Gelfand--Dorfman algebras, derived identities, and the Manin product of operads

TL;DR

This paper explores Gelfand--Dorfman algebras, their identities, and the Manin product of operads, focusing on their relation to differential Poisson algebras and formal calculus of variations.

Contribution

It provides a general description of identities for operations in differential algebras and connects GD-algebras with the Manin product of operads.

Findings

Identifies identities for operations in differential algebras with derivation d.

Describes the relation between GD-algebras and differential Poisson algebras.

Connects Gelfand--Dorfman algebras with the Manin product of operads.

Abstract

Gelfand--Dorfman bialgebras (GD-algebras) are nonassociative systems with two bilinear operations satisfying a series of identities that express Hamiltonian property of an operator in the formal calculus of variations. The paper is devoted to the study of GD-algebras related with differential Poisson algebras. As a byproduct, we obtain a general description of identities that hold for operations $a\succ b = d(a)b$ and $a\prec b = ad(b)$ on a (non-associative) differential algebra with a derivation~$d$.

Equations

$$(a\circ b)\circ c-a\circ (b\circ c)=(b\circ a)\circ c-b\circ (a\circ c),$$

$$(a\circ b)\circ c=(a\circ c)\circ b.$$

$$[a,b\circ c]-[c,b\circ a]+[b,a]\circ c-[b,c]\circ a-b\circ [a,c]=0.$$

$$\{x,yz\}=\{x,y\}z+y\{x,z\}.$$

$$x\circ y=xd(y).$$

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

$$d(f(d^{s_1}(x_1),\dots ,d^{s_n}(x_n))=\sum _{i=1}^{n}f(d^{s_1}(x_1),\dots ,d^{s_i+1}(x_i),\dots ,d^{s_n}(x_n)).$$

$$D(xy)=D(x)y+xD(y)+\lambda xy,$$

$$D\mathrm{Var}=\mathrm{Var}\circ \mathrm{Nov},D^{\ast }\mathrm{Var}=\mathrm{Var}\circ {\mathrm{GD}}^{!}.$$

$$\mathrm{Var}(n)=M_n(X)/M_n(X)\cap T_{\mathrm{Var}}(X),$$

$$O_3=\mathbb{k}{S}_3{\otimes }_{\mathbb{k}{S}_2}(\mathrm{GD}(2)\otimes \mathrm{GD}(2)),$$

$$a_1=1{\otimes }_{\mathbb{k}{S}_2}(\mu \otimes \mu )=[[x_1,x_2],x_3],$$

$$\displaystyle a_{3}=1\otimes_{\Bbbk S_{2}}\big{(}\mu\otimes\nu^{(12)}\big{)}=[x_{2}\circ x_{1},x_{3}],\quad$$

$$a_5=1{\otimes }_{\mathbb{k}{S}_2}(\nu \otimes \nu )=(x_1\circ x_2)\circ x_3,$$

$$\displaystyle a_{7}=1\otimes_{\Bbbk S_{2}}\big{(}\nu^{(12)}\otimes\mu\big{)}=x_{3}\circ[x_{1},x_{2}],\quad$$

$$\displaystyle a_{9}=1\otimes_{\Bbbk S_{2}}\big{(}\nu^{(12)}\otimes\nu^{(12)}\big{)}=x_{3}\circ(x_{2}\circ x_{1}),$$

$$\text{setcounter}MaxMatrixCols27\begin{array}{ccccccccccccccccccccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 0& 0& 1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 1& -1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& -1& 0& -1& 0& 0& 0& 0& 0\\ 0& 0& 1& -1& 0& 0& 0& 0& 0& 0& 0& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 1& 0& 1& 0& 0& 0& 0& 0& 0& 0& -1& 1& 0& 0& 0& 0& 0\end{array}$$

$$x\ast y=y\ast x,x\ast (y\ast z)=(x\ast y)\ast z,$$

$$(x\star y)\star z-x\star (y\star z)=(x\star z)\star y-x\star (z\star y),$$

$$x\star (y\star z)=y\star (x\star z),$$

$$x\star (y\ast z)=(x\star y)\ast z,$$

$$x\star (y\ast z)+y\star (x\ast z)=(x\ast y)\star z.$$

$$xy=x\ast y,x\circ y=y\star x$$

$$(x\odot y)\ast z=x\odot (y\ast z).$$

$$(x\odot y)\ast z=(x\ast y)\odot z,x\ast (y\odot z)=y\ast (x\odot z).$$

$$\begin{gathered} (x\odot y)\ast z=(y\odot x)\ast z=y\odot (x\ast z)=y\odot (z\ast x)=(y\odot z)\ast x \\ =(z\odot y)\ast x=z\odot (y\ast x)=(x\ast y)\odot z. \end{gathered}$$

$$x\ast (y\odot z)=(z\odot y)\ast x=z\odot (y\ast x)=z\odot (x\ast y)=(z\odot x)\ast y=y\ast (x\odot z).$$

$$x_1\ast \cdots \ast x_t\ast (y_1\odot \cdots \odot y_k),x_1\le \cdots \le x_t\le y_1\le \cdots \le y_k,$$

$$x_{i_1}\ast x_{i_2}\ast \dots \ast x_{i_t}\ast (y_{j_1}\odot \dots \odot y_{j_k}),x_i,y_j\in X.$$

$$u\ast v=l_1\ast \dots \ast l_{n+p+1}\ast (l_{n+p+2}\odot \dots \odot l_{n+p+m+q})$$

$$u\odot v=l_1\ast \dots \ast l_{n+p}\ast (l_{n+p+1}\odot \dots \odot l_{n+p+m+q})$$

$$x\star y=d(x)\odot y.$$

$$x\star (y\ast z)-(x\star y)\ast z=d(x)\odot (y\ast z)-(d(x)\odot y)\ast z=0,$$

$$\begin{gathered} x\star (y\ast z)+y\star (x\ast z)=d(x)\odot (y\ast z)+d(y)\odot (x\ast z)=(d(x)\odot y)\ast z+(d(y)\odot x)\ast z \\ =(d(x)\ast y)\odot z+(d(y)\ast x)\odot z=d(x\ast y)\odot z=(x\ast y)\star z \end{gathered}$$

Full text

1. Introduction

A series of non-associative structures related with the formal calculus of variations appeared in [9]. These structures are nowadays known as Novikov algebras (they were independently introduced in [2] as a tool for classification of linear Poisson brackets of hydrodynamic type), Novikov–Poisson algebras, and Gelfand–Dorfman bialgebras. The latter class of systems was initially invented in [9] as a source of Hamiltonian operators: the structure constants of a Gelfand–Dorfman bialgebra may be used as coefficients of a differential operator.

By definition, a linear space $A$ equipped with one bilinear operation $\circ:A\times A\rightarrow A$ is said to be a Novikov algebra (the term was proposed in [17]) if it satisfies the following identities:

[TABLE]

A Gelfand—Dorfmann bialgebra [21] is a system $(A,\circ,[\cdot,\cdot])$ with two bilinear operations such that $(A,\circ)$ is a Novikov algebra, $(A,[\cdot,\cdot])$ is a Lie algebra, and the following additional identity holds:

[TABLE]

In order to avoid a collision with the well-known notion of a bialgebra as an algebra equipped with a coproduct, we will use the term GD-algebra for a Gelfand—Dorfman bialgebra.

It turned out [21] (see also [11]) that GD-algebras are closely related with conformal Lie algebras appeared in the theory of vertex operators [13]. Conformal algebras and their generalizations (pseudo-algebras) also turn to be useful for the classification of Poisson brackets of hydrodynamic type [1]. It worth mentioning that Poisson algebras are also closely related with representations of Lie conformal algebras [14].

