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     \noindent\logo
     \hspace{80mm}\raisebox{1ex}{ISSN 1813-3304 }

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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
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     {\flushleft\it Том 12, стр. 144--144 (2015) \hspace{65mm}{\rm\small УДК \udcs}}
     \newline
        {\rm\small DOI~10.17377/semi.2015.12.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
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\begin{document}
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\title[Dual coalgebra of differential]{Dual coalgebra of the differential polynomial algebra in one variable and related coalgebras}
\author{{V.~N.~Zhelyabin, P.S. Kolesnikov }}%
\address{Victor N. Zhelyabin  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia
}%
\email{vicnic@math.nsc.ru}%
\address{Pavel S. Kolesnikov % Контактные данные всех авторов и место работы указываются только на английском языке
	\newline\hphantom{iii} Sobolev Institute of Mathematics,
	\newline\hphantom{iii} pr. Koptyuga, 4,
	\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{pavelsk77@gmail.com}%


\thanks{\sc Zhelyabin, V.N., Kolesnikov, P.S.   Dual coalgebra of the differential polinomial algebra  in one variable and related coalgebras}
\thanks{\copyright \ 2022 Zhelyabin V.N., Kolesnikov P.S.
}
\thanks{\rm Supported by RAS Fundamental Research Program, project  FWNF-2022-0002
}
\thanks{\it Поступила 1 января 2015 г., опубликована 31 декабря 2015 г.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. }  We show that the  dual  coalgebra of the polynomial algebra in one variable is the space of
linearly recursive sequences over an arbitrary field. Moreover, this coalgebra is a differential one relative to the dual standard derivation and  does not contain  nonzero finite-dimensional diffe\-rentially closed subcoalgebras if the   characteristic of the ground field is zero. 
We construct a Novikov co\-al\-geb\-ra which is the dual coalgebra
of the left-symmetric Witt algebra of index one.  
Also, we construct a Jordan supercoalgebra which is dual to 
the Jordan superalgebra of vector type of the polynomial algebra in one variable. 
All these  coalgebras do not contain non-zero finite-dimensional subcoalgebras  if the characteristic of ground field is zero. 
It is  shown that over a  field 
of characteristic different from~2
the adjoint Lie coalgebra of the dual coalgebra of the left-symmetric Witt algebra of index one is isomorphic to the  dual coalgebra of the Witt algebra of index one.
\medskip

\noindent 
{\bf Key words:} coalgebra, coderivation, associative commutative algebra, differential algebra, Novikov algebra, Lie algebra, Witt algebra,  Jordan superalgebra, locally finite coalgebra
 \end{quote}
}
In general, an algebra is a vector space $A$ over a ground field $F$ equipped with a linear map
$m: A\otimes A \to A$.
The dual notion is known as a coalgebra.
The theory of coalgebras
has been  initially developed within the framework of the theory of Hopf algebras \cite{Sweedler1969}.
The main result of the theory of associative coalgebras is the Fundamental Theorem on Coalgebras, which asserts that every associative coalgebra over a field is locally finite.
The latter means that every finitely generated coalgebra is finite-dimensional.

The interest to nonassociative coalgebras is related to the notion of a quantum group introduced by Drinfeld \cite{Drinfeld1987}.
Initially, it was associated with the notion of a Lie bialgebra.
The latter was introduced in \cite{Drinfeld1983} as one of the most important notions of quantum group theory.
A Lie bialgebra is simultaneously a Lie algebra and a Lie coalgebra.
In contrast to associative coalgebras
there exist Lie coalgebras that are not locally finite~\cite{Michaelis1980}.

It is  known (see \cite{Sweedler1969, Michaelis1980}) that the dual algebra of an associative coalgebra or a Lie coalgebra is associative or Lie, respectively. In \cite{Anquella1994} (1994),
 the notion of a coalgebra related to some variety of algebras was introduced.
In particular, alternative and Jordan coalgebras were defined, and their local  finiteness  was proved.
An analogue of this result is true for structurable  coalgebras \cite{Zhelyabin1996},
for Jordan copairs \cite{Zhelyabin2007},
for right alternative Malcev admissible coalgebras,
and for binary $(-1,1)$-coalgebras  \cite{Santos2021}.
In  \cite{Slinko1995}
(1995), some necessary and sufficient conditions for a Lie coalgebra to be locally finite
were  found.

In the papers \cite{Zhelyabin1996b, Zhelyabin2003} the first author found the  connection between Jordan and Lie (super)co\-al\-geb\-ras,
which is an analogue of the well-known Kantor--Koecher--Tits construction
for usual (super)\-algebras.
In  \cite{Goncharov2012, Goncharov2013}, it was shown that every Malcev coalgebra is embedded into
a Lie coalgebra with triality.

As mentioned above, there exist non-locally finite Lie co\-al\-geb\-ras.
In \cite{Kozybaev2000}, an example of a non-locally finite right-symmetric coalgebra was constructed.
In cont\-rast to Jordan coalgebras, non-locally finite Jordan supercoalgebras exist~\cite{Zhelyabin2003}.

In \cite{Kozybaev2022}, it was constructed an example of a non-locally finite differential  co\-al\-geb\-ra.
On this differential coalgebra we can define a Lie comultiplication so that the  obtained Lie coalgebra
 coincides with the example of Michaelis  presented in~\cite{Michaelis1980}.
The dual analogue of the Gelfand--Dorfman construction was proposed in \cite{Kozybaev2022},
which implied the construction of  Novikov
coalgebras based on differential associative
commutative coalgebras.
Using this construction,  an example of a non-locally finite Novikov coalgebra
was built.  A dual analogue of the Kantor construction
for usual Jordan superalgebra
was also presented in \cite{Kozybaev2022},
as a corollary, a new example of a non-locally finite Jordan supercoalgebra
was constructed.

In   \cite{Kozybaev2022} 
and \cite{Santos2021b},
examples non-locally finite right alternative coalgebras
were constructed.

In \cite{Michaelis1990},
it was shown that the dual coalgebra $W_1^\circ$ of the Witt Lie algebra
$W_1$ is a non-zero Lie coalgebra which does not contain non-zero finite-dimensional
subcoalgebras. An analogue of this result for  Jordan  supercoalgebra was obtained in \cite{Zhelyabin2005}.
The structure of the dual Lie coalgebra of the Witt algebra over field of characteristic not 2 and zero 
 was also described in~\cite{Nichols1990, Nichols1993}. 
These results were generalized to the case of several derivations 
in \cite{Griffing2002}.
Namely in \cite{Nichols1990} was shown that $W_1^\circ$ is the space of
linearly recursive sequences,  if $\mathrm{char}\,F=0$  or $\mathrm{char}\,F=p> 2$.

