Algebraic Constructions for Novikov–Poisson Algebras

Abstract

A Novikov–Poisson algebra $(A,\circ ,·)$ is a vector space with a Novikov algebra structure $(A,\circ )$ and a commutative associative algebra structure $(A,·)$ satisfying some compatibility conditions. Give a Novikov–Poisson algebra $(A,\circ ,·)$ and a vector space V. A natural problem is how to construct and classify all Novikov–Poisson algebra structures on the vector space $E=A\oplus V$ such that $(A,\circ ,·)$ is a subalgebra of E up to isomorphism whose restriction on A is the identity map. This problem is called extending structures problem. In this paper, we introduce the definition of a unified product for Novikov–Poisson algebras, and then construct an object ${\mathcal{GH}}^2(V,A)$ to answer the extending structures problem. Note that unified product includes many interesting products such as bicrossed product, crossed product and so on. Moreover, the special case when $\dim \left(V\right)=1$ is investigated in detail.

1. Introduction

A Novikov algebra is a vector space A with a binary operation that satisfies

$$\begin{array}{ccc}& & (a\circ b)\circ c-a\circ (b\circ c)=(b\circ a)\circ c-b\circ (a\circ c),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & (a\circ b)\circ c=(a\circ c)\circ b,\hfill \end{array}$$

(2)

for all a, b, $c\in A$. Denote it by $(A,\circ )$. Novikov algebras appeared in connection with the Poisson brackets of hydrodynamic type [1] and Hamiltonian operators in the formal variational calculus [2,3]. As a special class of pre-Lie algebras, they are closely related to many fields in mathematics and physics, such as Lie groups, Lie algebras, affine manifolds, vertex algebras, Lie conformal algebras, quantum field theory and so on.

A Novikov–Poisson algebra $(A,\circ ,·)$ is a vector space with a Novikov algebra structure $(A,\circ )$ and a commutative associative algebra structure $(A,·)$ satisfying some compatibility conditions. Novikov–Poisson algebras were introduced by Xu in [4] in order to establish the tensor theory of Novikov algebras. Moreover, Xu in [5] classified Novikov–Poisson algebras whose Novikov algebras are simple with an idempotent element and a class of simple Novikov algebras without non-zero idempotent elements were constructed through Novikov–Poisson algebras. In addition, Novikov–Poisson algebras are closely related to a class of Novikov conformal algebras called quadratic Novikov conformal algebras [6], higher-dimensional Lie conformal algebras [7], transposed Poisson algebras [8] and so on. There have been many other works on Novikov–Poisson algebras. For example, the classifications of Novikov–Poisson algebras of low dimensions were carried out in [9,10,11], the relationships between Novikov–Poisson algebras and Jordan algebras were investigated in [12,13], embedding of Novikov–Poisson algebras in Novikov–Poisson algebras of vector type was studied in [14] and free Novikov–Poisson algebras were investigated in [15].

Since Novikov–Poisson algebras are related to kinds of other algebra structures, the constructions and classifications of Novikov–Poisson algebras become important. Based on this, we plan to study the following problem of Novikov–Poisson algebras:

Extending structures problem: Let $(A,\circ ,·)$ be a Novikov–Poisson algebra and E a vector space containing A as a subspace. Describe and classify all Novikov–Poisson algebra structures on E, such that $(A,\circ ,·)$ is a subalgebra of E up to isomorphism whose restriction on A is the identity map.

From the perspective of algebraic theory, this problem is natural, important and worth investigating, since it will tell us how to construct a larger algebra from a given algebra. Extending structures problems for groups, Lie algebras, associative algebras, Hopf algebras, and left-symmetric algebra were investigated in [16,17,18,19,20], respectively. In particular, the theories of extending structures for Novikov algebras and commutative associative algebras have been developed in [18,20], respectively. Motivated by these results, we introduce the definition of a unified product for Novikov–Poisson algebras and then construct an object ${\mathcal{GH}}^2(V,A)$ to give a theoretical answer for the extending structures problem of Novikov–Poisson algebras, where V is a complement of A in E. Furthermore, when $\dim \left(V\right)=1$, we calculate ${\mathcal{GH}}^2(V,A)$ and give a specific example. These results will be useful for studying the structure theory of Novikov–Poisson algebras and classifying low-dimensional Novikov–Poisson algebras.

This paper is organized as follows. In Section 2, we recall some definitions about Novikov–Poisson algebras and some results about extending structures for Novikov algebras and commutative associative algebras. In Section 3, we introduce the definition of a unified product for Novikov–Poisson algebras and construct an object ${\mathcal{GH}}^2(V,A)$ to give a theoretical answer for the extending structures problem. In Section 4, we investigate the unified products $A♮V$ in detail when $\dim \left(V\right)=1$, and finally provide an example to compute ${\mathcal{GH}}^2(V,A)$.