The following statement provides a series of examples of GD-algebras. Recall that a Poisson algebra is a linear space $V$ equipped with two bilinear operations, i.e., a system $(V,\cdot,\{\cdot,\cdot\})$, where $(V,\cdot)$ is an associative and commutative algebra, $(V,\{\cdot,\cdot\})$ is a Lie algebra, and the following Leibniz identity holds:

[TABLE]

Theorem 1**.**

Let $V$ be a Poisson algebra with a derivation $d\in\mathrm{Der}\,V$ relative to the both products. Define a new operation $\circ$ on $V$ in the following way:

[TABLE]

Then $(V,\circ,\{\cdot,\cdot\})$ is a GD-algebra.

The proof is straightforward. It is well-known [2, 9] that $x\circ y=xd(y)$ satisfies (1) and (2). It remains to show that (3) holds. Indeed, evaluate the left-hand side of (3) in a differential Poisson algebra: $[u,w\circ v]-[v,w\circ u]+[w,u]\circ v-[w,v]\circ u-w\circ[u,v]=\{u,wd(v)\}-\{v,wd(u)\}+\{w,u\}d(v)-\{w,v\}d(u)-wd(\{u,v\})=\{u,w\}d(v)+\{u,d(v)\}w-\{v,w\}d(u)-\{v,d(u)\}w+\{w,u\}d(v)-\{w,v\}d(u)-w\{d(u),v\}-w\{u,d(v)\}=0$.∎

Let us call a GD-algebra $A$ special if it can be embedded into a differential Poisson algebra with operations $\{\cdot,\cdot\}$ and $\circ$ given by (4). The class of all homomorphic images of all special GD-algebras is a variety (i.e., a class of algebras which is closed with respect to homomorphic images, subalgebras, and Cartesian products). Let us denote this variety by $\mathop{SGD}\nolimits$. The main purpose of this paper is to describe a way for finding the defining identities of $\mathop{SGD}\nolimits$.

In order to solve that problem, we need to study differential algebras and answer the following question which is of independent interest. Suppose $A$ is a (non-associative) algebra with a derivation $d$. Introduce the following new binary operations on $A$:

[TABLE]

Assuming $A$ satisfies a family of polynomial identities, what can be said about identities on $A^{(d)}=(A,\succ,\prec)$?

For example, if $A$ is associative and commutative then $a\succ b=b\prec a$ and the operation $\prec$ satisfies (1) and (2). It was shown in [7] that there are no more independent identities that hold on $A^{(d)}$ for all associative commutative algebras $A$ and for all their derivations $d$. Moreover, it was proved in [4] that every Novikov algebra may be embedded into a differential associative commutative algebra.

We find an explicit answer to the question about identities of $A^{(d)}$. It turns out that, given a variety $\mathop{Var}\nolimits$ defined by multilinear identities (multilinear variety, for short), the systems $A^{(d)}$, $A\in\mathop{Var}\nolimits$, $d\in\mathrm{Der}(A)$, generate a variety governed by the operad $\mathop{Var}\nolimits\circ\mathrm{Nov}$, where $\mathrm{Nov}$ is the operad of Novikov algebras and $\circ$ stands for the Manin white product of operads [10, 16]. Hereinafter we identify notations for multilinear varieties and the corresponding operads.

If $\mathop{Var}\nolimits$ is a variety of algebras then the free algebra in $\mathop{Var}\nolimits$ generated by a set $X$ is denoted $\mathop{Var}\nolimits\langle X\rangle$. By $\mathop{Var}\nolimits\mathrm{Der}$ we denote the class of all differential $\mathop{Var}\nolimits$-algebras with one derivation (this is a variety of algebraic systems with one additional unary operation $d$ in the language). The free algebra in $\mathop{Var}\nolimits\mathrm{Der}$ generated by a set $X$ is denoted $\mathop{Var}\nolimits\mathrm{Der}\langle X,d\rangle$. We will often need to know the structure of the free differential $\mathop{Var}\nolimits$-algebra. It can be easily derived in general (c.f. [8] for associative algebra case) that the following statement holds.

Lemma 1**.**

For a multilinear variety $\mathop{Var}\nolimits$, the free differential $\mathop{Var}\nolimits$-algebra generated by a set $X$ with one derivation $d$ coincides with $\mathop{Var}\nolimits\langle d^{\omega}X\rangle$, where $d^{\omega}X=\{d^{s}(x)\mid s\geq 0,x\in X\}$.

Indeed, consider the free nonassociative (magmatic) algebra $M\langle d^{\omega}X\rangle$ and define a linear map $d:F(X;d)\to F(X;d)$ in such a way that $d(d^{s}(x))=d^{s+1}(x)$, $d(uv)=d(u)v+ud(v)$. Denote by $T_{\mathop{Var}\nolimits}(X)$ the T-ideal of all identities in a set of variables $X$ that hold on $\mathop{Var}\nolimits$.

Since $\mathop{Var}\nolimits$ is defined by multilinear identities, the T-ideal $T_{\mathop{Var}\nolimits}(d^{\omega}X)$ of $M\langle d^{\omega}X\rangle$ is $d$-invariant:

[TABLE]

Hence, $U=F(X;d)/T_{\mathop{Var}\nolimits}(d^{\omega}X)$ is a $\mathop{Var}\nolimits$-algebra with a derivation $d$. Since all relations from $T_{\mathop{Var}\nolimits}(d^{\omega}X)$ hold in every differential $\mathop{Var}\nolimits$-algebra, $U$ is isomorphic to the free differential $\mathop{Var}\nolimits$-algebra. ∎

Remark 1**.**

In Lemma 1, a derivation $d$ may be replaced with a generalized derivation, i.e., a linear map $D$ such that

[TABLE]

where $\lambda$ is a fixed scalar in $\Bbbk$. Indeed, it is enough to note that $d=D-\lambda\mathrm{id}$ is an ordinary derivation.

In order to study the relations between GD-algebras and differential Poisson algebras we need more general settings. Namely, given an algebra $A$ with a derivation $d$, consider the system $A^{(d*)}=(A,\succ,\prec,\cdot)$, where $\succ$, $\prec$ are defined by (5), and $\cdot$ is the initial multiplication on $A$. How to describe the identities that hold on $A^{(d*)}$ provided that we know the identities of $A$? It turns out that the answer involves the operad $\mathop{GD}\nolimits$ of GD-algebras. Namely, for a multilinear variety $\mathop{Var}\nolimits$, the systems $A^{(d*)}$, $A\in\mathop{Var}\nolimits$, $d\in\mathrm{Der}(A)$, generate a variety governed by $\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!}$, where $\mathop{GD}\nolimits^{!}$ is the Koszul-dual of the operad $\mathop{GD}\nolimits$.

The paper is organized as follows. In Section 2, we calculate the operad $\mathop{GD}\nolimits^{!}$ and describe the free $\mathop{GD}\nolimits^{!}$-algebra. In fact, the variety of $\mathop{GD}\nolimits^{!}$-algebras is a proper sub-variety of the class of Novikov–Poisson algebras. The defining identities of $\mathop{GD}\nolimits^{!}$ were earlier obtained in [4], but it was assumed in [4] that a Novikov–Poisson algebra has an identity element. We can not use this assumption for our purpose, so we describe the free $\mathop{GD}\nolimits^{!}$-algebra explicitly and prove that it can be embedded into the free differential associative and commutative algebra.

In Section 3, we define (generalized) derived identities of a multilinear variety $\mathop{Var}\nolimits$ of algebras with binary operations as those identities that hold on all $A^{(d)}$ (resp., $A^{(d*)}$) for all $A\in\mathop{Var}\nolimits$ and for all $d\in\mathrm{Der}(A)$. Denote the operads defined by (generalized) derived identities by $D\mathop{Var}\nolimits$ (resp., $D^{*}\mathop{Var}\nolimits$). Then we prove

[TABLE]

In Section 4, we describe the linear basis of the free special GD-algebra $\mathop{SGD}\nolimits\langle X\rangle$ in terms of the enveloping differential Poisson algebra. This result generalizes what was done in [7]: here we have to add one more operation $[\cdot,\cdot]$ into consideration.