It is known that both the Witt Lie algebra $W_1$  and
the left-symmetric algebra
$\mathcal{L}_1$
can be  obtained  from the differential
polynomial algebra in one variable
by means of appropriate constructions.
It follows from \cite[Theorem 1]{Nichols1993} that the dual coalgebra 
$W_1^\circ $ is obtained from $P_1^\circ $
by means of the dual construction.
The purpose  of this paper is to obtain the dual  analogue of these results
for the  dual coalgebra $\mathcal{L}_1^\circ$.

In particular, we prove that the  dual coalgebra
$P_ 1^\circ$ of the  algebra $P_ 1$ in one variable
is  the space of linearly recursive sequences over an arbitrary field,  Moreover, if the characteristic of ground field is zero then the coalgebra
$P_1^\circ$  does not contain finite-dimensional differentially closed subcoalgebras.

We also  show that over field of characteristic different from 2 the  dual coalgebra of the left-symmetric Witt algebra of index one
can be obtained from the coalgebra  $P_1^\circ$ by means of the dual Gelfand--Dorfman construction.

Finally, we describe the  dual supercoalgebra of a Jordan superalgebra  obtained by the Kantor construction from the differential polynomial algebra in one variable.

\section{Coalgebras and coderivations}

Let $F$ be an arbitrary field.
Denote by  $\underbrace{V\otimes\ldots \otimes V}_n$
the $n$th tensor power of the vector space $V$ over~$F$.
Denote by $V^*$  the dual vector space of~$V$,  i.~e.,
$V^*={\rm Hom}_F(V,F)$.

The map
$$
\rho:\underbrace{ V^*\otimes\ldots\otimes  V^*}_n\to (\underbrace{V\otimes\ldots\otimes V}_n)^*
$$
defined by
$$
\rho(f_1\otimes\ldots\otimes f_n)\bigg(\sum_{i_1,\ldots, i_n}v_{i_1}\otimes\ldots\otimes  v_{i_n}\bigg )=\sum_{i_1,\ldots, i_n}f_1(v_{i_1})\ldots f_n(v_{i_n})
$$
is injective.
Therefore, we can assume that
$$
\underbrace{ V^*\otimes\ldots\otimes  V^*}_n\subseteq  (\underbrace{V\otimes\ldots\otimes V}_n)^*.
$$
If $\phi:V \to U$ is a linear map of vector spaces
then the transpose $\phi^*: U^*\to V^*$
of~$\phi$ is  defined by the rule
$\phi^*(u^*)(v)=u^*(\phi(v))$,
where $v\in V$, $u^*\in U^*$.

\begin{definition}
A pair  $(C,\Delta)$, where  $C$ is a vector space over~$F$ and
$\Delta: C\to C\otimes C$ is a linear map, is called {\it a
coalgebra}. The map  $\Delta$ is said to be a {\it comultiplication}
of~$C$. For an element  $a\in C$, we will use the Sweedler's
notation (see \cite{Sweedler1969}), namely, $\Delta(a)=\sum\limits_{(a)} a_{(1)}\otimes
a_{(2)}$.
\end{definition}

If  $(C,\Delta)$ is a coalgebra then
the rule $f\otimes g \mapsto fg$, where
$$
(fg)(a)=\rho(f\otimes g)(\Delta(a))=\sum_{(a)}f(a_{(1)})g(a_{(2)}),
\quad
f,g\in C^*,\ a\in C,
$$
defines a product on $C^*$, so that $C^*$ is an algebra.
The algebra $C^*$ is called the {\it dual algebra} of $(C,\Delta)$.

The dual algebra  $C^*$ has natural  left  and  right actions
on the initial coalgebra~$C$.
Namely,
$$
\alpha\cdot a=\sum_{(a)}a_{(1)}\alpha(a_{(2)}),
\quad
a\cdot \alpha=\sum_{(a)}\alpha(a_{(1)})a_{(2)},
$$
for $\alpha \in C^*$, $a\in C$.
Hence, $C$ is a $C^*$-bimodule.


A linear map $d:C\to C$ is called a {\it coderivation} of a coalgebra  $(C,\Delta)$ if
$$
\Delta d=(d\otimes id+id\otimes d)\Delta,
$$
i.~e., for every $a\in C$
$$
\Delta(d(a))=\sum_{(a)} d(a_{(1)})\otimes a_{(2)}+a_{(1)}\otimes
d(a_{(2)}).
$$

The following statement is well-known. 
\begin{lemma}
Let $d$ be a coderivation of a coalgebra  $(C,\Delta)$.
Then its transpose map  $d^*$  is a derivation of the dual algebra
$C^*$, i.~e.,
    $$
    d^*(fg)=d^*(f)g+fd^*(g)
    $$
holds in the algebra  $C^*$ for all $f,g\in C^*$.
\end{lemma}

%\begin{proof}
%Let  $f,g\in C^*$, $a\in C$. Then
%\begin{multline*}
 %   (d^*(fg))(a)=(fg)(d(a))=\rho(f\otimes %g)\Delta(d(a))=\\
  %  =\sum_af(d(a_{(1)}))g(a_{(2)})+f(a_{(1)})g(d(a%_{(2)}))=\\
   % =   \sum_a(d^*(f))(a_{(1)})g(a_{(2)})+f(a_{(1)%})(d^*(g))(a_{(2)})
    %=(d^*(f)g+fd^*(g))(a).
%\end{multline*}
    %Thus we obtain $d^*(fg)=d^*(f)g+fd^*(g)$.
%\end{proof}

A coalgebra equipped with a coderivation is called a {\it differential coalgebra}.
\medskip

A coalgebra $(C,\Delta)$ is said to be associative
(coassociative) if
$$
(\Delta\otimes id- id \otimes\Delta)\Delta=0,
$$
i.~e., for every
$a\in C$
$$
\sum_{(a)}(\Delta(a_{(1)})\otimes a_{(2)}- a_{(1)}
\otimes\Delta(a_{(2)}))=0.
$$

It is  known that a coalgebra  $(C,\Delta)$
is associative if and only if its dual algebra~$C^*$
is associative. By this reason, the following definition
of a coalgebra related  to some variety of algebras
was stated in~\cite{Anquella1994}.

\begin{definition}
Let  $\mathcal{M}$ be a variety of algebras.
Then a coalgebra  $(C,\Delta)$ is called  an
$\mathcal{M}$-{\it coalgebra} if its dual algebra
$C^*$  is an algebra of~$\mathcal{M}$.
\end{definition}

Let  $V$ be a vector space, and let the linear map
$\tau:V\otimes V\to V\otimes V$
is defined by
$\tau(x\otimes y)=y\otimes x$, $x,y\in V$.

A coalgebra $(C,\Delta)$ is commutative (cocommutative) if
$$
\Delta=\tau\Delta,
$$
i.~e.,
$$
\sum_{(a)} a_{(1)}\otimes a_{(2)}
=\sum_{(a)} a_{(2)}\otimes a_{(1)}
$$
for every $a\in C$.

Let $(C,\Delta)$ be an arbitrary coalgebra.
A vector subspace  $B$ of  $C$
is a {\it subcoalgebra} of
$(C,\Delta)$ if
$\Delta(B)\subseteq B\otimes B$.