Throughout this paper, $\mathbf{k}$ is an arbitrary field and $\mathbb{C}$ is the field of complex numbers. We also denote the set of non-zero elements in $\mathbf{k}$ by ${\mathbf{k}}^{*}$. All vector spaces, Novikov algebras, commutative associative algebras, Novikov–Poisson algebras, linear or bilinear maps are over $\mathbf{k}$.

2. Preliminaries

In this section, we recall some definitions of Novikov–Poisson algebras, and some results about extending structures for Novikov algebras and commutative associative algebras.

Firstly, we recall some basic facts about Novikov algebras [20].

Let $(A,\circ )$ be a Novikov algebra. Then, there is a natural Lie algebra structure on A:

$$\begin{array}{c}\hfill [a,b]=a\circ b-b\circ a,\forall a,b\in A.\end{array}$$

(3)

Denote this Lie algebra by $g\left(A\right)$.

Definition 1.

Let $(A,\circ )$ be a Novikov algebra, V be a vector space and $l_A$, $r_A:A\to gl\left(V\right)$ be two linear maps. If it satisfies

$$\begin{array}{ccc}& & l_A\left(a\right)l_A\left(b\right)v-l_A(a\circ b)v=l_A\left(b\right)l_A\left(a\right)v-l_A(b\circ a)v,\hfill \\ & & l_A\left(a\right)r_A\left(b\right)v-r_A\left(b\right)l_A\left(a\right)v=r_A(a\circ b)v-r_A\left(b\right)r_A\left(a\right)v,\hfill \\ & & l_A(a\circ b)v=r_A\left(b\right)l_A\left(a\right)v,\hfill \\ & & r_A\left(a\right)r_A\left(b\right)v=r_A\left(b\right)r_A\left(a\right)v,\hfill \end{array}$$

for all a, $b\in A$ and $v\in V$, then $(V,l_A,r_A)$ is called a bimoduleof $(A,\circ )$.

Definition 2

([20], Definition 3.1, Theorem 3.2 and Corollary 3.5]). Let $(A,\circ )$ be a Novikov algebra and V be a vector space. A Novikov extending datum of $(A,\circ )$ through V is a system $\mathrm{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*)$ consisting of four linear maps: $l_A,r_A:A\to gl\left(V\right)$, $l_V,r_V:V\to gl\left(A\right)$, and two bilinear maps: $f:V\times V\to A$, $*:V\times V\to V$ satisfying the following conditions:

$$\begin{array}{ccc}\hfill \left(N1\right)& & l_V\left(x\right)(a\circ b)=-l_V\left(l_A\left(a\right)x\right)b+l_V\left(r_A\left(a\right)x\right)b\hfill \\ & & +\left(l_V\left(x\right)a\right)\circ b-\left(r_V\left(x\right)a\right)\circ b+r_V\left(r_A\left(b\right)x\right)a+a\circ \left(l_V\left(x\right)b\right),\hfill \\ \hfill \left(N2\right)& & l_A\left(a\right)r_A\left(b\right)x-r_A\left(b\right)l_A\left(a\right)x=r_A(a\circ b)x-r_A\left(b\right)r_A\left(a\right)x,\hfill \\ \hfill \left(N3\right)& & r_V\left(x\right)(a\circ b)-r_V\left(x\right)(b\circ a)=r_V\left(l_A\left(b\right)x\right)a-r_V\left(l_A\left(a\right)x\right)b\hfill \\ & & +a\circ \left(r_V\left(x\right)b\right)-b\circ \left(r_V\left(x\right)a\right),\hfill \\ \hfill \left(N4\right)& & l_A(a\circ b-b\circ a)x=l_A\left(a\right)l_A\left(b\right)x-l_A\left(b\right)l_A\left(a\right)x,\hfill \\ \hfill \left(N5\right)& & r_V(x*y)a=r_V\left(y\right)\left(r_V\left(x\right)a\right)-r_V\left(y\right)\left(l_V\left(x\right)a\right)+l_V\left(x\right)r_V\left(y\right)a\hfill \\ & & +f(l_A\left(a\right)x,y)+f(x,l_A\left(a\right)y)-a\circ f(x,y)-f(r_A\left(a\right)x,y),\hfill \\ \hfill \left(N6\right)& & l_A\left(a\right)(x*y)=-l_A(l_V\left(x\right)a-r_V\left(x\right)a)y+(l_A\left(a\right)x-r_A\left(a\right)x)*y\hfill \\ & & +r_A\left(r_V\left(y\right)a\right)x+x*\left(l_A\left(a\right)y\right),\hfill \\ \hfill \left(N7\right)& & l_V(x*y-y*x)a=(l_V\left(x\right)l_V\left(y\right)-l_V\left(y\right)l_V\left(x\right))a\hfill \\ & & -(f(x,y)-f(y,x))\circ a+f(x,r_A\left(a\right)y)-f(y,r_A\left(a\right)x),\hfill \\ \hfill \left(N8\right)& & r_A\left(a\right)(x*y-y*x)=r_A\left(l_V\left(y\right)a\right)x-r_A\left(l_V\left(x\right)a\right)y\hfill \\ & & +x*\left(r_A\left(a\right)y\right)-y*\left(r_A\left(a\right)x\right),\hfill \\ \hfill \left(N9\right)& & f(x*y,z)-f(x,y*z)-f(y*x,z)+f(y,x*z)\hfill \\ & & +r_V\left(z\right)(f(x,y)-f(y,x))-l_V\left(x\right)f(y,z)+l_V\left(y\right)f(x,z)=0,\hfill \\ \hfill \left(N10\right)& & (x*y)*z-x*(y*z)-(y*x)*z+y*(x*z)\hfill \\ & & +l_A(f(x,y)-f(y,x))z-r_A\left(f(y,z)\right)x+r_A\left(f(x,z)\right)y=0,\hfill \\ \hfill \left(N11\right)& & \left(l_V\left(x\right)a\right)\circ b+l_V\left(rA\left(a\right)x\right)b=\left(l_V\left(x\right)b\right)\circ a+l_V\left(r_A\left(b\right)x\right)a,\hfill \\ \hfill \left(N12\right)& & r_A\left(b\right)r_A\left(a\right)x=r_A\left(a\right)r_A\left(b\right)x,\hfill \\ \hfill \left(N13\right)& & \left(r_V\left(x\right)a\right)\circ b+l_V\left(l_A\left(a\right)x\right)b=r_V\left(x\right)(a\circ b),\hfill \\ \hfill \left(N14\right)& & r_A\left(b\right)l_A\left(a\right)x=l_A(a\circ b)x,\hfill \\ \hfill \left(N15\right)& & r_V\left(y\right)r_V\left(x\right)a+f(l_A\left(a\right)x,y)=r_V\left(x\right)r_V\left(y\right)a+f(l_A\left(a\right)y,x),\hfill \\ \hfill \left(N16\right)& & l_A\left(r_V\left(x\right)a\right)y+\left(l_A\left(a\right)x\right)*y=l_A\left(r_V\left(y\right)a\right)x+\left(l_A\left(a\right)y\right)*x,\hfill \\ \hfill \left(N17\right)& & r_V\left(y\right)\left(l_V\left(x\right)a\right)+f(r_A\left(a\right)x,y)=f(x,y)\circ a+l_V(x*y)a,\hfill \\ \hfill \left(N18\right)& & l_A\left(l_V\left(x\right)a\right)y+\left(r_A\left(a\right)x\right)*y=r_A\left(a\right)(x*y),\hfill \\ \hfill \left(N19\right)& & r_V\left(z\right)f(x,y)+f(x*y,z)=r_V\left(y\right)f(x,z)+f(x*z,y),\hfill \\ \hfill \left(N20\right)& & l_A\left(f(x,y)\right)z+(x*y)*z=l_A\left(f(x,z)\right)y+(x*z)*y,\hfill \end{array}$$

for all a, $b\in A$, x, y, $z\in V$.

Let $\mathrm{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*)$ be a Novikov extending datum of $(A,\circ )$ through V and $A♮V$ be the vector space $A\oplus V$ with the bilinear map “∘ " defined by:

$$\begin{array}{c}\hfill (a+x)\circ (b+y)=(a\circ b+l_V\left(x\right)b+r_V\left(y\right)a+f(x,y))+(x*y+l_A\left(a\right)y+r_A\left(b\right)x),\end{array}$$

for all a, $b\in A$, x, $y\in V$. Then, $(A♮V,\circ )$ is a Novikov algebra by [20] (Theorem 3.2), which is called theunified productof $(A,\circ )$ and $\mathrm{\Omega }(A,V)$.

Next, we recall the definition of a unified product for commutative associative algebras.