Section 5 is devoted to the description of defining relations of the operad SGD governing the variety generated by all special GD-algebras. It is easy to see that the operad SGD is a sub-operad of $D^{*}\mathrm{Pois}$ generated by a proper subspace of binary operations. We find the entire family of defining relations for $D^{*}\mathrm{Pois}$, so all defining identities of SGD occur among their corollaries. In particular, we are interested in special identities of GD-algebras, i.e., those identities that hold on all SGD-algebras but do not hold on the class of all GD-algebras. There are no special identities of degree 3, but we can find two independent special identities of degree 4. The complete list of special identities of GD-algebras remains unknown.

2. Dual Gelfand—Dorfmann algebras

Suppose $\mathop{Var}\nolimits$ is a multilinear variety of algebras. Fix a countable set of variables $X=\{x_{1},x_{2},\dots\}$ and denote

[TABLE]

where $M_{n}(X)$ is the space of all (non-associative) multilinear polynomials of degree $n$ in $x_{1},\dots,x_{n}$, $T_{\mathop{Var}\nolimits}(X)$ is the T-ideal of all identities that hold in $\mathop{Var}\nolimits$.

The collection of spaces $(\mathop{Var}\nolimits(n))_{n\geq 1}$ forms a symmetric operad relative to the natural composition rule and symmetric group action (see, e.g., [16]). We will denote this operad by the same symbol $\mathop{Var}\nolimits$. The operad $\mathop{Var}\nolimits$ is a binary one, i.e., it is generated (as a symmetric operad) by the elements of $\mathop{Var}\nolimits(2)$. In the generic case of algebras with one binary operation of multiplication, $\mathop{Var}\nolimits(2)$ is spanned by $x_{1}x_{2}$ and $x_{2}x_{1}$.

The variety of GD-algebras is defined by identities of degree 2 or 3. Therefore, the corresponding operad is a quadratic one [10]. Let us compute the Koszul dual operad $\mathop{GD}\nolimits^{!}$.

Denote two operations of a GD-algebra by $\mu$ and $\nu$, namely, $\mu=[x_{1},x_{2}]$, $\nu=x_{1}\circ x_{2}$. Then $\mathop{GD}\nolimits(2)=\mathrm{Span}\{\mu,\nu,\nu^{(12)}\}$, where $\nu_{2}^{(12)}=x_{2}\circ x_{1}$.

As in [10], the $S_{3}$-space of multilinear terms of degree 3 is identified with

[TABLE]

where the action of $(12)\in S_{2}$ on $\mathop{GD}\nolimits(2)^{\otimes 2}$ is given by $\text{id}\otimes(12)$.

We may choose a basis $a_{1},\dots,a_{27}$ of $O_{3}$ as follows:

[TABLE]

and $a_{9+i}=(13)a_{i}$, $a_{18+i}=(23)a_{i}$ for $i=1,\dots,9$.

Relative to the basis $a_{1},\dots,a_{27}$, the vectors representing the defining relations of $\mathop{GD}\nolimits$ may be presented by the rows of the following matrix:

[TABLE]

Compute the sign-twisted orthogonal complement $U^{\perp}\subset O_{3}^{\vee}$ [10] for the $S_{3}$-module $U$ spanned by these vectors to obtain the defining relations of the Koszul dual operad $\mathop{GD}\nolimits^{!}$. It turns out that a $\mathop{GD}\nolimits^{!}$-algebra has two operations $\mu^{\vee}(x_{1},x_{2})=x_{1}\ast x_{2}$ and $\nu^{\vee}(x_{1},x_{2})=x_{1}\star x_{2}$, where

[TABLE]

i.e., $\ast$ is associative and commutative,

[TABLE]

i.e., $\star$ is a right Novikov product, and

[TABLE]

Remark 2**.**

Relations (6)–(10) up to a change of notations

[TABLE]

coincide with the identities defining a differential GDN-Poisson algebra [4]. This is a proper sub-variety of the variety of Novikov—Poisson algebras introduced in [20]. However, all algebras in [4] are assumed to be unital, i.e., with an identity $e$ relative to the associative and commutative operation $\ast$. This is an essential restriction since it is unclear in general how to join a unit to a $\mathop{GD}\nolimits^{!}$-algebra. This is why in this section we have to study the free $\mathop{GD}\nolimits^{!}$-algebra without an identity assumption.

Let us find a normal form of elements in the free $\mathop{GD}\nolimits^{!}$-algebra, i.e., a linear basis of $\mathop{GD}\nolimits^{!}\langle X\rangle$.

Denote by $\mathrm{BiCom}$ the variety of algebras with two associative and commutative operations $*$ and $\odot$ such that

[TABLE]

It is easy to see that the following identities hold in $\mathrm{BiCom}$:

[TABLE]

Indeed, apply commutativity of $\ast$ and (11) to obtain

[TABLE]

Similarly,

[TABLE]

Theorem 2**.**

A linear basis of the free $\mathrm{BiCom}$-algebra $\mathrm{BiCom}\langle X\rangle$ generated by a linearly ordered set $X$ consists of the words

[TABLE]

$x_{i},y_{j}\in X$.

Proof.

It can be easily seen that every element of a $\mathrm{BiCom}$ algebra generated by a set $X$ can be transformed to a linear combination of words that have the following form:

[TABLE]

Commutativity of $\ast$ and $\odot$ allows us to assume $x_{i_{1}}\leq\dots\leq x_{i_{t}}$ and $y_{j_{1}}\leq\dots\leq y_{j_{k}}$. If $x_{i_{t}}>y_{j_{1}}$ then we may interchange them by means of the second relation in (12). Hence, every element of $\mathrm{BiCom}$ can be transformed to a linear combination of monomials (14).

To prove linear independence of (14), consider a linear space $B^{\prime}$ spanned by (14) and define multiplications $\ast$ and $\odot$ on $B^{\prime}$ in the following way. If $u=x_{1}\ast\ldots\ast x_{n}\ast(y_{1}\odot\ldots\odot y_{m})$, $v=z_{1}\ast\ldots\ast z_{p}\ast(t_{1}\odot\ldots\odot t_{q})$ then

[TABLE]

[TABLE]

where the sequence $(l_{1},\ldots,l_{n+p+m+q})$ is just the ordering of the sequence $(x_{1},\dots,x_{n},$ $y_{1},\dots,y_{m},z_{1},\dots,z_{p},t_{1},\dots,t_{q})$. It is easy to check that $B^{{}^{\prime}}$ is a $\mathrm{BiCom}$ algebra. Hence, $B^{{}^{\prime}}\simeq\mathrm{BiCom}\langle X\rangle$. ∎

Let $\mathrm{BiCom}\mathrm{Der}\langle X,d\rangle$ stand for the free differential $\mathrm{BiCom}$-algebra generated by a set $X$ with a derivation $d$ (i.e., $d$ is a derivation relative to the both operations $\ast$ and $\odot$).

Theorem 3**.**

Let $A$ be a $\mathrm{BiCom}$-algebra with a derivation $d$. Then $A$ is a $\mathop{GD}\nolimits^{!}$-algebra relative to the operations $*$ and $\star$ defined by

[TABLE]

Proof.