It is  known (see \cite{Anquella1994}) that a vector space
$B$ of $C$ is  a subcoalgebra  if and only if
$B$ is a submodule of the  $C^*$-bimodule~$C$.
Therefore, the intersection of a family of subcoalgebras
of $(C,\Delta)$ is again a subcoalgebra of~$C$.

Recall   that the orthogonal complement
$$
B^\perp=\{\alpha \in C^* \mid B\subseteq \ker\, \alpha\}
$$
of a subcoalgebra  $B$ of $(C,\Delta)$
is an ideal of the algebra~$C^*$.
Conversely, the orthogonal complement
$I^\perp$ of an ideal~$I$  of~$C^*$
is a subcoalgebra of $(C,\Delta)$ (see \cite[Proposition 1.4.9]{Sweedler1969}).

Let  $S$ be a subset   of~$C$.
The smallest subcoalgebra which contains~$S$ is called the
{\it subcoalgebra generated by $S$} and denoted by $Coalg(S)$.
In   other words, $Coalg(S) $ is the  submodule of the
$C^*$-bimodule $C$, generated by~$S$.
If $S$ is a finite set then $Coalg(S)$ is called finitely generated coalgebra.

\begin{definition} A coalgebra  $(C,\Delta)$ is called
{\it locally finite} if every finitely generated subcoalgebra of~$C$
is finite-dimensional.
\end{definition}

Let $(C,\Delta)$ be a coalgebra.
Denote by  $Loc(C)$ the sum of all  locally finite subcoal\-gebras of~$C$.
It is clear that $Loc(C)$ is  a  locally finite coalgebra.

Let  $A$ be an algebra over a field~$F$ with a multiplication
$m: A\otimes A \to A$, i.~e.,
$m(a\otimes b)=ab$  for  $a,b\in A$.
Then $m^*:A^*\to (A\otimes A)^*$
is the transpose map of~$m$.
A~vector subspace  $V$ of~$A^*$
is called  {\it good} if $m^*(V)\subseteq \rho(V\otimes V)$.
If $V$ is a good subspace then we can define the comultiplication
$\Delta_V$ on $V$ by the rule $\Delta_V=\rho^{-1}m^*$.

Let  $A^\circ$ be the sum of all good subspaces of $A^*$.
Then  $A^\circ$ is the largest good subspace of $A^*$, and hence the pair $(A^\circ,\Delta^\circ)$
is a coalgebra with the comultiplication
$\Delta^\circ=\Delta_{A^\circ}$  \cite{Michaelis1980, Anquella1994}.
The coalgebra  $(A^\circ,\Delta^\circ)$
is called the {\it finite dual coalgebra}
(or simply {\it dual coalgebra})  of~$A$.
For every $a,b\in A$ and for every $f\in A^\circ$
we have
$f(ab)=\sum_f f_{(1)}(a)f_{(2)}(b)$,
where $\Delta^\circ(f)=\sum_ff_{(1)}\otimes f_{(2)}$.

If $A$ is finite dimensional then $A^*\otimes A^*\equiv  \rho(A^*\otimes A^*) = (A \otimes  A)^*$  and
the dual space $A^*$ is a coalgebra with the comultiplication $\Delta = \rho^{-1}m^*$. Therefore, $(A^\circ,\Delta^\circ)=(A^*,\Delta)$.

In \cite{Anquella1994}, it was shown that the  dual coalgebra $A^\circ$ is
a $\mathcal{M}$-coalgebra if $A$ is an algebra of variety~$\mathcal{M}$.

\begin{lemma}
Let  $A$ be an algebra over a field  $F$, and let $(V,\Delta)$
be a coalgebra, where  $V\subseteq A^*$.
If $f(a b)=\rho(\Delta(f))(a\otimes b)$ holds
for every $a,b\in A$ and for every  $f\in V$
then  $V$ is a good subspace of~$A^*$.
\end{lemma}

\begin{proof}
By the definition of $m^*$, we have
$m^*(f)(a\otimes b)=f(ab)$
for all $a,b\in A$, $f\in A^*$.
 If $f(a b)=\rho(\Delta(f))(a\otimes b)$ holds for  $f\in V$
 then
$$
m^*(f)(a\otimes b)=\rho(\Delta(f))(a\otimes b).
$$
Consequently, $m^*(f)=\rho(\Delta(f))$.
Thus  $m^*(V)\subseteq \rho(V\otimes V)$.
\end{proof}

Let  $\mathcal{I}$ be the set of finite-codimensional
ideals  of an algebra~$A$.
Let us define
$$
A^*_\mathcal{I} =
 \{\alpha\in A^* \mid \text{exists } I\in \mathcal{I} \text{ such that } I\subseteq \ker \alpha \}.
$$
As it was shown in \cite{Anquella1994},
$A^*_\mathcal{I}=Loc(A^\circ)$.
It is known that
$A^*_\mathcal{I}=Loc(A^\circ)=A^\circ$ for every 
associative algebra~$A$.

\section{Dual coalgebra of the differential polynomial algebra $F[x]$}

Let $P_1=F[x]$
be the polynomial algebra in one variable
equipped with the standard derivation $d=\frac{d}{dx}$.
Then
$$
P_1^\circ=\{\alpha\in P_1^* \mid \text{ exists } f(x)\in F[x], f(x)\neq 0 \text{ such that } f(x)F[x]\subseteq \ker \alpha \}.
$$
Let
$\Delta^\circ=\Delta_{P_1^\circ}$.
For $\alpha\in P_1^*$ and $a\in P_1$
put
$\langle \alpha, a\rangle =\alpha(a)$.
Also, let us denote $d^\circ=d^*
|_{P_1^\circ}$,
where $d^*$ is the
transpose map  of the derivation~$d$.
Let  $(P_1,d)^\circ$ be the sum of all good subspaces
of $P_1^*$ which are differential coalgebras with the coderivation $d^\circ$.

\begin{lemma}\label{lemma3}
     The inclusion  $d^\circ(P_1^\circ)\subseteq P_1^\circ$ holds.
     The triple $(P_1^\circ,\Delta^\circ,d^\circ) $ is an associ\-ative
     and commutative differential coalgebra with the coderivation~$d^\circ$. Moreover, $(P_1^\circ,\Delta^\circ,d^\circ)=(P_1,d)^\circ$.
\end{lemma}

\begin{proof}
Let us show that  $d^\circ(P_1^\circ)\subseteq P_1^\circ$.
Suppose $u\in P_1^\circ$, then there exists $I=f(x)P_1$, $f(x)\ne 0$, such that $\langle u,I\rangle=0$. Consider $J=I^2$: this is an ideal of finite codimension in $P_1$, and
$\langle d^*(u), J\rangle = \langle u, d(J)\rangle \subseteq \langle u, I\rangle =0$ since $d(I^2)\subseteq Id(I)\subseteq I$.

