash x_{2}+(x_{1}\dashv x_{3})\vdash x_{2},$
$\displaystyle(x_{1}\dashv x_{2})\dashv x_{3}-x_{1}\dashv(x_{2}\dashv
x_{3})-x_{1}\dashv(x_{2}\vdash x_{3})=(x_{2}\vdash x_{1})\dashv
x_{3}-x_{2}\vdash(x_{1}\dashv x_{3}),$ $\displaystyle(x_{1}\vdash x_{3})\dashv
x_{2}-x_{1}\vdash(x_{2}\vdash x_{3})=(x_{2}\vdash x_{3})\dashv
x_{1}-x_{2}\vdash(x_{1}\vdash x_{3}).$ (4.2)

The formal change of operations $x\dashv y=x\prec y$, $x\vdash y=y\succ x$
turns (4.2) exactly into (2.2). Hence, the operad
$\mathrm{pre}\mathrm{Nov}=\mathrm{pre}D\mathrm{Com}$ defines the same class of
algebras as $D\mathrm{Zinb}=D\mathrm{pre}\mathrm{Com}$.

######  Remark 4.1.

This is not hard to compute that
$\mathrm{pre}D\mathrm{Lie}=D\mathrm{pre}\mathrm{Lie}$ and
$\mathrm{pre}D\mathrm{As}=D\mathrm{pre}\mathrm{As}$. In general, for every
binary operad $\mathrm{Var}$ there exists a morphism of operads
$\mathrm{pre}D\mathrm{Var}\to D\mathrm{pre}\mathrm{Var}$ (i.e., every
$D\mathrm{pre}\mathrm{Var}$-algebra is a $\mathrm{pre}D\mathrm{Var}$-algebra).
We do not know an example when this morphism is not an isomorphism, i.e., when
the operations $\mathrm{pre}$ and $D$ applied to a binary operad do not
commute.

An equivalent way to define the variety $\mathrm{pre}\mathrm{Var}$ was
proposed in [13]. Let $\mathrm{Perm}$ stand for the variety of associative
algebras that satisfy left commutativity

$x_{1}x_{2}x_{3}=x_{2}x_{1}x_{3}.$

An algebra $V$ with two operations $\dashv$, $\vdash$ is a
$\mathrm{pre}\mathrm{Var}$-algebra if and only if for every
$P\in\mathrm{Perm}$ the space $P\otimes V$ equipped with the single operation

$(p\otimes u)(q\otimes v)=pq\otimes(u\vdash v)+qp\otimes(u\dashv v),\qquad
p,q\in P,\quad u,v\in V,$ (4.3)

is a $\mathrm{Var}$-algebra. The same statement holds in the case when the
binary operad $\mathrm{Var}$ is generated by several operations.

######  Remark 4.2.

For an arbitrary binary operad, there is a morphism of operads
$\zeta\colon\mathrm{pre}\mathrm{Var}\to\mathrm{Zinb}\circ\mathrm{Var}$.
Namely, for every $A\in\mathrm{Var}$ and for every $Z\in\mathrm{Zinb}$ the
space $Z\otimes A$ equipped with two operations

$(z\otimes a)\vdash(w\otimes b)=(w\cdot z)\otimes ab,\qquad(z\otimes
a)\dashv(w\otimes b)=(z\cdot w)\otimes ab,$

for $z,w\in Z$, $a,b\in A$, is a $\mathrm{pre}\mathrm{Var}$-algebra.

However, $\mathrm{pre}\mathrm{Var}$ and $\mathrm{Zinb}\circ\mathrm{Var}$ are
not necessarily isomorphic. For example, if $\mathrm{Var}$ is defined by the
identity $(x_{1}\cdot x_{2})\cdot x_{3}=0$ then the kernel of $\zeta$ is
nonzero.

As a corollary, we obtain

######  Proposition 4.3.

The operad $\mathrm{pre}\mathrm{Nov}$ is isomorphic to the Manin white product
$\mathrm{Zinb}\circ\mathrm{Nov}$.

######  Remark 4.4.

Let $V$ be a $D\mathrm{Zinb}$-algebra. In terms of pre-Novikov operations
$\dashv$ and $\vdash$, the conformal algebra structure mentioned in Remark 2.6
is expressed as

$(u_{(\lambda)}v)=\partial(v\dashv u)+\lambda(u\vdash v+v\dashv u),\qquad
u,v\in V.$

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
$\mathrm{Perm}$-algebra $P$ the operation

$\displaystyle[(p\otimes u)_{(\lambda)}(q\otimes
v)]=pq\otimes(u_{(\lambda)}v)-qp\otimes(v_{(-\partial-\lambda)}u)$
$\displaystyle\qquad{}=pq\otimes(\partial(v\dashv u)+\lambda(u\vdash v+v\dashv
u))-qp\otimes(\partial(u\dashv v)-(\lambda+\partial)(v\vdash u+u\dashv v))$
$\displaystyle\qquad{}=\partial(qp\otimes v\vdash u+pq\otimes v\dashv
u)+\lambda(pq\otimes u\vdash v+qp\otimes u\dashv v+qp\otimes v\vdash
u+pq\otimes v\dashv u)$

is exactly the quadratic Lie conformal algebra structure [23] on
$\Bbbk[\partial]\otimes P\otimes V$ corresponding to the Novikov algebra
$P\otimes V$:

$[x_{(\lambda)}y]=\partial(yx)+\lambda(xy+yx)$

for $x=p\otimes u$, $y=q\otimes v$, and the product is given by (4.3).

Hence the construction of a $\mathrm{pre}\mathrm{Lie}$ conformal algebra from
a $D\mathrm{Zinb}$-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 $\mathrm{pre}\mathrm{Nov}$-algebras.
Although the defining identities of $\mathrm{pre}\mathrm{Nov}$ 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 $\Bbbk$ is not $2$ or $3$ then there
exists a $D\mathrm{Zinb}$-algebra which cannot be embedded into a differential
Zinbiel algebra.

###### Proof.

Consider the free Zinbiel algebra $F$ generated by

$\\{a,b\\}^{(\omega)}=\bigl{\\{}a,b,a^{\prime},b^{\prime},\dots,a^{(n)},b^{(n)},\dots\bigr{\\}}.$

This is the free differential Zinbiel algebra with two generators $a$, $b$,
its derivation $d$ maps $x^{(n)}$ to $x^{(n+1)}$ for $x\in\\{a,b\\}$. The
product of two elements $f,g\in F$ is denoted $f\cdot g$.

For every $f,g\in F$, define $f\prec g$, $f\succ g$ by the rule (2.3):

$f\prec g=f\cdot d(g),\qquad f\succ g=d(f)\cdot g.$

Then $(F,\prec,\succ)$ is a $D\mathrm{Zinb}$-algebra, and its subalgebra
generated by $a$, $b$ is isomorphic to the free algebra $D\mathrm{Zinb}\langle
a,b\rangle$.