The operation $x\star y=d(x)\odot y$ is known to be right Novikov. It remains to check the identities (9), (10):

[TABLE]

[TABLE]

by (12). ∎

Let us say that a $\mathop{GD}\nolimits^{!}$-algebra $B$ is special if it can be embedded into an appropriate differential $\mathrm{BiCom}$-algebra. Denote by $\mathop{SGD}\nolimits^{!}$ the variety generated by special $\mathop{GD}\nolimits^{!}$-algebras (all homomorphic images of special $\mathop{GD}\nolimits^{!}$-algebras). Obviously, the free $\mathop{SGD}\nolimits^{!}$-algebra $\mathop{SGD}\nolimits^{!}\langle X\rangle$ generated by a set $X$ is isomorphic to the subalgebra of $\mathrm{BiCom}\mathrm{Der}\langle X,d\rangle$ generated by $X$ relative to the operations $\star$ and $*$.

For every monomial $u\in\mathrm{BiCom}\mathrm{Der}\langle X,d\rangle$ define the weight of $u$ by the following induction rule:

[TABLE]

The space of differential $\mathrm{BiCom}$-monomials of weight $-1$ has the following basis:

[TABLE]

Theorem 4**.**

An element $f\in\mathrm{BiCom}\mathrm{Der}\langle X,d\rangle$ belongs to $\mathop{SGD}\nolimits^{!}\langle X\rangle$ if and only if all monomials that appear in $f$ have weight $-1$.

Proof.

The “only if” part can be easily proved by induction. For the converse (“if”) part, it is enough to consider the case when $f$ is a monomial of the form (16). Note that if $\mathop{wt}\nolimits(f)=-1$ then $f=x_{1}*\dots*x_{t}*w$, where $w=y_{1}\odot\dots\odot y_{i}\odot d^{r_{i+1}}(y_{i+1})\odot\dots\odot d^{r_{k}}(y_{k})$, $\mathop{wt}\nolimits(w)=-1$. As shown in [7], $w$ can be expressed via $\star$. Then $f\in\mathop{SGD}\nolimits^{!}\langle X\rangle$. ∎

Theorem 5**.**

For a linearly ordered set $X$, $\mathop{SGD}\nolimits^{!}\langle X\rangle$ is isomorphic to $\mathop{GD}\nolimits^{!}\langle X\rangle$.

Proof.

First, let us prove the following technical statement.

Lemma 2**.**

The following elements span the free $\mathop{GD}\nolimits^{!}$-algebra $\mathop{GD}\nolimits^{!}\langle X\rangle$ generated by a linearly ordered set $X$:

[TABLE]

where $w$ is a monomial from $\mathrm{RNov}\langle X\rangle$ relative to the operation $\star$.

Here $\mathrm{RNov}$ denotes the variety of right Novikov algebras defined by the identities (7) and (8).

Proof.

Let $f$ be a monomial in $\mathop{GD}\nolimits^{!}\langle X\rangle$. Proceed by induction on $n=\deg f$. If $n=1,2$ then the statement is clear. Suppose $n\geq 3$. Then $f=u*v$ or $f=u\star v$, $\deg u,\deg v<n$. By the inductive assumption,

[TABLE]

where $w_{1},w_{2}\in\mathrm{RNov}\langle X\rangle$.

Case 1: $f=u*v$. Then $f=x_{1}*\dots*x_{r}*y_{1}*\dots*y_{s}*(w_{1}*w_{2})$. If $\deg w_{1}=1$ or $\deg w_{2}=1$ then we are done. If $w_{i}=(w_{i1}\star w_{i2})$, $i=1,2$, then the inductive assumption implies

[TABLE]

where $z\in X$, $w^{\prime}\in\mathrm{RNov}\langle X\rangle$. Hence, $f=x_{1}*\dots*x_{r}*y_{1}*\dots*y_{s}*z*(w_{11}\star w^{\prime})$.

Case 2: $f=u\star v$. Then $f=(x_{1}*\dots*x_{r}*w_{1})\star(y_{1}*\dots*y_{s}*w_{2})$. If $\deg u=1$ then the statement follows from (9) and commutativity of $*$:

[TABLE]

For $\deg u>1$, we have to consider the following sub-cases: $r=0$ and $r>0$. The first one is similar to what is done for $\deg u=1$, in the second we have $u=x_{1}*u^{\prime}$, so by (10)

[TABLE]

The latter expression is of the type considered in Case 1. ∎

Suppose $X$ is linearly ordered. Consider the epimorphism

[TABLE]

given by $\varphi(x)=x$ for $x\in X$, $\varphi(u\star v)=d(\varphi(u))\odot\varphi(v)$, $\varphi(u*v)=\varphi(u)*\varphi(v)$.

Recall that elements from $\mathrm{RNov}\langle X\rangle$ may be presented as differential polynomials from the free differential commutative algebra $\mathrm{Com}\mathrm{Der}\langle X,d\rangle$ (relative to binary operation $\odot$ and derivation $d$) by means of $x\star y=d(x)\odot y$. Such a presentation is known to be unique [7]. Therefore, by Lemma 2 $\mathop{GD}\nolimits^{!}\langle X\rangle$ is spanned by elements

[TABLE]

where $x_{1}\leq\dots\leq x_{t}$, $s_{1}+\dots+s_{k}-k=-1$, and

[TABLE]

Note that if $u$ is of the form (17) then $\varphi(u)=u\in\mathrm{BiCom}\mathrm{Der}\langle X,d\rangle$. Moreover, $s_{1}=0$.

Relation (12) implies that we may add one more condition on the words (17): $x_{t}\leq y_{1}$.

Therefore, we have found a set of linear generators in $\mathop{GD}\nolimits^{!}\langle X\rangle$ whose images in $\mathop{SGD}\nolimits^{!}\langle X\rangle$ are linearly independent. Hence, $\mathop{GD}\nolimits^{!}\langle X\rangle\simeq\mathop{SGD}\nolimits^{!}\langle X\rangle$. In particular, the set of monomials (16) is a linear basis of $\mathop{GD}\nolimits^{!}\langle X\rangle$. ∎

Consider the class $\mathop{SNP}\nolimits$ of all algebras with two operations $\ast$ and $\star$ obtained from differential associative commutative algebras via

[TABLE]

Corollary 1**.**

The class of all homomorphic images of algebras from $\mathop{SNP}\nolimits$ coincides with the variety of $\mathop{GD}\nolimits^{!}$-algebras.

In other words, the defining identities of $\mathop{GD}\nolimits^{!}$ and their corollaries exhaust all identities that hold for operations (18) on commutative differential algebras.

Proof.

Consider the homomorphism of $\mathop{GD}\nolimits^{!}$-algebras

[TABLE]

defined as follows: $\psi(x)=x$ for $x\in X$, $\psi(f*g)=\psi(f)\psi(g)$, $\psi(f\star g)=d(\psi(f))\psi(g)$. We are interested in the kernel of $\psi$.

By Theorem 5, $\mathop{GD}\nolimits^{!}\langle X\rangle\subset\mathrm{BiCom}\mathrm{Der}\langle X,d\rangle$. Therefore, $\psi$ is the restriction of the following homomorphism of differential $\mathrm{BiCom}$-algebras:

[TABLE]

$\varphi(x)=x$ for $x\in X$, $\varphi(f*g)=\varphi(f\odot g)=\varphi(f)\varphi(g)$.

Suppose $X$ is equipped with a linear order and $w$ is a monomial in $\mathrm{BiCom}\mathrm{Der}\langle X,d\rangle$ of the form (16). Then

[TABLE]

where $x_{1}\leq\dots\leq x_{t}\leq y_{1}\leq\dots\leq y_{i}$, $r_{i+1}>0$, $(r_{i+1},y_{i+1})\leq\dots\leq(r_{k},y_{k})$ lexicographically. Therefore, the images of the basic monomials of $\mathop{GD}\nolimits^{!}\langle X\rangle$ are linearly independent, so

[TABLE]

∎

In particular, we may find a simple form for a basis of the free algebra $\mathop{GD}\nolimits^{!}\langle X\rangle$.