Obviously, $(P_1^\circ,\Delta^\circ,d^\circ) $
is a differential coalgebra with coderivation~$d^\circ$.
Hence,  $(P_1^\circ,\Delta^\circ,d^\circ)=(P_1,d)^\circ$.
\end{proof}

\begin{remark}
The proof of Lemma~\ref{lemma3} remains valid 
for an arbitrary associative commutative algebra
$A$ equipped with an arbitrary derivation $D$.
Hence, the equality $(A^\circ, \Delta^\circ, D^\circ) = (A,D)^\circ $ holds in general.
\end{remark}

\begin{theorem}
Let $F$ be a field  of characteristic  zero.
Then the coalgebra  $(P_1^\circ,\Delta^\circ,d^\circ) $
does not contain  non-zero finite-dimensional
differentially closed subcoalgebras.
\end{theorem}

\begin{proof}
Let  $B$ be a finite-dimensional differentially closed subcoalgebra of
the diffe\-ren\-tial coalgebra
$(P_1^\circ,\Delta^\circ,d^\circ) $.
Then the orthogonal complement  $B^\perp$ is a
differential\-ly closed ideal of~$P_1$.
Moreover, the ideal  $B^\perp$
has finite codimension. Since the characteristic of $F$ is zero then
$P_1$ is a differentially simple algebra.
Therefore, either  $B^\perp=0$ or $B^\perp=P_1$.
The first option is impossible since $P_1$ is an infinite-dimensional
algebra.  Consenquently, $B^\perp=P_1$ and $B=0$.
\end{proof}

Let $W_1$ be the Witt algebra of index one, i.~e., $W_1={\rm Der}_F
(P_1)$; $P_1^*$ and $W_1^*$ are isomorphic as vector spaces.


Recall that the vector space $P_1$ is a Lie algebra relative to the
operation
$$
[f,g]_d=fd(g)-d(f)g,
$$
this Lie algebra is isomorphic to the Witt algebra~$W_1$ under isomorphism $f\mapsto fd$. Thus, we can identify  $(P_1, [\,, ]_d)$ and $W_1$.

Define the new comultiplication $\Delta_{d^\circ}^{(-)}$ on the
vector space  $P_1^\circ$:
$$
\Delta_{d^\circ}^{(-)}(\alpha) =(id\otimes d^\circ - d^\circ\otimes
id)\Delta^\circ(\alpha).
$$
Then
$$
\langle \alpha, [f,g]_d)\rangle =\langle
\alpha,fd(g)-d(f)g\rangle=\rho(\Delta_{d^\circ}^{(-)}(\alpha))(f\otimes
g)
$$
for  $\alpha\in P_1^\circ$ and $f,g\in P_1$. Therefore, $P_1^\circ$
is a good subspace of $W_1^*$. Con\-sequent\-ly, we can
assume that $(P_1^\circ,\Delta_{d^\circ}^{(-)}) $ is a subcoalgebra
of
 $(W_1^\circ, \Delta_{W_1^\circ})$.


 In  the algebra
$P_1$ we put $x_i=x^{i+1}$,
$i=-1,0,1,\ldots $.
Then we have $x_i\cdot x_j=x_{i+j+1}$ in the algebra~$P_1$.


We identify elements of $P_1^*$  with
sequences of elements of $F$. Namely, every $\alpha \in  P_1^*$ corresponds to $(\alpha  (x_n))_{n \geq -1}$.


Following \cite{Nichols1990}, we say that a sequence $(a_n)_{n\geq-1}$
of elements of $F$ is   ($F$-){\it linearly recursive} if there exist
$\beta_0,\beta_1,\dots ,\beta_r\in F$, not all equal to zero,
and a number $k$ such that
$\sum_{i=0}^r\beta_ia_{n+i}=0$ for all $n\geq k$.


In the next statement, the base field $F$
is of arbitrary characteristic.


\begin{theorem} Let $V$ be the space of
    linearly recursive sequences. Then $P_1^\circ=V$. In particular, the space  $V$ is the dual  coalgebra of
$P_1$. Therefore, the coalgebra $(V,\Delta_{d^\circ}^{(-)})$  is a Lie subcoalgebra of
$(W_1)^\circ$.     \end{theorem}

\begin{proof} Let $\alpha \in P_1^* $ and $a_n=\alpha(x_n)$, where $n\geq -1.$


    Assume $\alpha \in P_1^\circ $. We show that the
    sequence $(a_n)_{n\geq-1}$  is  linearly recursive.
    There  exists $ f(x)\in P_1$, $f(x)\neq 0$,
    such that $\alpha(f(x)g(x))=0$ for all $g(x)\in P_1$.
    Let $f(x)=\beta_rx_{r-1}+\ldots +\beta_0x_{-1}$, where $\beta_0,\beta_1,\dots,\beta_r\in F$.
    Since $f(x)\neq 0$, not all   $\beta_0,\beta_1,\dots ,\beta_r$
    are  zero.
    Then for all $n\geq 0$ we have
    $$
    \sum_{i=0}^r\beta_ia_{n+i} =\sum_{i=0}^r\beta_i\alpha(x_{n+i}) =\sum_{i=0}^r\beta_i\alpha(x^{n+i+1})=\alpha(f(x)x^{n+1})=0.$$
    Hence, the sequence $(a_n)_{n\geq-1}$  is  linearly recursive.


    Let  $(a_n)_{n\geq-1}$  be  a  linearly recursive  sequence. Then there exist  $\beta_0,\beta_1,...,\beta_r\in F$, not all zero,
    and a number $k$, such that
    $\sum_{i=0}^r\beta_ia_{n+i}=0$ for all $n\geq k$. Put $f(x)=\beta_rx_{k+r}+\ldots+\beta_0x_{k}$. Then for all $j\geq -1$ we get
    $$
    \alpha(f(x)x_j)=\sum_{i=0}^r\beta_i\alpha(x_{k+i+j+1})=\sum_{i=0}^r\beta_ia_{k+i+j+1}=0,
    $$
    since $k+j+1\geq k$.  Consequently,  $f(x)P_1\subseteq \ker \alpha.$

Since $(P_1^\circ,\Delta_{d^\circ}^{(-)}) $ is a subcoalgebra of
 $(W_1^\circ, \Delta_{W_1^\circ})$ then the space V  is a Lie subcoalgebra of
$(W_1)^\circ$.
    \end{proof}

By \cite[Theorem 5]{Nichols1990} ,  if $F$ has characteristic $0$ or
characteristic $p\neq 2$, then $W_1^\circ$  is the space of
    linearly recursive sequences. Hence, $P_1^\circ= W_1^\circ$  by Theorem~2.