Denote $f=b\prec b=b\cdot b^{\prime}\in D\mathrm{Zinb}\langle
a,b\rangle\subset F$, and let $J$ stand for the ideal in $F$ generated by $f$
and all its derivatives:

$J=\bigl{(}f,f^{\prime},f^{\prime\prime},\dots\bigr{)}\triangleleft F,\qquad
d(J)\subseteq J.$

In particular,

$h=a\cdot\bigl{(}f^{\prime}\cdot b^{\prime}\bigr{)}-a\cdot\bigl{(}f\cdot
b^{\prime\prime}\bigr{)}\in J.$

Let us show that $h\in D\mathrm{Zinb}\langle a,b\rangle\subset J$. Indeed,

$\displaystyle h$ $\displaystyle=a\cdot\bigl{(}\bigl{(}b\cdot
b^{\prime}\bigr{)}^{\prime}\cdot
b^{\prime}\bigr{)}-a\cdot\bigl{(}\bigl{(}b\cdot b^{\prime}\bigr{)}\cdot
b^{\prime\prime}\bigr{)}$ $\displaystyle=a\cdot\bigl{(}\bigl{(}b^{\prime}\cdot
b^{\prime}\bigr{)}\cdot b^{\prime}\bigr{)}+a\cdot\bigl{(}\bigl{(}b\cdot
b^{\prime\prime}\bigr{)}\cdot b^{\prime}\bigr{)}-a\cdot\bigl{(}\bigl{(}b\cdot
b^{\prime}\bigr{)}\cdot
b^{\prime\prime}\bigr{)}=a\cdot\bigl{(}\bigl{(}b^{\prime}\cdot
b^{\prime}\bigr{)}\cdot b^{\prime}\bigr{)}$

due to the right commutativity of Zinbiel algebras. Next, $(b^{\prime}\cdot
b^{\prime})\cdot b^{\prime}=2b^{\prime}\cdot(b^{\prime}\cdot b^{\prime})$, so
$(b^{\prime}\cdot b^{\prime})\cdot b^{\prime}+b^{\prime}\cdot(b^{\prime}\cdot
b^{\prime})=\frac{3}{2}(b^{\prime}\cdot b^{\prime})\cdot b^{\prime}$.
Therefore,

$\displaystyle h$ $\displaystyle=a\cdot\bigl{(}\bigl{(}b^{\prime}\cdot
b^{\prime}\bigr{)}\cdot
b^{\prime}\bigr{)}=\frac{2}{3}a\cdot\bigl{(}\bigl{(}b^{\prime}\cdot
b^{\prime}\bigr{)}\cdot b^{\prime}+b^{\prime}\cdot\bigl{(}b^{\prime}\cdot
b^{\prime}\bigr{)}\bigr{)}=\frac{2}{3}\bigl{(}a\cdot\bigl{(}b^{\prime}\cdot
b^{\prime}\bigr{)}\bigr{)}\cdot b^{\prime}$
$\displaystyle=\frac{1}{3}\bigl{(}\bigl{(}a\cdot b^{\prime}\bigr{)}\cdot
b^{\prime}\bigr{)}\cdot b^{\prime}=\frac{1}{3}((a\prec b)\prec b)\prec
b=2\bigl{[}ab^{\prime}b^{\prime}b^{\prime}\bigr{]}\in D\mathrm{Zinb}\langle
a,b\rangle.$

As in Example 2.3, we denote by $[x_{1}x_{2}\cdots x_{n-1}x_{n}]$ the
following expression in a Zinbiel algebra:

$[x_{1}x_{2}\cdots x_{n-1}x_{n}]=x_{1}\cdot(x_{2}\cdot(\cdots(x_{n-1}\cdot
x_{n})\cdots)).$

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 $\mathrm{pre}\mathrm{Com}\langle X\rangle$).

Let $I$ be the ideal in $D\mathrm{Zinb}\langle a,b\rangle$ generated by $f$,
and let $V=D\mathrm{Zinb}\langle a,b\mid f\rangle=D\mathrm{Zinb}\langle
a,b\rangle/I$. Then $F/J$ is the universal differential Zinbiel envelope of
$V$. To prove the theorem, it remains to show that $h\notin I$: if so then
$h+I$ would lie in the kernel of every homomorphism from $V$ to the derived
algebra $Z^{(d)}$ constructed from a differential Zinbiel algebra $Z$ with a
derivation $d$.

Assume $[ab^{\prime}b^{\prime}b^{\prime}]\in I$. The specific of the Zinbiel
identity (2.1) is that the first letter remains unchanged in all terms. Hence,
$[ab^{\prime}b^{\prime}b^{\prime}]$ should be a linear combination of the
elements $(a\ast f\star b)$, $(a\ast b\star f)$, where
$\ast,\star\in\\{\prec,\succ\\}$, with two possible bracketing each, so we
have in total 16 terms under consideration. Let us write them all in the
normal form in $F$:

$\displaystyle(a\prec f)\prec
b=3\bigl{[}ab^{\prime}b^{\prime}b^{\prime}\bigr{]}+\bigl{[}ab^{\prime}bb^{\prime\prime}\bigr{]}+\bigl{[}abb^{\prime}b^{\prime\prime}\bigr{]}+\bigl{[}abb^{\prime\prime}b^{\prime}\bigr{]},$
$\displaystyle(a\prec f)\succ
b=\bigl{[}a^{\prime}bb^{\prime}b^{\prime}\bigr{]}+\bigl{[}a^{\prime}b^{\prime}bb^{\prime}\bigr{]}+\bigl{[}a^{\prime}b^{\prime}b^{\prime}b\bigr{]}+2\bigl{[}a^{\prime}bbb^{\prime\prime}\bigr{]}+\bigl{[}a^{\prime}bb^{\prime\prime}b\bigr{]}+\bigl{[}abb^{\prime\prime}b^{\prime}\bigr{]}+\bigl{[}ab^{\prime\prime}bb^{\prime}\bigr{]}$
$\displaystyle\hphantom{(a\prec f)\succ
b=}{}+\bigl{[}ab^{\prime\prime}b^{\prime}b\bigr{]}+2\bigl{[}abb^{\prime}b^{\prime\prime}\bigr{]}+2\bigl{[}ab^{\prime}bb^{\prime\prime}\bigr{]}+2\bigl{[}ab^{\prime}b^{\prime\prime}b\bigr{]}+2\bigl{[}abbb^{\prime\prime\prime}\bigr{]}+\bigl{[}abb^{\prime\prime\prime}b\bigr{]},$
$\displaystyle(a\succ f)\prec
b=\bigl{[}a^{\prime}b^{\prime}bb^{\prime}\bigr{]}+2\bigl{[}a^{\prime}bb^{\prime}b^{\prime}\bigr{]},$
$\displaystyle(a\succ f)\succ
b=2\bigl{[}a^{\prime\prime}bbb^{\prime}\bigr{]}+\bigl{[}a^{\prime\prime}bb^{\prime}b\bigr{]}+\bigl{[}a^{\prime}bb^{\prime}b^{\prime}\bigr{]}+\bigl{[}a^{\prime}b^{\prime}bb^{\prime}\bigr{]}+\bigl{[}a^{\prime}b^{\prime}b^{\prime}b\bigr{]}+2\bigl{[}a^{\prime}bbb^{\prime\prime}\bigr{]}+\bigl{[}a^{\prime}bb^{\prime\prime}b\bigr{]},$
$\displaystyle a\prec(f\prec
b)=2\bigl{[}ab^{\prime}b^{\prime}b^{\prime}\bigr{]}+2\bigl{[}abb^{\prime\prime}b^{\prime}\bigr{]}+2\bigl{[}abb^{\prime}b^{\prime\prime}\bigr{]},$
$\displaystyle a\prec(f\succ
b)=a\bigl{[}b^{\prime}b^{\prime}b\bigr{]}^{\prime}+a\bigl{[}b^{\prime}bb^{\prime}\bigr{]}^{\prime}+a\bigl{[}bb^{\prime\prime}b\bigr{]}^{\prime}+a\bigl{[}bbb^{\prime\prime}\bigr{]}^{\prime}=\bigl{[}ab^{\prime\prime}b^{\prime}b\bigr{]}+2\bigl{[}ab^{\prime}b^{\prime\prime}b\bigr{]}+2\bigl{[}ab^{\prime}b^{\prime}b^{\prime}\bigr{]}$
$\displaystyle\hphantom{a\prec(f\succ
b)=}{}+\bigl{[}ab^{\prime\prime}bb^{\prime}\bigr{]}+2\bigl{[}ab^{\prime}bb^{\prime\prime}\bigr{]}+\bigl{[}abb^{\prime\prime\prime}b\bigr{]}+\bigl{[}abb^{\prime\prime}b^{\prime}\bigr{]}+\bigl{[}abb^{\prime}b^{\prime\prime}\bigr{]}+\bigl{[}abbb^{\prime\prime\prime}\bigr{]},$
$\displaystyle a\succ(f\prec
b)=2\bigl{[}a^{\prime}bb^{\prime}b^{\prime}\bigr{]},$ $\displaystyle
a\succ(f\succ
b)=\bigl{[}a^{\prime}b^{\prime}b^{\prime}b\bigr{]}+\bigl{[}a^{\prime}b^{\prime}bb^{\prime}\bigr{]}+\bigl{[}a^{\prime}bb^{\prime\prime}b\bigr{]}+\bigl{[}a^{\prime}bbb^{\prime\prime}\bigr{]},$
$\displaystyle(a\prec b)\prec
f=3\bigl{[}ab^{\prime}b^{\prime}b^{\prime}\bigr{]}+\bigl{[}ab^{\prime}bb^{\prime\prime}\bigr{]}+\bigl{[}abb^{\prime}b^{\prime\prime}\bigr{]}+\bigl{[}abb^{\prime\prime}b^{\prime}\bigr{]},$
$\displaystyle(a\prec b)\succ
f=\bigl{[}a^{\prime}b^{\prime}bb^{\prime}\bigr{]}+2\bigl{[}a^{\prime}bb^{\prime}b^{\prime}\bigr{]}+\bigl{[}ab^{\prime\prime}bb^{\prime}\bigr{]}+\bigl{[}abb^{\prime\prime}b^{\prime}\bigr{]}+\bigl{[}abb^{\prime}b^{\prime\prime}\bigr{]},$
$\displaystyle(a\succ b)\prec
f=\bigl{[}a^{\prime}bb^{\prime}b^{\prime}\bigr{]}+\bigl{[}a^{\prime}b^{\prime}bb^{\prime}\bigr{]}+\bigl{[}a^{\prime}b^{\prime}b^{\prime}b\bigr{]}+2\bigl{[}a^{\prime}bbb^{\prime\prime}\bigr{]}+\bigl{[}a^{\prime}bb^{\prime\prime}b\bigr{]},$
$\displaystyle(a\succ b)\succ
f=2\bigl{[}a^{\prime\prime}bbb^{\prime}\bigr{]}+\bigl{[}a^{\prime\prime}bb^{\prime}b\bigr{]}+\bigl{[}a^{\prime}b^{\prime}bb^{\prime}\bigr{]}+2\bigl{[}a^{\prime}bb^{\prime}b^{\prime}\bigr{]},$
$\displaystyle a\prec(b\prec
f)=\bigl{[}ab^{\prime}b^{\prime}b^{\prime}\bigr{]}+\bigl{[}abb^{\prime\prime}b^{\prime}\bigr{]}+2\bigl{[}abb^{\prime}b^{\prime\prime}\bigr{]}+\bigl{[}ab^{\prime}bb^{\prime\prime}\bigr{]}+\bigl{[}abbb^{\prime\prime\prime}\bigr{]},$
$\displaystyle a\prec(b\succ
f)=\bigl{[}ab^{\prime\prime}bb^{\prime}\bigr{]}+\bigl{[}ab^{\prime}b^{\prime}b^{\prime}\bigr{]}+\bigl{[}ab^{\prime}bb^{\prime\prime}\bigr{]},$
$\displaystyle a\succ(b\prec
f)=\bigl{[}a^{\prime}bb^{\prime}b^{\prime}\bigr{]}+\bigl{[}a^{\prime}bbb^{\prime\prime}\bigr{]},$
$\displaystyle a\succ(b\succ
f)=\bigl{[}a^{\prime}b^{\prime}bb^{\prime}\bigr{]}.$

Arrange the normal Zinbiel words in the following order:
$[ab^{\prime}b^{\prime}b^{\prime}]$, $[ab^{\prime}bb^{\prime\prime}]$,
$[abb^{\prime}b^{\prime\prime}]$, $[abb^{\prime\prime}b^{\prime}]$,
$[ab^{\prime\prime}bb^{\prime}]$, $[ab^{\prime\prime}b^{\prime}b]$,
$[ab^{\prime}b^{\prime\prime}b]$, $[abbb^{\prime\prime\prime}]$,
$[abb^{\prime\prime\prime}b]$, $[a^{\prime\prime}bb^{\prime}b]$,
$[a^{\prime\prime}bbb^{\prime}]$, $[a^{\prime}bb^{\prime}b^{\prime}]$,
$[a^{\prime}b^{\prime}bb^{\prime}]$, $[a^{\prime}b^{\prime}b^{\prime}b]$,
$[a^{\prime}bbb^{\prime\prime}]$, $[a^{\prime}bb^{\prime\prime}b]$. Then the
assumption $[ab^{\prime}b^{\prime}b^{\prime}]\in I$ is equivalent to the
condition that the row vector $e_{1}=(1,0,\dots,0)\in\Bbbk^{16}$ belongs to
the row space of the matrix