Let $w$ be a monomial in $\mathrm{Com}\mathrm{Der}\langle X,d\rangle$. Define the weight of $w$ by induction as follows: $\mathop{wt}\nolimits(x)=-1$, $x\in X$; $\mathop{wt}\nolimits(uv)=\mathop{wt}\nolimits(u)+\mathop{wt}\nolimits(v)$; $\mathop{wt}\nolimits(d(u))=\mathop{wt}\nolimits(u)+1$.

Corollary 2**.**

A linear basis of $\mathop{GD}\nolimits^{!}\langle X\rangle$ consists of all monomials in $\mathrm{Com}\mathrm{Der}\langle X,d\rangle$ of weight $\leq-1$.

Remark 3**.**

For unital $\mathop{GD}\nolimits^{!}$-algebras, a more precise result was obtained in [4]: if $V$ is a $\mathop{GD}\nolimits^{!}$-algebra with an identity $e$ relative to the operation $*$ then $V$ coincides with an associative commutative algebra $A$ with a derivation $d\in\mathrm{Der}(A)$ equipped with the operations $x\ast y=xy$, $x\star y=d(x)y$.

3. Derived identities

Let $\mathcal{P}$ and $\mathcal{Q}$ be two operads. Then the family of spaces $(\mathcal{P}(n)\otimes\mathcal{Q}(n))_{n\geq 1}$ is an operad relative to the componentwise composition and symmetric group action. This operad, denoted $\mathcal{P}\otimes\mathcal{Q}$, is known as the Hadamard product of $\mathcal{P}$ and $\mathcal{Q}$. If $\mathcal{P}$ and $\mathcal{Q}$ are binary operads then $\mathcal{P}\otimes\mathcal{Q}$ may be non-binary. The sub-operad of $\mathcal{P}\otimes\mathcal{Q}$ generated by $\mathcal{P}(2)\otimes\mathcal{Q}(2)$ is known as the Manin white product of $\mathcal{P}$ and $\mathcal{Q}$, it is denoted $\mathcal{P}\circ\mathcal{Q}$ [10].

Example 1**.**

The operad $\mathrm{Lie}\circ\mathrm{Nov}$ is isomorphic to the operad governing the class of all algebras (magmatic operad).

Indeed, both $\mathrm{Lie}$ and $\mathrm{Nov}$ are quadratic operads, and so is $\mathrm{Lie}\circ\mathrm{Nov}$ [10]. Let $[x_{1},x_{2}]$ and $x_{1}\circ x_{2}$ be the generators of $\mathrm{Lie}$ and $\mathrm{Nov}$. It is enough to find the defining identities of $\mathrm{Lie}\circ\mathrm{Nov}$ that are quadratic with respect to the generators.

Identify $[x_{1},x_{2}]\otimes(x_{1}\circ x_{2})$ with $[x_{1}\prec x_{2}]$, then $[x_{1},x_{2}]\otimes(x_{2}\circ x_{1})=([x_{2},x_{1}]\otimes(x_{1}\circ x_{2}))^{(12)}=-[x_{2}\prec x_{1}]$. Hence, $(\mathrm{Lie}\circ\mathrm{Nov})(n)$ is an image of the space $M_{n}(X)$ of all multilinear non-associative polynomials of degree $n$ in $X=\{x_{1},x_{2},\dots\}$ relative to the operation $[\cdot\prec\cdot]$. Calculating the compositions in the componentwise way, we obtain

[TABLE]

It remains to find the intersection of the $S_{3}$-submodule generated by $m_{1}$ and $m_{2}$ in $M_{3}(X)\otimes M_{3}(X)$ with the kernel of the projection $M_{3}(X)\otimes M_{3}(X)\to\mathrm{Lie}(3)\otimes\mathrm{Nov}(3)$. Straightforward calculation shows the intersection is zero. Hence, the operation $[\cdot\prec\cdot]$ satisfies no identities.

Example 2**.**

The operad $\mathop{As}\nolimits\circ\mathrm{Nov}$ is generated by 4-dimensional space $(\mathop{As}\nolimits\circ\mathrm{Nov})(2)$ spanned by

[TABLE]

relative to the following identities:

[TABLE]

This result may be checked with a straightforward computation. Namely, one has to find the intersection of the $S_{3}$-submodule in $M_{3}(X)\otimes M_{3}(X)$ generated by

[TABLE]

with the kernel of the projection $M_{3}(X)\otimes M_{3}(X)\to\mathop{As}\nolimits(3)\otimes\mathrm{Nov}(3)$.

Given a (non-associative) algebra $A$ with a derivation $d$, denote by $A^{(d)}$ the same linear space $A$ considered as a system with two binary linear operations of multiplication $\prec$ and $\succ$ defined by

[TABLE]

In the same settings, denote by $A^{d*}$ the system $(A,\prec,\succ,\cdot)$, where $\cdot$ is the initial multiplication on $A$.

Definition 1**.**

Let $\mathop{Var}\nolimits$ be a multilinear variety of algebras. A non-associative polynomial $f(x_{1},\dots,x_{n})$ in two operations of multiplication $\prec$ and $\succ$ is called a derived identity of $\mathop{Var}\nolimits$ if for every $A\in\mathop{Var}\nolimits$ and for every derivation $d\in\mathrm{Der}(A)$ the algebra $A^{(d)}$ satisfies the identity $f(x_{1},\dots,x_{n})=0$. A generalized derived identity is a polynomial in three operations $\prec$, $\succ$, and $\cdot$ that turns into an identity on $A^{d*}$ for every $A\in\mathop{Var}\nolimits$ and for every $d\in\mathrm{Der}(A)$.

Obviously, the set of all (generalized) derived identities is a T-ideal of the algebra of non-associative polynomials in two (three) operations.

For example, $(x\succ y)=(y\prec x)$ is a derived identity of $\mathrm{Com}$. Moreover, the operation $(\cdot\prec\cdot)$ satisfies the axioms of Novikov algebras (1) and (2). It was actually shown in [7] that the entire T-ideal of derived identities of $\mathrm{Com}$ is generated by these identities. Similarly, Corollary 1 implies that the identities of a $\mathop{GD}\nolimits^{!}$-algebra (6)–(10) present the complete list of generalized derived identities of $\mathrm{Com}$ relative to the following change of notations:

[TABLE]

Remark 4**.**

For the variety of associative algebras, it was mentioned in [15] that (19) and (20) are derived identities of $\mathop{As}\nolimits$.

Our aim in this section is to show that (generalized) derived identities of $\mathop{Var}\nolimits$ are exactly the same as defining relations of $\mathop{Var}\nolimits\circ\mathrm{Nov}$ ($\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!}$). We will focus on the proof of the description of generalized derived identities.

Let $\mathcal{N}_{\mathop{Var}\nolimits}$ be the class of all differential $\mathop{Var}\nolimits$-algebras with one locally nilpotent derivation, i.e., for every $A\in\mathcal{N}_{\mathop{Var}\nolimits}$ and for every $a\in A$ there exists $n\geq 1$ such that $d^{n}(a)=0$.

Note that $\mathcal{N}_{\mathop{Var}\nolimits}$ and the entire class $\mathop{Var}\nolimits\mathrm{Der}$ satisfy the same (differential) identities: we prove that within the following statement.

Lemma 3**.**

Suppose $f=f(x_{1},\dots,x_{n})$ is a multilinear identity that holds on $A^{(d*)}$ for all $A\in\mathcal{N}_{\mathop{Var}\nolimits}$. Then $f$ is a generalized derived identity of $\mathop{Var}\nolimits$.

Proof.