Therefore, the following statement holds.

\begin{corollary}[see also \cite{Nichols1993}] Define   the comultiplication
    $$
    \Delta_{d^\circ}^{(-)}(\alpha)=(id\otimes d^\circ - d^\circ\otimes id)\Delta^\circ(\alpha).
    $$
    on the vector space  $P_1^\circ$. If $F$ is  a field of characteristic $\neq 2$ then $(P_1^\circ,\Delta_{d^\circ}^{(-)})=(W_1^\circ, \Delta_{W_1^\circ}).$
\end{corollary}



Following \cite{Michaelis1990}, we define elements $y_n$, of $W_1^*$, $n\geq -1$, by $y_n(x_i) =\delta_{n,i}$. Define
$Y=span(y_n\mid n\geq -1)$,
and let $\Delta_{Y}$ be the restriction
of $m^*$ in $P_1^*$ on the space~$Y$. Then $$\Delta_Y(y_n)=\sum_{i+j=n-1}y_i\otimes y_j.$$
By Lemma 2, $Y $  is a good subspace of $ P_1^*$. Consequently, $(Y,\Delta_{Y})$
is a subcoalgebra of $(P_1^\circ,\Delta^\circ)$.




Now following  \cite{Nichols1990},
define $y_{a,j}\in P_1^*$,
for $0\neq a\in F$, $j\geq -1$,
by putting
$y_{a,j}(x_i)=
a^i\begin{pmatrix}  i+1 \\  j+1 \\ \end{pmatrix}$.
Define
$Y_a=span(y_{a,j}\mid j\geq -1)$,
and let $\Delta_{Y_a}$ be the restriction
of $m^*$ in $P_1^*$ on the space~$Y_a$.
For the sake of uniformity, we set $Y_0 = Y$.
Let us turn $Y_a$ into a differential 
subcoalgebra of $P_1^\circ $.

\begin{lemma}
The space $Y_a$ is a good subspace  of
$P_1^*$, and
$(Y_a,\Delta_{Y_a},d^\circ)$
is a differential coalgebra.
Consequently,
$(Y_a,\Delta_{Y_a},d^\circ)\subseteq (P_1,d)^\circ$. If $F$ is an algebraically closed field of characteristic zero then $P_1^\circ=\oplus_{a\in F}Y_a$
\end{lemma}

\begin{proof} First let  $a=0$. We prove that    $d^\circ(y_k)=(k+2)y_{k+1}$ for $k\geq -1$. Indeed, if $i>-1$ then $$ d^\circ(y_k)(x_i)= y_k(d((x_i))=(i+1)y_k(x_{i-1})=(i+1)y_{k+1}(x_{i})=(k+2)y_{k+1}(x_{i}).$$ If $i=-1$ then
$ d^\circ(y_k)(x_{-1})= y_k(d((1))=0.$ On the other hand $y_{k+1}(x_{-1})=\delta_{k+1,-1}=0.$ Hence, $ d^\circ(y_k)(x_{-1})=(k+2)y_{k+1}(x_{-1}).$

 It is clearly that $d^\circ$ is a coderivation of $(Y,\Delta_Y)$. Therefore,  $(Y,\Delta_Y,d^ \circ ) $
is a good subspace of $(P_1,d)^*$.
Consequently,
$(Y,\Delta_{Y},d^\circ)\subseteq (P_1,d)^\circ$.



    Let $0\neq a\in F$.
Recall that for binomial coefficients we have
$$\begin{pmatrix}
    r+s+2 \\
    k+1
\end{pmatrix}
=\sum_{i+j=k-1,\, i,j\geq   -1}
\begin{pmatrix}
    r+1 \\
    i+1
\end{pmatrix}
\begin{pmatrix}
s+1 \\
j+1
\end{pmatrix},
$$
where $r,s,k\geq -1$.

The comultiplication $\Delta_{Y_a}$ on the space
$Y_a$ is given by
$$
\Delta_{Y_a}(y_{a,n})=\sum_{i+j=n-1,\, i,j\geq-1 } ay_{a,i}\otimes y_{a,j}.
$$
Indeed,
\begin{multline*}
\rho(\Delta_{Y_a}(y_{a,k}))(x_r\otimes x_s)
=\sum_{i+j=k-1,\,i,j\geq -1}ay_{a,i}(x_r)y_{a,j}(x_s)= \\
\sum_{i+j=k-1,\,i,j\geq -1}a^{r+s+1}
\begin{pmatrix}
    r+1 \\
    i+1
\end{pmatrix}
\begin{pmatrix}
    s+1 \\
    j+1
\end{pmatrix}
=a^{r+s+1}
\begin{pmatrix}
    r+s+2 \\
    k+1
\end{pmatrix}  = \\
= y_{a,k}(x_{r+s+1}) = y_{a,k}(x_r\cdot x_s).
\end{multline*}
%
Therefore, $Y_a$ is a good subspace of~$P_1^*$.
Consequently, $(Y_a,\Delta_{Y_a})$
is a subcoalgebra of $(P_1^\circ,\Delta^\circ)$.

Let us show that for the coderivation  $d^\circ$ we have  $d^\circ(y_{a,k})=a^{-1}(k+2)y_{a,k+1}$.
Indeed, if $i>-1$
then
\begin{multline*}
d^\circ(y_{a,k})(x_i)=y_{a,k}(d(x_i))=(i+1)y_{a,k}(x_{i-1})=a^{i-1}(i+1)
\begin{pmatrix}
    i \\
    k+1
\end{pmatrix} = \\
a^{i-1}(i+1)\frac{i!}{(k+1)!(i-k-1)!}=
a^{-1}(k+2)a^i\frac{(i+1)!}{(k+2)!(i+1-k-2)!}= \\
= a^{-1}(k+2)y_{a,k+1}(x_i).
\end{multline*}
 %
If $i=-1$ then  $d^\circ(y_{a,k})(x_{-1})=y_{a,k}(d(1))=0$.
On the other hand,
$y_{a,k+1}(x_{-1})=a^{-1}
\begin{pmatrix}
    0 \\
    k+2 \\
\end{pmatrix} =0$.
Therefore,
$d^\circ(y_{a,k})(x_{-1})=a^{-1}(k+2)y_{a,k+1}(x_{-1})$.

Consequently,  $(Y_a,\Delta_{Y_a},d^\circ)$ is a good subspace of $(P_1,d)^*$, so
$(Y_a,\Delta_{Y_a},d^\circ)\subseteq (P_1,d)^\circ$.



If $F$ is an algebraically closed field of characteristic zero then
Theorems~4 and~5 from \cite{Nichols1990}
imply
$W_1^\circ=\oplus_{a\in F}Y_a$.
Hence,
$P_1^\circ=\oplus_{a\in F}Y_a$.
\end{proof}

An algebra $(A,\circ)$ with a  multiplication operation
$\circ $ is called a {\it Novikov algebra} 
(see \cite{Gelfand1979, Balinskii1985})
if   $A$ satisfies  the following identities:
\[
x\circ(y\circ z) - (x\circ y)\circ z = y\circ(x\circ
z)-(y\circ x)\circ z \text{ (left symmetry)},
\]
\[
(x\circ y)\circ z =(x\circ z)\circ y \text{ (right commutativity )}.
\]

Define the operation of  multiplication  $\circ$ on  $P_1$  by
the rule
$f\circ g=fd(g)$, $f,g\in P_1$.
Then  $(P_1,\circ)$ is a Novikov algebra  (see \cite{Gelfand1979}).
Following  \cite{Kozybaev2016}, we will call it left-symmetric Witt algebra of index~1 and denote by~$\mathcal{L}_1$.
Let  $x_i=x^{i+1}$, $i=-1,0,1,\ldots$.
Then the equality  $x_i\circ x_j=(j+1)x_{i+j}$ holds in the  algebra  $\mathcal{L}_1$.
It is known that the algebra  $\mathcal{L}_1$ is simple, if $F$ is field of characteristic zero.