$\left(\begin{array}[]{cccccccccccccccc}3&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&2&2&1&1&1&2&2&1&0&0&1&1&1&2&1\\\ 0&0&0&0&0&0&0&0&0&0&0&2&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&1&2&1&1&1&2&1\\\ 2&0&2&2&0&0&0&0&0&0&0&0&0&0&0&0\\\
2&2&1&1&1&1&2&1&1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&2&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1\\\ 3&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&1&1&1&0&0&0&0&0&0&2&1&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&1&1&1&2&1\\\
0&0&0&0&0&0&0&0&0&1&2&2&1&0&0&0\\\ 1&1&2&1&0&0&0&1&0&0&0&0&0&0&0&0\\\
1&1&0&0&1&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&1&0&0&1&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\end{array}\right).$

This matrix may be transformed by elementary row and column transformations
with integer coefficients to the triangular form with $1$, $2$, or $0$ on the
diagonal, so that its rank equals 10 for $\operatorname{char}\Bbbk\neq 2$.
Adding one more row $e_{1}$ increases the rank, so
$[ab^{\prime}b^{\prime}b^{\prime}]\notin I$. ∎

As a corollary, we obtain the following observation which is in some sense
converse to Corollary 3.6. Suppose $W$ is a Novikov algebra equipped with a
Rota–Baxter operator, that is, $R\colon W\to W$ is a linear operator such that

$R(u)R(v)=R(uR(v)+R(u)v),\qquad u,v\in W.$

Then, in general, there is no commutative algebra $(A,*)$ with a derivation
$d$ and a Rota–Baxter operator $\rho$ such that $W\subseteq A$, $uv=u*d(v)$,
$R(u)=\rho(u)$, for $u,v\in W$, and $\rho d=d\rho$. 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 $(A,*,d,\rho)$ exists for every Novikov algebra
with a Rota–Baxter operator. Every pre-Novikov algebra $V$ with operations
$\vdash$ and $\dashv$ can be embedded into a Novikov algebra $W=\widehat{V}$
with a Rota–Baxter operator so that $u\vdash v=R(u)v$, $u\dashv v=uR(v)$,
$u,v\in V$ (see, e.g., [13]). We may further embed this $W$ into a
differential commutative Rota–Baxter algebra $(A,*,d,\rho)$ in which $a\vdash
b=\rho(a)*d(b)$, $a\dashv b=a*d(\rho(b))=a*\rho(d(b))$, for $a,b\in W$. On the
other hand, the new operation $\cdot$ on $A$ given by $a\cdot b=a*\rho(b)$
turns $A$ into a Zinbiel algebra, $d$ remains a derivation relative to this
new operation, and $u\vdash v=d(v)\cdot u=v\succ u$, $u\dashv v=u\cdot
d(v)=u\prec v$ for $u,v\in V$. Therefore, we would embed a
$D\mathrm{Zinb}$-algebra into a differential Zinbiel algebra which is not the
case.

### Acknowledgments

F. Mashurov and B. Sartayev were supported by the Science Committee of the
Ministry of Education and Science of the Republic of Kazakhstan (Grant No.
AP14870282). P. Kolesnikov was supported by the Program of Fundamental
Research RAS (project FWNF-2022-0002). The authors are grateful to the
referees for useful comments.

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toltxlabel

# Two-step targeted minimum-loss based estimation for non-negative two-part
outcomes

Nicholas T. Williams  Department of Epidemiology, Mailman School of Public
Health, Columbia University. Richard Liu  Division of Biostatistics,
Department of Population Health, New York University Grossman School of
Medicine.  Katherine L. Hoffman  Department of Epidemiology, Mailman School of
Public Health, Columbia University. Sarah Forrest  Department of Epidemiology,
Mailman School of Public Health, Columbia University. Kara E. Rudolph KER and
ID are both senior authors Department of Epidemiology, Mailman School of
Public Health, Columbia University. Iván Díaz∗ Division of Biostatistics,
Department of Population Health, New York University Grossman School of
Medicine.

###### Abstract

Non-negative two-part outcomes are defined as outcomes with a density function
that have a zero point mass but are otherwise positive. Examples, such as
healthcare expenditure and hospital length of stay, are common in healthcare
utilization research. Despite the practical relevance of non-negative two-part
outcomes, very few methods exist to leverage knowledge of their semicontinuity
to achieve improved performance in estimating causal effects. In this paper,
we develop a nonparametric two-step targeted minimum-loss based estimator
(denoted as hTMLE) for non-negative two-part outcomes. We present methods for
a general class of interventions referred to as modified treatment policies,
which can accommodate continuous, categorical, and binary exposures. The two-
step TMLE uses a targeted estimate of the intensity component of the outcome
to produce a targeted estimate of the binary component of the outcome that may
improve finite sample efficiency. We demonstrate the efficiency gains achieved
by the two-step TMLE with simulated examples and then apply it to a cohort of
Medicaid beneficiaries to estimate the effect of chronic pain and physical
disability on days’ supply of opioids.

##  1 Introduction

Non-negative, right-skewed variables with a point mass at zero are commonly
encountered in health research (e.g., healthcare expenditures (Smith et al.,
2014), hospital length of stay (Xie et al., 2004), daily alcohol consumption
(Olsen and Schafer, 2001; Liu et al., 2016)). These outcomes can often be
characterized as arising from two distinct processes: the first controlling
whether an observation is zero (the binary component), and the second
controlling the value conditional on the observation not being zero (the
intensity component). Consider, for example, the days supply of opioids
prescribed to a patient. The binary component controls whether or not a
patient receives a prescription for opioids, while the intensity component
controls the days supply of opioids prescribed conditional on the patient
receiving an opioid prescription.

Approaches to modeling non-negative two-part outcomes include only modeling
the binary process or restricting the analysis to observations where the
outcome is greater than zero — both of which may lead to erroneous
conclusions. For instance, an exposure might not affect the binary component
but could influence the intensity component. Consider a hypothetical scenario
where there is no difference in the probability of receiving an opioid
prescription between patients with physical disability and those with chronic
pain. Nonetheless, among the subset of patients who do receive an opioid
prescription, individuals with chronic pain are more likely to be prescribed a
greater supply of opioids than those with physical disability. This effect
would be overlooked if receiving an opioid prescription or not was the sole
outcome considered. Conversely, if we limit the analysis to patients who
received an opioid prescription, we are assuming that chronic pain and
physical disability have no effect on the likelihood of receiving an opioid
prescription; our effect estimate may be biased if this assumption does not
hold true.