Consider the free differential $\mathop{Var}\nolimits$-algebra $U_{n}$ generated by the set $X=\{x_{1},x_{2},\dots\}$ with one derivation $d$ modulo defining relations $d^{n}(x)=0$, $x\in X$.

Then $U_{n}\in\mathcal{N}_{\mathop{Var}\nolimits}$. As a differential $\mathop{Var}\nolimits$-algebra, $U_{n}$ is a homomorphic image of the free magmatic algebra $M\langle d^{\omega}X\rangle$. Denote by $I_{n}$ the kernel of the homomorphism $M\langle d^{\omega}X\rangle\to U_{n}$. As an ideal of $M\langle d^{\omega}X\rangle$, $I_{n}$ is a sum of two ideals: $I_{n}=T_{\mathop{Var}\nolimits}(d^{\omega}X)+N_{n}$, where $N_{n}$ is generated by $d^{n+t}(x)$, $x\in X$, $t\geq 0$. Note that the last relations form a $d$-invariant subset of $M\langle d^{\omega}X\rangle$, so the ideal $N_{n}$ is $d$-invariant.

If a polynomial $F\in M\langle d^{\omega}X\rangle$ belongs to $I_{n}$ and its degree in $d$ is less than $n$ then $F$ belongs to $T_{\mathop{Var}\nolimits}(d^{\omega}X)$. If $f=f(x_{1},\dots,x_{n})$ is a multilinear identity in $U_{n}^{(d)}$ then the image $F$ of $f$ in $M\langle d^{\omega}X\rangle$ has degree $n-1$ in $d$, so $F\in T_{\mathop{Var}\nolimits}(d^{\omega})$. Hence, $f$ is a derived identity of $\mathop{Var}\nolimits$. ∎

Theorem 6**.**

If a multilinear identity $f$ holds on the variety governed by the operad $\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!}$ then $f$ is a generalized derived identity of $\mathop{Var}\nolimits$.

Proof.

For every $A\in\mathcal{N}_{\mathop{Var}\nolimits}$ we may construct an algebra in the variety $\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!}$ as follows. Consider the linear space $H$ spanned by elements $x^{(n)}$, $n\geq 0$, with multiplication

[TABLE]

This is a $\mathop{GD}\nolimits^{!}$-algebra (in characteristic 0, this is just the ordinary polynomial algebra with $x^{(n)}=x^{n}/n!$). Denote $\hat{A}=A\otimes H$ and define operations $\prec$, $\succ$, and $\cdot$ on $\hat{A}$ by

[TABLE]

The algebra $\hat{A}$ obtained belongs to the variety governed by the operad $\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!}$. There is an injective map

[TABLE]

which preserves operations $\prec$, $\succ$, and $\cdot$. Indeed, let us check that $\Phi$ preserves $\prec$. Apply (24) to $a\prec b$, $a,b\in A$:

[TABLE]

Since $\Phi$ is injective, $A^{(d*)}$ is isomorphic to a subalgebra of a $(\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!})$-algebra, so all defining relations of $\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!}$ hold on $A^{(d)}$. Lemma 3 completes the proof. ∎

It is easy to see that if we forget about the operation $\cdot$ on $\hat{A}$ then we obtain a proof of the following

Theorem 7**.**

If a multilinear identity $f$ holds on the variety governed by the operad $\mathop{Var}\nolimits\circ\mathrm{Nov}$ then $f$ is a derived identity of $\mathop{Var}\nolimits$. ∎

It remains to prove the converse: why all generalized derived identities of $\mathop{Var}\nolimits$ hold on all $(\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!})$-algebras.

Theorem 8**.**

Let $f$ be a multilinear generalized derived identity of $\mathop{Var}\nolimits$. Then $f$ holds on all $(\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!})$-algebras.

Proof.

Let $X=\{x_{1},x_{2},\dots\}$, and let $f(x_{1},\dots,x_{n})$ be a non-associative multilinear polynomial in three operations of multiplication $\cdot$, $\prec$, and $\succ$. By the construction of the Manin white product, if an identity holds on $\mathop{Var}\nolimits\langle X\rangle\otimes\mathop{GD}\nolimits^{!}\langle X\rangle$ equipped with operations (21)–(23) then it holds on all $(\mathop{Var}\nolimits\circ\mathop{GD}\nolimits^{!})$-algebras.

By Corollary 2, $\mathop{GD}\nolimits^{!}\langle X\rangle$ is a subalgebra of $C^{(d*)}$ for the free differential commutative algebra $C=\mathrm{Com}\mathrm{Der}\langle X,d\rangle$. Therefore, $\mathop{Var}\nolimits\langle X\rangle\otimes\mathop{GD}\nolimits^{!}\langle X\rangle$ is a subalgebra of $(\mathop{Var}\nolimits\langle X\rangle\otimes C)^{(D*)}$, where $D(u\otimes f)=u\otimes d(f)$ for $u\in\mathop{Var}\nolimits\langle X\rangle$, $f\in C$. Since $A=\mathop{Var}\nolimits\langle X\rangle\otimes C\in\mathop{Var}\nolimits$, the identity $f(x_{1},\dots,x_{n})=0$ holds on $A^{(D*)}$, so it holds on $\mathop{Var}\nolimits\langle X\rangle\otimes\mathop{GD}\nolimits^{!}\langle X\rangle$. ∎

Similarly, we have

Theorem 9**.**

Let $f$ be a multilinear derived identity of $\mathop{Var}\nolimits$. Then $f$ holds on all $(\mathop{Var}\nolimits\circ\mathrm{Nov})$-algebras.

Remark 5**.**

In this section, we have considered algebras with one binary operation, so the operad $\mathop{Var}\nolimits$ has only one generator. However, Theorems 6–9 are straightforward to generalize for algebras with multiple binary operations, i.e., for an arbitrary binary operad $\mathop{Var}\nolimits$. In the next sections, we apply these results to Poisson algebras.

4. Free special GD-algebra

Denote by $\mathrm{Pois}$ the variety of all Poisson algebras, and let $\mathrm{Pois}\mathrm{Der}$ stand for the variety of differential Poisson algebras. Theorem 1 determines a functor from $\mathrm{Pois}\mathrm{Der}$ to $\mathop{GD}\nolimits$ corresponding to the following morphism of operads:

[TABLE]

Let us say that a $\mathop{GD}\nolimits$-algebra $A$ is special if there exists $V\in\mathrm{Pois}$ such that $A$ is a subalgebra of $V^{(\delta)}$. Denote by $\mathop{SGD}\nolimits$ the variety generated by special $\mathop{GD}\nolimits$-algebras, i.e., the class of all homomorphic images of all special $\mathop{GD}\nolimits$-algebras. The main purpose of this section is to describe the free $\mathop{SGD}\nolimits$-algebra.

Let $X$ be a linearly ordered set of generators. Denote by $\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$ the free Poisson differential algebra with a derivation $d$. Consider the set $d^{\omega}X=\{d^{n}(x)\mid n\geq 0,x\in X\}$. Let us consider the elements $d^{n}(x)\in d^{\omega}X$ as pairs $(n,x)$ and compare them lexicographically. Then $\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$ is isomorphic to $\mathrm{Pois}\langle d^{\omega}X\rangle$ as a Poisson algebra.

As a commutative algebra, $\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$ is isomorphic to the symmetric algebra over the space $\mathrm{Lie}\langle d^{\omega}X\rangle$. Recall that a linear basis of the free Lie algebra $\mathrm{Lie}\langle Y\rangle$ generated by a linearly ordered set $Y$ may be chosen as follows. An associative word $U\in Y^{*}$ is said to be a Lyndon–Shirshov word if for every presentation $U=U_{1}U_{2}$, $U_{i}\in Y^{*}$, we have $U>U_{2}U_{1}$ lexicographically. This definition goes back to [5] and [19] (see also [3]). For every Lyndon–Shirshov word $U$ there is a unique (canonical) bracketing $[U]$ such that $[U]\in\mathrm{Lie}\langle Y\rangle$ and the set of all Lyndon–Shirshov words with canonical bracketing forms a linear basis of $\mathrm{Lie}\langle Y\rangle$.