Define the comultiplication  $\Delta_N$ on  $P_1^\circ$  by
the rule
$$
\Delta_N(\alpha)=(id\otimes d^\circ)\Delta^\circ(\alpha).
$$
As it was shown in  \cite{Kozybaev2022},
the coalgebra  $(P_1^\circ,\Delta_N)$ is a Novikov coalgebra.
Obviously, $P_1^\circ $
is a good subspace of  $\mathcal{L}_1^*$.
Therefore,
$(P_1^\circ, \Delta_N)\subseteq (\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ})$.


The following Theorem holds.

\begin{theorem}  Let
$\Delta_N=(id\otimes d^\circ)\Delta^\circ$.
Then
$(P_1^\circ, \Delta_N)\subseteq (\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ})$.
Therefore, the  dual coalgebra
    $(\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ})$
    of $\mathcal{L}_1$  is a non-zero Novikov coalgebra.
    If $F$ is a field  of characteristic zero then
 the coalgebra
    $(\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ})$
    does not contain  non-zero  proper finite-dimensional subcoalgebras.
\end{theorem}

Let $f,g\in \mathcal{L}_1$.
It is easy to see that in the algebra $\mathcal{L}_1$ we have
$$
[f,g]=f\circ g-g\circ f=fd(g)-d(f)g=[f,g]_d.
$$
Consequently, the algebra
$(\mathcal{L}_1,[\,,\,])$ is isomorphic to the Witt algebra~$W_1$.

Define the new comultiplication
$\Delta_{\mathcal{L}_1^\circ}^{(-)} = (1-\tau)\Delta_{\mathcal{L}_1^\circ}$
on the vector space  $\mathcal{L}_1^\circ$.
Then for  every
$f, g \in  P_1$ and $\alpha \in \mathcal{L}_1^\circ$
we have
$$
\langle\alpha,[f,g]\rangle
=\langle \alpha, f\circ g - f\circ g\rangle
= \rho(\Delta_{\mathcal{L}_1^\circ}^{(-)}(\alpha))(f\otimes g).
$$
Therefore,
$(\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ}^{(-)})$
is a good subspace of
$(\mathcal{L}_1,[\,,\,])^*$,
and  we can assume that
$(\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ}^{(-)})$
is a good subspace of $W_1^*$.
Consequently,
$(\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ}^{(-)})\subseteq (W_1^\circ,\Delta_{W_1^\circ})$. Theorem~3 implies $(P_1^\circ,\Delta_{d^\circ}^{(-)})\subseteq (\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ}^{(-)}).$

The following statement holds.


\begin{theorem}
Let $F$ be  a field of characteristic $\neq 2$.
Then
$(P_1^\circ,\Delta_N)=(\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ})$
and
$(W_1^\circ,\Delta_{W_1^\circ})
=(\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ}^{(-)})$,
where
$\Delta_{\mathcal{L}_1^\circ}^{(-)}=(1-\tau)\Delta_{\mathcal{L}_1^\circ}$.
\end{theorem}

\begin{proof}
By Theorem~3, we have $P_1^\circ \subseteq \mathcal{L}_1^\circ$.
It was shown that $\mathcal{L}_1^\circ \subseteq W_1^\circ$.
Let $F$ be  a field of characteristic $\ne 2$.  Then, by Theorem~2,
$P_1^\circ=W_1^\circ$. Consequently, $P_1^\circ=\mathcal{L}_1^\circ = W_1^\circ$.
Therefore,  $(W_1^\circ,\Delta_{W_1^\circ})
=(\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ}^{(-)})$.


Let us prove that
$\Delta_{\mathcal{L}_1^\circ}=\Delta_N$.
Suppose $f$, $g$ are arbitrary polynomials in $P_1$,
and  $\alpha\in \mathcal{L}_1^\circ$.
Then
$$
\langle \alpha,  fd(g)\rangle=\rho(\Delta^\circ (\alpha))(f\otimes d(g))=\rho((id\otimes d^\circ)\Delta^\circ (\alpha))(f\otimes g)=\rho(\Delta_{N}(\alpha))(f\otimes g).
$$
On the other hand,
$$
\langle \alpha,  fd(g)\rangle=\langle \alpha,  f\circ g\rangle=\rho(\Delta_{\mathcal{L}_1^\circ} (\alpha))(f\otimes g).
$$
Hence, $\Delta_N=\Delta_{\mathcal{L}_1^\circ}$, so
$(\mathcal{L}_1^\circ,\Delta_{\mathcal{L}_1^\circ}) =(P_1^\circ,\Delta_N)$.

\end{proof}

\section{Dual coalgebra of the Kantor construction of polynomial algebra in
one variable}

Let  $G$ be the Grassmann algebra with identity.
The algebra  $G=G_0+ G_1$
is a
$\mathbb Z_2$-graded algebra.
Let $J=J_0+J_1$ be a $\mathbb Z_2$-graded algebra.
Then  $G(J)=J_0\otimes G_0+J_1\otimes G_1$
is a subalgebra of  $J\otimes G$ and it is called  the
{\it Grassmann  envelope} of~$J$.

An algebra  $J$ is called  a {\it Jordan superalgebra}
 if its Grassmann envelope is a Jordan algebra, i.~e.,
 $G(J)$ satisfies the identities
\[
xy=yx,\quad (x^2y)x=x^2(xy).
\]

Recall the Kantor constuction \cite{Kantor1992}.
Let $A$ be an associative commutative algebra  over a field~$F$
with a derivation~$D$.
Denote by $\overline{A}$
an isomorphic copy of the vector space~$A$
with an isomorphism $a\mapsto\overline{a}$.
On the direct sum of the vector spaces  $J(A,D) = A+\overline{A}$
define a product $(\cdot)$ by
$$
a \cdot  b = ab,\quad
a \cdot \overline{b} = \overline{ab},
\quad
\overline{a}\cdot b = \overline{ab},
\quad
\overline{a}\cdot \overline{ b} = aD(b)-D(a)b,
$$
where $a, b\in A$ and
$ab$ is the product of elemets in~$A$.
Then  $J(A,D)$ is an Jordan superalgebra.
The superalgebra  $J(A,D)$ is said to be a superalgebra  {\it of vector type}.