Instead, the standard approach for modeling non-negative two-part data is with
parametric two-part mixture models, often referred to as hurdle models, with
the binary component modeled using logistic or probit regression and the
intensity component modeled using linear regression on the $\log$ scale (Duan
et al., 1983; Smith et al., 2014). However, this approach results in two sets
of covariate coefficients; if the goal is to conduct causal inference on the
marginal mean, parametric hurdle models do not yield a simple interpretation
of causal effects. In response, Smith et al. (2014) developed a parametric
marginalized two-part model that yields more interpretable estimates but still
relies on correct parametric specification. More recently, several non-
parametric estimators for causal effects with non-negative two-part outcomes
have been proposed. Oganisian et al. (2021) developed a non-parametric
Bayesian estimator and demonstrated its improved performance compared to a
parametric hurdle model and other non-parametric estimators that do not model
the outcome as two parts. Huling et al. (2021) developed a non-parametric
estimator for individual treatment rules with semicontinuous outcomes.

We add to this recent literature by proposing a crossfitted, two-step targeted
minimum-loss based estimator (TMLE) for non-negative two-part outcomes. Our
approach is non-parametric, doubly-robust, and capable of incorporating data-
adaptive estimators of nuisance parameters while achieving $n^{1/2}$
consistency. Improving upon other estimators for causal effects for non-
negative two-part outcomes, we define our proposed estimator in terms of
modified treatment policies, allowing effects to be estimated for binary,
categorical, and continuous treatments/exposures (Haneuse and Rotnitzky, 2013;
Young et al., 2014; Díaz and van der Laan, 2012; Díaz et al., 2023).

In Section 2, we define notation, the target parameter, and its identification
result; in Section 3, we present the efficiency bound in the non-parametric
model; in Section 4, we discuss the proposed two-step TMLE, followed by the
results of a brief numerical study in Section 5. Lastly, as a motivating
example, in Section 6, we apply our method to estimate the associations
between having a physical disability and/or chronic pain condition vs. neither
at the time of Medicaid enrollment on the number of days prescribed an opioid
during the subsequent 18 months following enrollment among non-elderly, adult
Medicaid beneficiaries.

##  2 Notation and Definition of the Target Parameter

Let $O_{1},...,O_{n}$ denote $n$ i.i.d. observations of observed data
$O=(X,T,Y=\Delta S)$, drawn from some distribution $P$, where $X$ denotes a
vector of observed covariates that precede a binary, categorical, or
continuous exposure $T$, and a non-negative outcome $Y=\Delta S$ such that
$\Delta=\mathbbm{1}(Y>0)$ and $S$ is the value of $Y$ if $Y>0$. Let
$\mathsf{g}(t,X)$ denote $\mathsf{P}(T=t\mid X)$; $\mathsf{q}(t,X)$ denote
$\mathsf{P}(\Delta=1|T=t,X)$; and $\mathsf{m}(t,X)$ denote $\mathsf{E}\\{Y\mid
T=t,X,\Delta=1\\}$. Note that $\mathsf{E}\\{Y\mid T,X\\}=\mathsf{E}\\{Y\mid
T,X,\Delta=1\\}\mathsf{P}(\Delta=1\mid T,X)+\mathsf{E}\\{Y\mid
T,X,\Delta=0\\}\mathsf{P}(\Delta=0\mid T,X)$ and that $\mathsf{E}\\{Y\mid
T,X,\Delta=0\\}=0$. Thus, $\mathsf{E}\\{Y\mid T,X\\}=\mathsf{E}\\{Y\mid
T,X,\Delta=1\\}\mathsf{P}(\Delta=1\mid T,X)=\mathsf{m}(T,X)\mathsf{q}(T,X)$.

We formalize the definition of causal effects using a non-parametric
structural equation model (NPSEM): assume the existence of deterministic
functions $f_{X}$, $f_{T}$, and $f_{Y}$ such that $X=f_{X}(U_{X})$;
$T=f_{T}(X,U_{T})$; $Y=f_{Y}(X,T,U_{Y})$; and $U=(U_{X},U_{T},U_{Y})$ is a
vector of exogenous variables. In what follows, the interventions of interest
will be characterized by a function $\mathsf{d}(t,x,\epsilon)$ that maps $T$,
$X$, and potentially a randomizer $\epsilon$ to a new value of $T$. Our target
parameter is $\psi=\mathsf{E}\\{Y(T^{\mathsf{d}})\\}$; the variable
$Y(T^{\mathsf{d}})$ is the counterfactual value of $Y$ that would be observed
if the structural equation model for the exposure, $f_{T}$, was replaced with
the output of the user-defined function $\mathsf{d}$, i.e., we define
$T^{\mathsf{d}}=\mathsf{d}(T,X,\epsilon)$ and
$Y(T^{\mathsf{d}})=f_{Y}(X,T^{\mathsf{d}},U_{Y})$. Defining our target
parameter in terms of $\mathsf{d}(t,x,\epsilon)$ allows us to estimate a
variety of interventions from the causal inference literature, including
static and dynamic interventions; stochastic interventions, such as
incremental propensity score interventions; and modified treatment policies
(Díaz et al., 2023; Hoffman et al., 2023a). For example, if $T$ is a binary
exposure and $\mathsf{d}(t,X,\epsilon)=1$ (representing a static intervention
of $T=1$) then our target parameter would be the expected value of $Y$ if all
members in the population received treatment. We refer readers to Hoffman et
al. (2023a) for further background.

We make the following set of standard assumptions for estimating the causal
parameter $\mathsf{E}\\{Y(T^{\mathsf{d}})\\}$ from observational data:

######  Assumption 1Positivity.

If $(t,x)\in\text{supp}(T,X)$ then
$(\mathsf{d}(t,x,\epsilon),x))\in\text{supp}(T,X)$.

######  Assumption 2No Unmeasured Confounding.

$T\perp Y(t)\mid X$ $\forall$ $t$.

Under assumptions 1 and 2, $\mathsf{E}\\{Y(T^{\mathsf{d}})\\}$ is identified
from observed data and equal to

$\psi=\int\mathsf{m}^{\mathsf{d}}(t,x)\mathsf{q}^{\mathsf{d}}(t,x)\mathsf{p}(t,x)d\nu(x),$
(1)

where $\mathsf{m}^{\mathsf{d}}(T,X)=\mathsf{m}(T^{\mathsf{d}},X)$ and
$\mathsf{q}^{\mathsf{d}}(T,X)=\mathsf{q}(T^{\mathsf{d}},X$).

##  3 Efficient Influence Function

Parametric model-based estimators rely on the assumption that data were
generated from the specified parametric distribution. In contrast,
semiparametric estimators make few, or possibly, no assumptions about the
underlying distribution. Key to constructing semiparametric estimators is the
efficient influence function (EIF).