Definition 2**.**

Let $u$ be a monomial in $\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$. Define the weight function $\mathop{wt}\nolimits(u)\in\mathbb{Z}$ by induction in the following way:

[TABLE]

Since all defining identities of Poisson algebras are homogeneous relative to $\{\cdot,\cdot\}$ and $\cdot$, the function $\mathop{wt}\nolimits$ is well-defined.

By the general algebra arguments (see, e.g., [12]) $\mathop{SGD}\nolimits\langle X\rangle$ is isomorphic to the subalgebra of $\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$ generated by $X$ relative to the operations $[u,v]=\{u,v\}$ and $u\circ v=ud(v)$.

Consider a linear map

[TABLE]

By Theorem 1, $\phi$ is an epimorphism of $\mathop{GD}\nolimits$-algebras.

For a monomial $u\in\mathop{GD}\nolimits\langle X\rangle$, denote by $m(u)$ the number of Novikov multiplications in $u$ or, which is the same, the number of commutative multiplications in $\phi(u)$. For a monomial $a\in\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$, denote $D(a)=n$ the total number of derivations $d$ in $a$. It is easy to see that $D(\phi(u))=m(u)$.

Lemma 4**.**

Let $a=x_{1}\dots x_{t}[U]\in\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$, where $x_{i}\in X$, $U$ is a Lyndon–Shirshov word in $d^{\omega}X$, $D([U])=t$. Then there exists $f\in\mathop{GD}\nolimits\langle X\rangle$ such that $\phi(f)=a$.

Proof.

Let us prove the statement by induction on $D([U])$ and $|U|$, where $|U|$ is the length of $U$ as of a word in $d^{\omega}X$.

If $D([U])=t=0$ then the desired pre-image may be obtained just by replacing $\{\cdot,\cdot\}$ with $[\cdot,\cdot]$. We do not distinguish notations for $[U]\in\mathrm{Lie}\langle X\rangle\subset\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$ and its pre-image in $\mathop{GD}\nolimits\langle X\rangle$.

Suppose $D(U)=1$. If $|U|=1$ then the result is obvious. Let $|U|=2$. Then

[TABLE]

so $[x_{1}\circ y_{1},y_{2}]-[x_{1},y_{2}]\circ y_{1}$ is a desired pre-image.

If $|U|=l>2$ then $[U]=\{[d(y_{1})u],[v]\}$ for some words $u$, $|u|=i<l-1$, and $v$, $|v|=j<l$, by the definition of a Lyndon–Shirshov word ($d(y_{1})$ is the greatest letter in $U$). Assume there exists

[TABLE]

for all Lyndon–Shirshov words $W$ such that $D(W)=1$, $|W|=j<l$. Then

[TABLE]

Therefore,

[TABLE]

is a desired pre-image of $x_{1}[U]$.

Suppose $D(U)=t>1$ and $|U|=1$, i.e., $U=d^{t}(y)$, $y\in X$. Then there exists a pre-image of $x_{1}\dots x_{t}d^{t}(y)$

[TABLE]

as shown in [7].

Assume $D(U)=t>1$, $|U|=l>1$, and

[TABLE]

exists for all Lyndon–Shirshov words $W$ such that either $i=D(W)<t$ or $D(W)=t$, $j=|W|<l$. Since

[TABLE]

where

[TABLE]

we have

[TABLE]

where $\hat{x}_{k}$ means that the factor $x_{k}$ is omitted. By the inductive assumption, there exist

[TABLE]

and

[TABLE]

where $\overline{f}$ stands for the linear combination of Lyndon–Shirshov words with canonical bracketing representing a Lie polynomial $f\in\mathrm{Lie}\langle d^{\omega}X\rangle$. Obviously, we may choose

[TABLE]

Here we consider the variable $z$ in the right-hand side of (27) as a new one, and assume $z<x$ for all $x\in X$. ∎

Theorem 10**.**

Differential Poisson monomials of weight $-1$ span the special GD-algebra $\mathop{SGD}\nolimits\langle X\rangle\subset\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$.

Proof.

Obviously, $\phi(f)$ is a linear combination of differential Poisson monomials of weight $-1$ for every $f\in\mathop{GD}\nolimits\langle X\rangle$.

Conversely, let $a$ be a monomial in $\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$ such that $\mathop{wt}\nolimits(a)=-1$. We have to show that there exists $\phi^{-1}(a)\in\mathop{GD}\nolimits\langle X\rangle$. It is enough to prove it for basic elements of $\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$, i.e., elements of the form

[TABLE]

where $X_{i}$ are Lyndon–Shirshov words in $d^{\omega}X$, $[\cdot]$ denotes the canonical bracketing, $X_{1}\leq\dots\leq X_{n}$.

Let us proceed by induction on $D(a)$. If $D([X_{i}])=k_{i}$ then $\mathop{wt}\nolimits([X_{i}])=k_{i}-1$, so $\mathop{wt}\nolimits([X_{i}])\geq-1$. Note that $\mathop{wt}\nolimits(a)=\mathop{wt}\nolimits([X_{1}])+\dots+\mathop{wt}\nolimits([X_{n}])=D(a)-n$. Hence, if $D(a)=0$ then $n=1$ and $a=[X_{1}]$ is a pure Lie element in $X$ which belongs to $\mathop{GD}\nolimits\langle X\rangle$.

Assume that for some $m>0$ all monomials $b\in\mathrm{Pois}\mathrm{Der}\langle X,d\rangle$ such that $\mathop{wt}\nolimits(b)=-1$, $D(b)<m$ have pre-images in $\mathop{GD}\nolimits\langle X\rangle$. Suppose $D(a)=m$ then, without loss of generality, we may assume $D([X_{1}])=k>0$, $k\leq m$. Then there must exist $k$ differential-free factors, say $[X_{2}],\dots,[X_{k+1}]$ of weight $-1$. Lemma 4 allows us to replace $[X_{1}]\cdots[X_{k+1}]$ with a new variable $y$, $\mathop{wt}\nolimits(y)=-1$, find a pre-image $f(y)$ of $y[X_{k+2}]\cdots[X_{n}])\in\mathop{GD}\nolimits\langle X\cup\{y\}\rangle$, and find the desired pre-image of $a$ as a substitute $f(y)|_{y\in\phi^{-1}([X_{1}]\cdots[X_{k+1}])}$. ∎

In order to calculate $\dim\mathop{SGD}\nolimits(n)$, we need to count Poisson differential monomials in $x_{1},\dots,x_{n}$ of weight $-1$. Let us introduce the following function on $n,k\in\mathbb{Z}$, $n,k\geq 1$:

[TABLE]

In particular, $H(n,2)$ is the $n$th partial sum of the harmonic series.

Corollary 3**.**

For the operad $\mathop{SGD}\nolimits$, we have

[TABLE]

Proof.