Let us give the dual  analogue of the Kantor costruction for coalgebras.
Suppose
$(C,\Delta,d)$ is an associative commutative differential
coalgebra with a coderivation~$d$.
Let  $\overline{C}$ be an isomorphic copy of the vector space~$C$
with an isomorphism
$c\mapsto \overline{c}$.
Define the comultiplication  $\Delta_J$
on the direct sum of vector spaces $J^c(C,\Delta_J,d)=
C+\overline{C}$  by
$$
\begin{gathered}
\Delta_J(c)=\sum_{(c)}c_{(1)}\otimes c_{(2)}
+\overline{ c_{(1)}}\otimes \overline{d(c_{(2)})}-\overline{d(c_{(1)})}\otimes
\overline{c_{(2)}},\\
\Delta_J(\overline{c})
=\sum_{(c)}\overline{c_{(1)}}\otimes
c_{(2)}+c_{(1)}\otimes \overline{c_{(2)}},
\end{gathered}
$$
where $c\in C$ and
$\Delta(c)=\sum_{(c)}c_{(1)}\otimes c_{(2)}$.
It is shown in  \cite{Kozybaev2022}  that the dual algebra
of the coalgebra $J^c(C,\Delta_J,d)$ is
a Jordan superalgebra of vector type  $J(C^*,d^*)$,
where~$d^*$ is the transpose map to~$d$.
Therefore,  $J^c(C,\Delta_J,d)$ is a Jordan super\-co\-al\-geb\-ra.

\begin{lemma}
Let $(P_1^\circ, \Delta^\circ,   d^\circ)$ be the  dual coalgebra of $(P_1,d)$. Then    $J^c(P_1^\circ,\Delta^\circ_J, d^\circ)=J(P_1,d)^\circ$.
\end{lemma}

\begin{proof}
Set  $J=J(P_1,d)$, and  let $(J^\circ,\Delta_{J^\circ})  $ be the  dual supercoalgebra of~$J$. It was noted above that
$J^c=J^c(P_1^\circ,\Delta^\circ_J,d^\circ)$
is a Jordan supercoalgebra.
It is clear that $J^c$ is a good subspace of~$J^*$.
Therefore,
$J^c\subseteq J^\circ$.

Let us put $\langle\alpha,a\rangle=\alpha(a)$
for  $\alpha\in J^*$, $a\in J$, as above.
Since
$J^c=P_1^\circ+\overline{P_1^\circ}$,
where $\overline{P_1^\circ}$ is an
isomorphic copy of the vector space  $P_1^\circ$ with the isomorphism
$a\mapsto \overline{a}$,
we can assume
$\langle\alpha,a\rangle=\langle\overline{\alpha},\overline{a}\rangle$
for $\alpha\in P_1^\circ$ and $a\in P_1$.

The algebra $J$ acts on $J^*$ as on a bimodule by the rule
 $$
 \langle\alpha\cdot a,b\rangle=\langle\alpha ,ab\rangle,
 \quad
 \langle a\cdot \alpha, b \rangle=\langle  \alpha,ba\rangle,
 $$
 where $\alpha\in J^*$, $a,b\in J$.

 Since $J=P_1\oplus \overline{P_1}$, then we can assume that $J^*=P_1^*\oplus \overline{P_1}^*$
 (e.g., $P_1^*\cong \overline{P_1}^\perp \subset J^*$).
 Moreover, $P_1^*$ is a $P_1$-subbimodule of~$J^*$. Indeed, let
 $\alpha\in P_1^*$  and $f\in P_1$.
 Then
 \[
 \langle \alpha\cdot f, \overline{P_1}\rangle
  = \langle \alpha , f\overline{P_1} \rangle
  \subseteq \langle \alpha, \overline{P_1}\rangle =0.
 \]
 Consequently, $\alpha\cdot f\in P_1^*$.
 Similarly,
 $f\cdot \alpha\in P_1^*$.  Hence, $P_1^*$ is a $P_1$-subbimodule of $J^*$.
 In the same way,
 one may show that  $\overline{P_1}^*
 \cong P_1^\perp \subset J^*$
 is a $P_1$-subbimodule of $J^*$.

 Denote by $V$ and $W$ the projections of
 $J^\circ$ on the spaces $P_1^*$ and
 $\overline{P_1}^*$, respectively.
 Since $J^\circ$ is a $P_1$-subbimodule of~$J^*$ then $V$ and $W$ are also $P_1$-subbimodules of~$J^*$.

 Let us show that $V$ is a good subspace of~$P_1^*$.
 For $\alpha \in V$ there exist $\gamma\in J^\circ$ and $\beta\in W$ such that
 $\gamma=\alpha+\beta$.
 Let $\Delta_{J^\circ}(\gamma)=\sum_\gamma\gamma_{(1)}\otimes\gamma_{(2)}.$  Then $\gamma\cdot P_1\subseteq \sum_\gamma\langle\gamma_{(1)},P_1\rangle\gamma_{(2)}.$ Consequently,  the space $\gamma \cdot P_1$ is finite-dimensional.
 The space $\alpha \cdot P_1$ is the projection of
 $\gamma \cdot P_1$ to $V$, therefore,
  $\alpha \cdot P_1$ is finite-dimensional.
  Similarly   the space  $P_1 \cdot \alpha$ is finite-dimensional. By Corollary 2.5 from \cite{Anquella1994} we get   that $V$ a good subspace of $P_1^*$.

Consequently, $V\subseteq P_1^\circ.$
  Since $P_1^\circ \subseteq J^c\subseteq J^\circ$ then $P_1^\circ \subseteq V$. Therefore, $P_1^\circ = V$. Then $V\subseteq J^\circ$ and $W\subseteq J^\circ$.  Hence, the  coalgebra  $J^\circ$ is a  $\mathbb Z_2$-graded space,
where $V=(J^\circ)_0$, $W=(J^\circ)_1$. Moreover, $\overline{V}=\overline{P_1^\circ}\subseteq W.$

Assume $w\in W$. Then  $\Delta_{J^\circ}(w)=\sum_i v_i\otimes
w_i+w_i\otimes v_i$,
where
$v_i\in V$, $w_i\in W$, and for all $a\in P_1$ we have
$$
\langle w,\overline{a}\rangle=
\langle w,\overline{1}\cdot a\rangle=
\bigg \langle \sum_i\langle w_i,\overline{1}\rangle
v_i,a \bigg \rangle
= \bigg \langle \sum_i\langle w_i,\overline{1}\rangle
\overline{v_i},\overline{a}\bigg \rangle.
$$
Since $W\subseteq \overline{P_1}^*$, we have
$w=\sum_i\langle w_i,\overline{1}\rangle \overline{v_i}.$
Consequently, $W=\overline{V}$,
and thus
$J^c(P_1^\circ,\Delta^\circ_J,d^\circ)= J^\circ$.
\end{proof}

It is  known (see \cite{Zhelyabin2005})
that a Jordan superalgebra is simple  if and only if it is simple
as an algebra.
Also, recall that the superalgebra  $J(A,D)$
is simple if and only if the algebra
$A$ is a differentially simple algebra~\cite{King1995}.