Define $\mathsf{g}(T,X)$ as the density function of $T$ conditional on $X$. We
can then specify the density ratio
$\mathsf{r}(t,X)=\frac{\mathsf{g}^{\mathsf{d}}(T,X)}{\mathsf{g}(T,X)}$, where
$\mathsf{g}^{\mathsf{d}}(T,X)$ is the post-intervention density function of
$T$. From standard results (Díaz and van der Laan, 2012; Díaz et al., 2023),
the EIF for $\psi$ is equal to

$D_{\eta,\psi}(O)=\mathsf{r}(T,X)\\{Y-\mathsf{q}(T,X)\mathsf{m}(T,X)\\}+\mathsf{q}^{\mathsf{d}}(T,X)\mathsf{m}^{\mathsf{d}}(T,X)-\psi,$
(2)

where we have denoted $\eta=(\mathsf{m},\mathsf{q},\mathsf{r})$ to represent
the set of nuisance parameters. The EIF can alternatively be represented as

$D_{\eta,\psi}(O)=\mathsf{r}(T,X)\mathsf{m}(T,X)(\Delta-\mathsf{q}(T,X))+\Delta\mathsf{r}(T,X)(S-\mathsf{m}(T,X))+\mathsf{q}^{\mathsf{d}}(T,X)\mathsf{m}^{\mathsf{d}}(T,X)-\psi.$
(3)

The proof is provided in the supplementary materials.

##  4 Targeted minimum-loss based estimation

We propose a two-step targeted minimum-loss based estimator (TMLE),
$\hat{\psi}_{\text{hTMLE}}$, for $\psi$. The targeted minimum-loss based
estimator is a substitution estimator that uses estimates of $\hat{\eta}$ that
are constructed to solve the efficient influence function estimating equation

$n^{-1}\sum_{i=1}^{n}D_{\hat{\eta},h\hat{\psi}_{\text{hTMLE}}}(O_{i})=0.$

We propose using a two-step TMLE since it may improve finite sample efficiency
compared to standard TMLE. We use cross-fitting to avoid imposing entropy
conditions on the estimates $\hat{\eta}$. (Klaassen, 1987; Zheng and van der
Laan, 2011; Chernozhukov et al., 2018). Let ${\cal V}_{1},\ldots,{\cal V}_{J}$
denote a random partition of data with indices $i\in\\{1,\ldots,n\\}$ into $J$
prediction sets of approximately the same size such that
$\bigcup_{j=1}^{J}{\cal V}_{j}=\\{1,\ldots,n\\}$. For each $j$, the training
sample is given by ${\cal T}_{j}=\\{1,\ldots,n\\}\setminus{\cal V}_{j}$.
$\hat{\eta}_{j}$ denotes the estimator of $\eta$, obtained by training the
corresponding prediction algorithm using only data in the sample ${\cal
T}_{j}$, and $j(i)$ denotes the index of the validation set which contains
observation $i$. The algorithm is as follows:

  1. 1.

Initialize.

The EIF $D_{\eta\psi}(O)$ is a function of the nuisance parameters
$\eta=\\{\mathsf{r},\mathsf{q},\mathsf{m}\\}$, all of which may be estimated
by any appropriate regression method in the statistics and machine learning
literature. For example, $\mathsf{m}^{\mathsf{d}}$ may be estimated by
regressing $Y_{i}$ on $T_{i}$ and $X_{i}$ on the subset of observations where
$Y_{i}>0$ and generating predictions for all observations with $T_{i}$ set to
the output of $\mathsf{d}(T_{i},X_{i},\epsilon)$. We use density ratio
estimation by classification to estimate $\hat{\mathsf{r}}$ (Qin, 1998; Cheng
and Chu, 2004; Díaz et al., 2023).

  2. 2.

Fit the tilting model:

$\operatorname{logit}\hat{\mathsf{m}}^{\hat{\epsilon}_{\mathsf{m}}}(T_{i},X_{i})=\hat{\epsilon}_{\mathsf{m}}+\operatorname{logit}\hat{\mathsf{m}}(T_{i},X_{i}).$
(4)

Fit an intercept-only logistic regression of $Y_{i}$ with offset equal to
$\text{logit }\hat{\mathsf{m}}(T_{i},X_{i})$ and weights
$\hat{\mathsf{r}}_{i}$ among the subset of observations where $Y_{i}>0$.

  3. 3.

Update
$\hat{\mathsf{m}}(T_{i},X_{i})=\mathsf{m}^{\hat{\epsilon}_{\mathsf{m}}}(T_{i},X_{i})$.

  4. 4.

Fit the tilting model:

$\operatorname{logit}\hat{\mathsf{q}}^{\hat{\epsilon}_{\mathsf{q}}}(T_{i},X_{i})=\hat{\epsilon}_{\mathsf{q}}+\operatorname{logit}\hat{\mathsf{q}}(T_{i},X_{i}).$
(5)

Using logistic regression, regress $\mathbbm{1}(Y_{i}>0)$ on
$\operatorname{logit}\hat{\mathsf{q}}(T_{i},X_{i})$ and weights
$\hat{\mathsf{r}}_{i}(T_{i},X_{i})\times\hat{\mathsf{m}}^{\hat{\epsilon}_{\mathsf{m}}}(T_{i},X_{i})$.

  5. 5.

Update
$\hat{\mathsf{q}}(T_{i},X_{i})=\hat{\mathsf{q}}^{\hat{\epsilon}_{\mathsf{q}}}(T_{i},X_{i})$.

  6. 6.

The estimate of $\hat{\psi}_{\text{hTMLE}}$ is equal to
$\frac{1}{n}\sum_{i=1}^{n}\hat{\mathsf{q}}^{\mathsf{d},\hat{\epsilon}_{\mathsf{q}}}(T_{i},X_{i})\hat{\mathsf{m}}^{\mathsf{d},\hat{\epsilon}_{\mathsf{m}}}(T_{i},X_{i})$.

The variance of $\hat{\psi}_{\text{hTMLE}}$ can be calculated as the empirical
variance of Equation 2. However, in finite samples, using the empirical
variance of the estimated influence function as a variance estimator for
substitution-based estimators, such as $\psi_{\text{hTMLE}}$, often results in
anti-conservative variance estimates. We instead estimate the variance of
$\psi_{\text{hTMLE}}$ using a non-parametric bootstrap of steps 2-6, such that
the variance of $\hat{\psi}_{\text{hTMLE}}$ is equal to the variance of
$b=1,2,...,B$ bootstrapped estimates of $\hat{\psi}_{\text{hTMLE}}$ (Tran et
al., 2023).

The two-step TMLE is doubly-robust to nuisance parameter model
misspecification; it is possible to construct a consistent estimator of
$\psi_{\text{hTMLE}}$ if $\hat{\mathsf{r}}$, or
$(\hat{\mathsf{q}},\hat{\mathsf{m}})$ is consistently estimated. A software
implementation of the algorithm is available for download from GitHub as an R
package.