Definition 3

([18], Definition 3.11, Theorem 2.2). Let $(A,·)$ be a commutative associative algebra and V be a vector space. A commutative associative extending datum of $(A,·)$ through V is a system $\mathsf{\Omega }(A,V)=(◃,▹,g,•)$ consisting of four bilinear maps: $◃:V\times A\to V$, $▹:V\times A\to A,$$g:V\times V\to A,$$•:V\times V\to V$ such that g and • are symmetric and the following conditions hold:

$$\begin{array}{ccc}\hfill \left(C1\right)& & x•(y•z)-(x•y)•z=z◃g(x,y)-x◃g(y,z),\hfill \\ \hfill \left(C2\right)& & (x•y)◃a=x◃(y▹a)+x•(y◃a),\hfill \\ \hfill \left(C3\right)& & x▹(a·b)=a·(x▹b)+(x◃b)▹a,\hfill \\ \hfill \left(C4\right)& & x◃(a·b)=(x◃a)◃b,\hfill \\ \hfill \left(C5\right)& & (x•y)▹a=x▹(y▹a)+g(x,y◃a)-g(x,y)·a,\hfill \\ \hfill \left(C6\right)& & g(x,y•z)-g(x•y,z)=z▹g(x,y)-x▹g(y,z),\hfill \end{array}$$

for all a, $b\in A$, x, y, $z\in V$.

Let $\mathsf{\Omega }(A,V)=(◃,▹,g,•)$ be a commutative associative extending datum of $(A,·)$ through V and let $A♮V$ be the vector space $A\oplus V$ with the linear map “· " defined by:

$$\begin{array}{c}\hfill (a+x)·(b+y)=(a·b+y▹a+x▹b+g(x,y\left)\right)+(y◃a+x◃b+x•y)\end{array}$$

for all a, $b\in A$, x, $y\in V$. Then, $A♮V$ is a commutative associative algebra by [18] (Theorem 2.2), which is called theunified productof $(A,·)$ and $\mathsf{\Omega }(A,V)$.

Remark 1.

Note that $\left(C4\right)$ means that $(V,◃)$ is a right module of $(A,·)$.

Finally, we recall the definition of Novikov–Poisson algebra.

Definition 4

([4]). A Novikov–Poisson algebra is a vector space A with two operations $“\circ ”$, $“·”$, such that $(A,\circ )$ forms a Novikov algebra, $(A,·)$ forms a commutative associative algebra and they satisfy

$$\begin{array}{ccc}& & (a·b)\circ c=a·(b\circ c),\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & (a\circ b)·c-a\circ (b·c)=(b\circ a)·c-b\circ (a·c),\hfill \end{array}$$

(5)

for all a, b and $c\in A$.

Let $(A,\circ ,·)$ and $(B,\circ ,·)$ be two Novikov–Poisson algebras. If a linear map $\phi :A\to B$ is both a homomorphism of Novikov algebras and a homomorphism of commutative associative algebras, then $\phi$ is called a homomorphism of Novikov–Poisson algebras.

Let $I\subseteq A$ be a subspace of A. If $(I,\circ )$ is a Novikov subalgebra (a two-sided ideal) of $(A,\circ )$ and $(I,·)$ is a subalgebra (an ideal) of $(A,·)$, then $(I,\circ ,·)$ is called a subalgebra (an ideal) of Novikov–Poisson algebra $(A,\circ ,·)$.

Definition 5.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra. If $(V,l_A,r_A)$ is a bimodule of $(A,\circ )$, $(V,⊳)$ is a left module of $(A,·)$ and they satisfy

$$\begin{array}{ccc}& & r_A\left(b\right)(a⊳v)=a⊳\left(r_A\left(b\right)v\right)=(a\circ b)⊳v,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & l_A(a·b)v=a⊳\left(l_A\left(b\right)v\right),\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & (a\circ b)⊳v-l_A\left(a\right)(b⊳v)=(b\circ a)⊳v-l_A\left(b\right)(a⊳v),\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & b⊳\left(l_A\left(a\right)v\right)-l_A\left(a\right)(b⊳v)=b⊳\left(r_A\left(a\right)v\right)-r_A(a·b)v,\hfill \end{array}$$

(9)

for all a, $b\in A$ and $v\in V$, then $(V,l_A,r_A,⊲)$ is called a left moduleof $(A,\circ ,·)$.