Therefore, the following Theorem is true.

\begin{theorem}
Let  $P_1=F[x]$ be the polynomial algebra in the variable~$x$,
and  $d =\frac{ d}{dx}$ is the derivation  with respect to the variable~$x$. Consider the Jordan superalgebra $J(P_1,d)$ of vector type.
Then $J(P_1,d)^\circ=J^c(P_1^\circ,\Delta^\circ_J,d^\circ)$. Moreover, if $F$ is a field of characteristic  $0$ then $J(P_1,d)^\circ $ does not contain non-zero finite-dimensional subcoalgebras.
\end{theorem}

\begin{thebibliography}{99}

\bibitem{Sweedler1969}
M. Sweedler, {\it Hopf algebras}, W.A. Benjamin, Inc., New York, 1969.

\bibitem{Drinfeld1987}  V.G. Drinfeld,  {\it  Quantum groups},
In Proc. Int. Cong. Math. (Berkely, 1986), page 798-820. Amer. Math. Soc., Providence, RI, 1987.

\bibitem{Drinfeld1983} V.G. Drinfeld, {\it Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations,}
    Dokl. Akad. Nauk SSSR \textbf{ 268 }:2 (1983), 285--287.

\bibitem{Michaelis1980}  W. Michaelis, {\it Lie coalgebras},
Adv. Math., {\bf 38} (1980), 1-54.

\bibitem{Anquella1994} J. Anquella,  T. Cortes, F. Montaner,
{\it Nonassociative coalgebras},
Comm. in Algebra,  \textbf{22}:12  (1994),  4693-4716.

\bibitem{Zhelyabin1996} V.N. Zhelyabin,
{\it Structurable coalgebras},
Algebra i Logika \textbf{35}:5 (1996),  529--542; translation in Algebra and Logic \textbf{35}:5 (1996),  296--304.

\bibitem{Zhelyabin2007} V.N. Zhelyabin,  {\it Embedding of Jordan copairs into Lie coalgebras},
    Comm. in Algebra, \textbf{35}:2 (2007),  561-576.

\bibitem{Santos2021}  G. Santos Filho, L. Murakami, I. Shestakov,
{\it Locally finite coalgebras and the locally nilpotent
    radical I}, Linear Algebra and its Applications, {\bf 621} (2021), 235-253.

\bibitem{Slinko1995} A. M. Slinko, {\it Local finiteness of coalgebraic Lie coalgebras}, Comm. Algebra. \textbf{23}:5 (1995), 1165-1170.

\bibitem{Zhelyabin1996b} V.N. Zhelyabin,
{\it The Kantor-Koecher-Tits construction for Jordan coalgebras},
Algebra i Logika \textbf{35}:2 (1996), 173--189;
translation in Algebra and Logic \textbf{35}:2 (1996),  96--104.

\bibitem{Zhelyabin2003}  V.N. Zhelyabin,
{\it Jordan (super)coalgebras and Lie (super)coalgebras},
Sib. Mat. Zh. \textbf{44}:1 (2003), 87--111; translation in Siberian Math. J. \textbf{44}:1 (2003), 73--92.

\bibitem{Goncharov2012}  M.E.  Goncharov, V.N. Zhelyabin,
{\it  Mikheev's construction for Mal'tsev coalgebras}, Algebra Logika \textbf{51}:5 (2012),  668--671; translation in Algebra Logic \textbf{51}:5 (2012), 445--447.

\bibitem{Goncharov2013}  M.E.  Goncharov, V.N. Zhelyabin,
{\it Embedding of Mal'tsev coalgebras into Lie coalgebras with triality}, Algebra Logika \textbf{52}:1 (2013), 34--56; translation in Algebra Logic \textbf{52}:1 (2013), 24--40.

\bibitem{Kozybaev2000}  D.Kh. Kozybaev, {\it Right-alternative and right-symmetric coalgebras},
South-Kazakhstan Science and Education. \textbf{19}:12 (2000), 155--163.

\bibitem{Kozybaev2022} D. Kozybaev, U. Umirbaev, V. Zhelyabin,
{\it Some examples of nonassociative coalgebras and
supercoalgebras},
Linear Algebra and its Applications {\bf 643 } (2022), 235-257.

\bibitem{Santos2021b} G. Santos Filho, L. Murakami, and I. Shestakov,
{\it Locally finite coalgebras and the locally nilpotent radical II},
Comm. in Algebra,  \textbf{49}:12 (2021), 5472--5482.

\bibitem{Michaelis1990}
W. Michaelis,
{\it An example of a non-zero Lie coalgebra M for which Loc(M) = 0},
J. Pure Appl. Algebra. {\bf 68} (1990),  341-348.

\bibitem{Nichols1990} W.D.  Nichols,
{\it The structure of the dual Lie coalgebra of the Witt algebra},
J. Pure Appl. Algebra. {\bf  68} (1990),  359-364.

\bibitem{Nichols1993} W.D.  Nichols,
{\it On Lie and associative duals},
J. Pure Appl. Algebra. {\bf  87} (1993), 313-320.

\bibitem{Griffing2002} G. Griffing, 
{\it The dual coalgebra of certain infinite-dimensional Lie algebras},
Comm. Algebra \textbf{30}:12 (2002), 5715--5724.

\bibitem{Zhelyabin2005}
V.N. Zhelyabin, {\it Dual coalgebras of Jordan bialgebras and superalgebras}, Sib. Mat. Zh. \textbf{46}:6 (2005), 1302--1315; translation in Siberian Math. J. \textbf{46}:6 (2005), 1050--1061.

\bibitem{Gelfand1979} I. M. Gel'fand,  I. Ya. Dorfman,
{\it  Hamiltonian operators and algebraic structures related to
them},
Funktsional. Anal. i Prilozhen \textbf{13}:4  (1979),  3--30.

\bibitem{Balinskii1985}  I.M.  Balinskii, S.P. Novikov,
{\it   Poisson brackets of hydrodynamic type, Frobenius algebras and Lie
algebras},  Dokl. Akad. Nauk SSSR \textbf{283}:5 (1985),  1036--1039.

\bibitem{Kozybaev2016} D. Kozybaev, U. Umirbaev,
{\it Identities of the left-simmtric Witt algebras},
International Journal of algebra and computation,
\textbf{26}:2 (2016),  435-450.

\bibitem{Kantor1992} I.L. Kantor,
{\it  Jordan and Lie superalgebras determined by a Poisson algebra},
Algebra and Analysis, Tomsk, 1989; translation  Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, {\bf 151}
    (1992),  55–80.

\bibitem{King1995}  D. King, K. McCrimmon,
{\it The Kantor doubling process revisited}, Comm. in Algebra,
\textbf{23}:1 (1995), 357–372.

\end{thebibliography}








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