##  5 Monte Carlo Unit Testing

We performed a brief simulation study to illustrate the performance of the
two-step TMLE and to ensure correct implementation. We conducted 1000
simulations for samples size $n\in(200,500,1000,5000)$ and compared the
performance of the proposed estimator to two other semiparametric estimators
commonly used in the causal inference literature: standard TMLE, and augmented
inverse probability weight (AIPW); all estimators were estimated using cross-
fitting with 10-folds. The target parameter was defined as the mean
counterfactual value of $Y$ if $T=1$: $\mathsf{d}(t,X,\epsilon)=1$. Estimator
performance was evaluated in terms of absolute bias, Monte Carlo variance,
mean squared error (MSE), and nominal 95% confidence interval coverage. We
considered the following data generating mechanism:

$X=(X_{1},X_{2},X_{3},X_{4})^{\top}$ | $\sim N(0,I_{4})$
---|---
$P(T=1\mid X)$ | $=\operatorname{logit}^{-1}(\beta_{p}-X_{1}+0.5X_{2}-0.25X_{3}-0.1X_{4})$
$P(\Delta=1\mid T,X)$ | $=\operatorname{logit}^{-1}(-0.4X_{1}^{2}+0.1X_{2}+0.8X_{3}-0.3X_{4}+2T)$
$U$ | $\sim\operatorname{exp}(\lambda=1)$
$S\mid T,X$ | $\sim e^{(0.1+0.2X_{1}+0.4X_{2}+0.8X_{3}+0.3X_{4}+2T)}+U$
$Y$ | $=\Delta S$

where $\beta_{p}$ controls positivity. Data were generated under two
parameterizations of $\beta_{p}$: -3, severe positivity violations; and 0,
minimal positivity violations.

Nuisance parameters were fit using an ensemble of a main-effects GLM and
multivariate adaptive regression splines (Friedman, 1991) using the super
learner algorithm (van der Laan et al., 2007). The super learner optimally
combines predictions from a library of candidate algorithms.

###  5.1 Results

Simulation results are shown in Table 1. In general, estimator performance
improved as sample size increased. Regardless of the degree of severity of the
practical positivity violations, TMLE and AIPW remained more biased than our
proposed estimator hTMLE. The discrepancy in bias becomes more pronounced with
increasing severity of practical positivity violations. This is expected and
it illustrates the importance of using separate models for the point-mass and
for the continuous part of the model.

The two-step TMLE outperformed TMLE and AIPW in terms of MSE and variance
across all scenarios. When data were generated with severe positivity
violations, all estimators exhibited larger variance, MSE, and suffered from
less than nominal confidence interval coverage. However, only the two-step
TMLE recovered in terms of nominal confidence interval coverage with
increasing sample size; for $n=5000$, the absolute bias of the two-step TMLE
was $0.19$ and confidence interval coverage was $0.95$, compared to the
standard TMLE which has an absolute bias of $0.32$ and $0.86$ confidence
interval coverage.

Table 1: Simulation results for the two-step TMLE (hTMLE), TMLE, and augmented inverse probability weighting (AIPW) estimators. The target parameter is the expected value under a hypothetical intervention where all observations receive treatment, $\mathsf{E}\\{Y(T^{\mathsf{d}})\\}\approx 11.99$. $n$ |  | hTMLE | TMLE | AIPW
---|---|---|---|---
$\beta_{p}=0$ | $\beta_{p}=-3$ | $\beta_{p}=0$ | $\beta_{p}=-3$ | $\beta_{p}=0$ | $\beta_{p}=-3$
200 | $|\text{Bias}|$ | 0.01 | 0.12 | 0.15 | 0.29 | 0.11 | 1.15
Var. | 1.86 | 9.68 | 2.64 | 33.97 | 25.01 | 320.49
MSE | 1.86 | 9.68 | 2.66 | 34.02 | 24.99 | 321.44
Coverage | 0.92 | 0.88 | 0.91 | 0.90 | 0.92 | 0.89
500 | $|\text{Bias}|$ | 0.05 | 0.01 | 0.20 | 0.41 | 0.09 | 0.12
Var. | 0.69 | 4.27 | 0.94 | 15.98 | 0.99 | 39.26
MSE | 0.69 | 4.26 | 0.98 | 16.13 | 1.00 | 39.24
Coverage | 0.93 | 0.90 | 0.92 | 0.87 | 0.93 | 0.87
1000 | $|\text{Bias}|$ | 0.07 | 0.00 | 0.18 | 0.39 | 0.08 | 0.12
Var. | 0.30 | 2.22 | 0.39 | 5.78 | 0.43 | 10.01
MSE | 0.31 | 2.22 | 0.42 | 5.92 | 0.43 | 10.01
Coverage | 0.95 | 0.92 | 0.94 | 0.87 | 0.95 | 0.87
5000 | $|\text{Bias}|$ | 0.04 | 0.19 | 0.10 | 0.32 | 0.06 | 0.34
Var. | 0.07 | 0.46 | 0.09 | 1.02 | 0.09 | 1.07
MSE | 0.07 | 0.49 | 0.10 | 1.12 | 0.10 | 1.18
Coverage | 0.94 | 0.95 | 0.93 | 0.86 | 0.94 | 0.86

##  6 Illustrative application

We applied the two-step TMLE to estimate the association between physical
health status at the time of Medicaid enrollment and the number of days of
prescribed opioids among a cohort of 2,440,932 beneficiaries. Physical health
status at the time of Medicaid enrollment was operationalized as four mutually
exclusive categories defined during a 6-month look-back period: (1) physical
disability and co-occurring chronic pain (n = 6,717), (2) physical disability
without chronic pain (n = 50,999), (3) chronic pain without physical
disability (n = 77,696), and (4) neither physical disability nor chronic pain
(n = 2,305,520). Physical disability was identified based on meeting criteria
for Medicaid enrollment related to disability status and excluding alternative
qualifying disabilities, as described in Hoffman et al. (2023b). Chronic pain
status was identified by ICD-10 codes for non-cancer diagnoses typically
associated with chronic pain. A beneficiary met the criteria if they had two
codes for the same condition occurring during the 6-month look-back period
with a minimum separation of 90 days.

A rolling window of 6-months days supply of opioids was calculated by summing
the days supply that providers marked for the opioid prescription from the
Medicaid pharmacy files. The average of all 6-month rolling windows throughout
the study duration was taken for the outcome of average 6-month days supply of
opioids prescription. Days of opioids prescribed can be decomposed into two
distinct components: whether or not an observation was prescribed any opioids
and the number of days opioids were prescribed among those with a
prescription.

We considered the following variables as potential confounders: age in years
at the time of Medicaid enrollment, sex, race/ethnicity, English as the
primary language, marriage/partnership status, household size, veteran status,
income likely $>$133% of the Federal Poverty Level, any inpatient or
outpatient diagnosis of bipolar disorder, any anxiety disorder, attention
deficit hyperactivity disorder (ADHD), any depressive disorder, or other
mental disorders (e.g., anorexia, personality disorders).

Nuisance parameters were fit using the H2O AutoML algorithm (LeDell and
Poirier, 2020; Fryda et al., 2023).