Consider the set $X=\{x_{1},x_{2},\dots\}$ which is linearly ordered as above: $x_{1}<x_{2}<\dots$. Let us first calculate the number $L(n,k)$ of homogeneous Poisson monomials $u_{1}\dots u_{k}\in\mathrm{Pois}(n)$ that split into $k$ Lyndon—Shirshov factors $u_{i}\in\mathrm{Lie}\langle X\rangle$, $u_{1}\leq\dots\leq u_{k}$ lexicographically. If $k=1$, $L(n,1)=(n-1)!$. If $k>1$ then $u_{k}$ starts with the greatest letter $x_{n}$. Suppose the length of $u_{k}$ is $d$, $1\leq d\leq n-1$. The number of such Lyndon—Shirshov words is $(d-1)!\binom{n-1}{d-1}$. The remaining $n-d$ letters form a product of $k-1$ Lyndon—Shirshov words $u_{1},\dots,u_{k-1}$. Hence,

[TABLE]

Therefore,

[TABLE]

where the sum ranges over all $(d_{1},\dots,d_{k-1})$ such that $d_{i}\geq 1$ and $d_{1}+\dots+d_{k-1}<n$. It is easy to see that

[TABLE]

for $n,k\geq 2$.

In order to get a differential Poisson monomial of weight $-1$, we have to apply $k-1$ derivations $d$ to the letters of a Poisson monomial $u_{1}\dots u_{k}$. There are $n$ letters $x_{1},\dots,x_{n}$, and we may differentiate every letter more than once. Hence, the number of all possible combinations is

[TABLE]

∎

In particular, for $n=1,\dots,7$ we have the following values:

[TABLE]

5. Defining identities of special GD-algebras

In this section, we describe the defining relations of the operad $\mathop{SGD}\nolimits$ and derive examples of special identities, i.e., those identities that distinguish $\mathop{GD}\nolimits$ and $\mathop{SGD}\nolimits$.

By definition, the operad $\mathop{SGD}\nolimits$ is the sub-operad of $\mathrm{Pois}\circ\mathop{GD}\nolimits^{!}$ generated by $x_{1}x_{2}\otimes x_{2}\star x_{1}$ and $\{x_{1},x_{2}\}\otimes x_{1}\ast x_{2}$.

Proposition 1**.**

The variety of $(\mathrm{Pois}\circ\mathop{GD}\nolimits^{!})$-algebras consists of linear spaces $A$ equipped with four bilinear operations $*$, $\circ$, $[\cdot,\cdot]$, and $[\cdot\succ\cdot]$ such that $(A,*,[\cdot,\cdot])$ is a Poisson algebra, $(A,\circ)$ is a Novikov algebra, and the following identities hold:

[TABLE]

[TABLE]

Proof.

Apply the computation process similar to that in Examples 1, 2 relative to the following notations for the generators of $\mathrm{Pois}\circ\mathop{GD}\nolimits^{!}$ in $\mathrm{Pois}\otimes\mathop{GD}\nolimits^{!}$:

[TABLE]

Straightforward solution of the corresponding linear algebra problem leads to the required identities. ∎

Corollary 4**.**

The T-ideal $T_{\mathop{SGD}\nolimits}(X)$ coincides with the intersection of $T_{\mathrm{Pois}\circ\mathop{GD}\nolimits^{!}}(X)$ with the space of nonassociative polynomials on operations $\circ$ and $[\cdot,\cdot]$. In other words, to get the defining relations of the operad $\mathop{SGD}\nolimits$ it is enough to find all those corollaries of the identities stated in Proposition 1 that contain only the operations $\circ$ and $[\cdot,\cdot]$.

According to Corollary 4, defining identities of $\mathop{SGD}\nolimits$ may be derived from the identities of $\mathrm{Pois}\circ\mathop{GD}\nolimits^{!}$ given by Proposition 1. For example, we may use (35) to eliminate the operation $\succ$ in (36). As a result, we obtain (3).

Finding the complete list of defining relations for the operad $\mathop{SGD}\nolimits$ seems to be a hard problem. We will deduce some of these relations below.

Consider the monomial $[y\succ x]*(z\circ u)$. Apply (32) and then (35) to get

[TABLE]

On the other hand, apply (35) to get

[TABLE]

Therefore, the following identity holds on $\mathop{SGD}\nolimits$-algebras:

[TABLE]

(It is also easy to check in a straightforward way that the corresponding identity holds on differential Poisson algebras.)

Proposition 2**.**

The identity (40) does not hold on the class of all $\mathop{GD}\nolimits$-algebras.

Proof.

Without loss of generality we may assume the base field $\Bbbk$ is algebraically closed. We will find an example of a $\mathop{GD}\nolimits$-algebra which does not satisfy (40) by means of algebraic geometry arguments using the approach of [18].

Let us fix a structure of a Lie algebra $\mathfrak{g}$ on a $n$-dimensional linear space spanned by $x_{1},\dots,x_{n}$. Namely, we fix a multiplication table of this Lie algebra

[TABLE]

Consider the class of all Novikov products $\circ$ on the space $\mathfrak{g}$ satisfying (3) as a Zariski closed set $\mathcal{V}(\mathfrak{g})$ in the $n^{3}$-dimensional affine space assuming

[TABLE]

A point $(a^{ij}_{k})_{i,j,k=1,\dots,n}\in\Bbbk^{n^{3}}$ belongs to $\mathcal{V}(\mathfrak{g})$ if and only if

[TABLE]

for all $i,j,p,l=1,\dots,n$. These relations represent the defining identities of GD-algebras.

Denote by $I(\mathfrak{g})$ the ideal in $\Bbbk[a^{ij}_{k}\mid i,j,k=1,\dots,n]$ generated by the left-hand sides of (41)–(43). If we assume that (40) holds for all GD-algebras in the class $\mathcal{V}(\mathfrak{g})$ then the corresponding polynomials

[TABLE]

belong to the radical of $I(\mathfrak{g})$.

For dimensions $n=2,3$ it is possible to compute the radical of $I(\mathfrak{g})$ in a straightforward way by means of the computer algebra system Singular [6]. These computations show that for every 2-dimensional Lie algebra $\mathfrak{g}$ all polynomials (44) belong to $\sqrt{I(\mathfrak{g})}$.

However, even for the 3-dimensional Heisenberg algebra $\mathfrak{h}_{3}$ spanned by $x_{1},x_{2},x_{3}$, $[x_{1},x_{2}]=x_{3}$, $x_{3}$ is central, there exist polynomials of the form (44) that do not belong to $\sqrt{I(\mathfrak{h}_{3})}$. ∎

Let us find one more (independent) special identity for $\mathop{GD}\nolimits$-algebras. Consider the expression

[TABLE]

Apply (38) first, then (32) and commutativity of $*$:

[TABLE]

Relation (35) and right commutativity of $\circ$ imply

[TABLE]

On the other hand, it follows from (35) that

[TABLE]

Finally, using (40) we obtain

[TABLE]

Proposition 3**.**

The identity (48) does not hold on the class of all $\mathop{GD}\nolimits$-algebras satisfying (40).

Proof.

Let us consider the class of algebras $(V,\circ,[\cdot,\cdot])$ satisfying the following identities:

[TABLE]

Obviously, all these algebras belong to $\mathop{GD}\nolimits$ and (40) holds. If we assume that (48) holds on all these algebras then

[TABLE]

has to be a corollary of (49). If $\mathrm{char}\,\Bbbk\neq 2$, relation (50) may be transformed to

[TABLE]

modulo (49). However, if we write down all those corollaries of (49) that contain two operations $\circ$ and one operation $[\cdot,\cdot]$ then we see that (51) does not belong to this list. ∎

Remark 6**.**

We do not suppose that (40) and (48) exhaust all special identities of $\mathop{GD}\nolimits$-algebras. Finding the complete list of special identities (or at least determining whether it is finite) is an interesting problem.

Remark 7**.**

If $(A,\circ)$ is a Novikov algebra then $(A,\circ,[\cdot,\cdot])$ with

[TABLE]

is a GD-algebra [9]. It is easy to check that (40) and (48) hold for such a GD-algebra. We conjecture that all GD-algebras obtained in this way are special.

Acknowledgements

The first author was supported by the Program of fundamental scientific researches of the Siberian Branch of Russian Academy of Sciences, I.1.1, project 0314-2016-0001.