Results are presented in Table 2. After adjusting for baseline confounders and
informative right censoring, the estimated average number of days of opioids
prescribed was 3.72 (95% CI: 3.69 to 3.75) among individuals without a
physical disability and co-occurring chronic pain, 9.78 (95% CI: 9.42 to
10.14) among individuals with physical disability and without co-occurring
chronic pain, 23.97 (95% CI: 23.53 to 24.41) among individuals with chronic
pain and without co-occurring physical disability, and 44.59 (95% CI: 41.59 to
47.54) among individuals with physical disability and co-occurring chronic
pain. Co-occurring chronic pain and physical disability resulted in 40.84 (95%
CI: 37.87 to 43.82) more days of prescribed opioids compared to having neither
chronic pain nor disability. Chronic pain alone resulted in a 20.26 (95% CI:
19.82 to 20.7) increase in the number of days of prescribed opioids compared
to neither, while physical disability alone resulted in a 6.06 (95% CI: 5.7 to
6.42) increase in the number of days of prescribed opioids compared to
neither.

Estimates were similar between hTMLE and standard TMLE, except for the
estimated average number of days of opioids prescribed among individuals with
physical disability and co-occurring chronic pain. It’s important to note that
the data do exhibit practical positivity violations. For example, conditional
on baseline covariates, the minimum estimated probability of having co-
occurring physical disability and chronic pain was 0.0000174. This may
partially explain the discrepancy in results between the hTMLE algorithm and
standard TMLE.

Table 2: Estimates of the expected number of days prescribed opioids for each physical health status category, and estimates of the average causal effects comparing physical disability and co-occurring chronic pain, only chronic pain, and only physical disability to having neither chronic pain or physical disability on average number of days prescribed opioids. | hTMLE | TMLE
---|---|---
| Unadjusted |  $\hat{\psi}$ | 95% CI |  $\hat{\psi}$ | 95% CI
Population Estimate
PD/CP1 | $67.43$ |  $44.56$ |  $41.59$ \- $47.54$ |  $57.37$ |  $56.55$ \- $58.19$
CPO2 | $28.21$ |  $23.97$ |  $23.53$ \- $24.41$ |  $24.40$ |  $23.94$ \- $24.86$
PDO3 | $14.39$ |  $9.78$ |  $9.42$ \- $10.14$ |  $10.30$ |  $9.94$ \- $10.67$
Neither | $3.61$ |  $3.72$ |  $3.69$ \- $3.75$ |  $3.71$ |  $3.69$ \- $3.74$
Average Causal Effect
PD/CP1 vs. Neither | $63.82$ |  $40.84$ |  $37.87$ \- $43.82$ |  $53.65$ |  $52.83$ \- $54.47$
CPO2 vs. Neither | $24.60$ |  $20.26$ |  $19.82$ \- $20.70$ |  $20.68$ |  $20.22$ \- $21.15$
PDO3 vs. Neither | $10.78$ |  $6.06$ |  $5.70$ \- $6.42$ |  $6.59$ |  $6.22$ \- $6.95$
1Physical disability and chronic pain; 2chronic pain only; 3physical disability only. |

##  7 Conclusion

We proposed a crossfitted two-step TMLE for non-negative two-part outcomes.
Our method is non-parametric and capable of incorporating highly flexible
data-adaptive algorithms while remaining $n^{1/2}$ consistent. The target
parameter was defined in terms of modified treatment policies, allowing
effects to be estimated for binary, categorical, and continuous exposures. We
demonstrated that the two-step TMLE can improve finite sample performance
compared to other commonly used nonparametric estimators when the goal is to
estimate the causal effect of an exposure on a non-negative two-part outcome,
and that this increase in performance may be more pronounced when there are
violations of the positivity assumption.

We applied the hTMLE algorithm to a cohort of Medicaid beneficiaries to
estimate the association between physical health status at the time of
Medicaid enrollment and the number of days of prescribed opioids. We found
that co-occurring physical disability and chronic pain, physical disability
alone, and chronic pain alone were all associated with an increase in the
number of days of prescribed opioids. Future improvements could include
extending the hTMLE algorithm to the longitudinal setting.

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##  Appendix A Alternate representation of the EIF

$\mathbb{IF}(\psi_{1}\psi_{2})=\mathbb{IF}(\psi
1)\psi_{2}+\psi_{1}\mathbb{IF}(\psi_{2})$

Define $P(\Delta=1\mid a,w)=\mathsf{q}(a,w)$ and
$\mathsf{E}(Y\mid\Delta=1,a,w)=\mathsf{m}(a,w)$

$\displaystyle\mathbb{IF}$
$\displaystyle=\mathbb{IF}\bigg{[}\sum_{w}\mathsf{q}(a,w)\mathsf{m}(a,w)\mathsf{p}(w)\bigg{]}$
$\displaystyle=\sum_{w}\bigg{\\{}\mathbb{IF}\big{[}\mathsf{q}(a,w)\big{]}\mathsf{m}(a,w)\mathsf{p}(w)+\mathsf{q}(a,w)\mathbb{IF}\big{[}\mathsf{m}(a,w)\big{]}\mathsf{p}(w)+\mathsf{q}(a,w)\mathsf{m}(a,w)\mathbb{IF}\big{[}\mathsf{p}(w)\big{]}\bigg{\\}}$
$\displaystyle=\frac{\mathbbm{1}(A=a)}{\mathsf{g}(a\mid
w)}(\Delta-\mathsf{q}(a,w))\mathsf{m}(a,w)+\mathsf{q}(a,w)\frac{\Delta\mathbbm{1}(A=a)}{P(\Delta=1,A=a\mid
w)}(Y-\mathsf{m}(a,w))+\mathsf{q}(a,w)\mathsf{m}(a,w)$
$\displaystyle=\frac{\mathbbm{1}(A=a)\mathsf{m}(a,w)}{\mathsf{g}(a\mid
w)}(\Delta-\mathsf{q}(a,w))+\frac{\Delta\mathbbm{1}(A=a)}{\mathsf{g}(a\mid
w)}(Y-\mathsf{m}(a,w))+\mathsf{q}(a,w)\mathsf{m}(a,w)$

$\displaystyle\mathsf{q}(a,w)\frac{\mathbbm{1}(\Delta=1,A=a)}{P(\Delta=1,A=a\mid
w)}$
$\displaystyle=\mathbbm{1}(\Delta=1,A=a)\frac{\frac{P(\Delta=1,A=a,w)}{P(A=a,w)}}{\frac{P(\Delta=1,A=a,w)}{P(w)}}$
$\displaystyle=\mathbbm{1}(\Delta=1,A=a)\frac{P(\Delta=1,A=a,w)P(w)}{P(\Delta=1,A=a,w)P(A=a,w)}$
$\displaystyle=\mathbbm{1}(\Delta=1,A=a)\frac{P(w)}{P(A=a,w)}$
$\displaystyle=\mathbbm{1}(\Delta=1,A=a)\frac{P(w)}{P(A=a\mid w)P(w)}$
$\displaystyle=\frac{\mathbbm{1}(\Delta=1,A=a)}{P(A=a|w)}$