Similarly, $(V,l_A,r_A,⊲)$ is called a right moduleof $(A,\circ ,·)$, if $(V,l_A,r_A)$ is a bimodule of $(A,\circ )$, $(V,⊲)$ is a right module of $(A,·)$, and they satisfy

$$\begin{array}{ccc}& & r_A\left(b\right)(v⊲a)=v⊲(a\circ b)=\left(r_A\left(b\right)v\right)⊲a,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & l_A(a·b)v=\left(l_A\left(b\right)v\right)⊲a,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & v⊲(a\circ b)-l_A\left(a\right)(v⊲b)=v⊲(b\circ a)-l_A\left(b\right)(v⊲a),\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& & \left(l_A\left(a\right)v\right)⊲b-l_A\left(a\right)(v⊲b)=\left(r_A\left(a\right)v\right)⊲b-r_A(a·b)v,\hfill \end{array}$$

(13)

for all a, $b\in A$ and $v\in V$.

Finally, we introduce the following definition, which is important for studying the extending structures problem.

Definition 6.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra and let E be a vector space containing A as a subspace. Let $(E,\circ ,·)$ and $(E,{\circ }^{{}^{\prime }},{·}^{{}^{\prime }})$ be two Novikov–Poisson algebra structures on E, such that $(A,\circ ,·)$ is a subalgebra of them. If there is an isomorphism of Novikov–Poisson algebras $\phi :(E,\circ ,·)\to (E,{\circ }^{{}^{\prime }},{·}^{{}^{\prime }})$ whose restriction on A is the identity map, then we say that $(E,\circ ,·)$ and $(E,{\circ }^{{}^{\prime }},{·}^{{}^{\prime }})$ areequivalent. We denote it by $(E,\circ ,·)\equiv (E,{\circ }^{{}^{\prime }},{·}^{{}^{\prime }})$.

Let S be the set of all Novikov–Poisson algebra structures on E which contain $(A,\circ ,·)$ as a subalgebra. Obviously, $“\equiv ”$ is an equivalence relation on S. Denote the set of all equivalence classes via $“\equiv ”$ by $\mathrm{Extd}(E,A)$. Therefore, to study the extending structures problem, we only need to characterize $\mathrm{Extd}(E,A)$.

3. Unified Products for Novikov–Poisson Algebras

In this section, we will introduce a notion of unified product for Novikov–Poisson algebras. Through this, we can characterize $\mathrm{Extd}(E,A)$ to answer the extending structures problem.

Definition 7.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra and V a vector space. Anextending datumof $(A,\circ ,·)$ through V is a system $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ consisting of four linear maps and six bilinear maps, as follows:

$$\begin{array}{ccc}& & f:V\times V\to A,*:V\times V\to V,\hfill \\ & & l_A,r_A:A\to gl\left(V\right),l_V,r_V:V\to gl\left(A\right),\hfill \\ & & ◃:V\times A\to V,▹:V\times A\to A,g:V\times V\to A,•:V\times V\to V.\hfill \end{array}$$

Let $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ be an extending datum of a Novikov–Poisson algebra $(A,\circ ,·)$ through V. Denote the vector space $A\oplus V$ by $A{♮}_{\mathsf{\Omega }(A,V)}V=A♮V$. Define two bilinear operations $“\circ ”$ and $“·”$ on $A♮V$ as follows:

$$\begin{array}{ccc}\hfill & & (a+x)\circ (b+y)=(a\circ b+l_V\left(x\right)b+r_V\left(y\right)a+f(x,y))\hfill \\ & & +(x*y+l_A\left(a\right)y+r_A\left(b\right)x),\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill & & (a+x)·(b+y)=(a·b+x▹b+y▹a+g(x,y\left)\right)+(x◃b+y◃a+x•y),\hfill \end{array}$$

(15)

for all a, $b\in A$, x, $y\in V$. If $(A\oplus V,\circ ,·)$ is a Novikov–Poisson algebra, then we call that $A♮V$ is aunified productof $(A,\circ ,·)$ and $\mathsf{\Omega }(A,V)$.

Definition 8.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra, V be a vector space and $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ be an extending datum of $(A,\circ ,·)$ through V. If $\mathsf{\Omega }(A,V)$ satisfies the following compatibility conditions:

$$\begin{array}{ccc}\hfill \left(G0\right)& & (l_A,r_A,l_V,r_V,f,*)\mathit{is}a\mathit{Novikov}\mathit{extending}\mathit{datum}\mathit{of}(A,\circ )\mathit{through}V,\hfill \\ & & (◃,▹,g,•)\mathit{is}a\mathit{commutative}\mathit{associative}\mathit{extending}\mathit{datum}\mathit{of}(A,·)\mathit{by}V,\hfill \\ \hfill \left(G1\right)& & r_V\left(z\right)(a·b)=a·\left(r_V\left(z\right)b\right)+\left(l_A\left(b\right)z\right)▹a,\hfill \\ \hfill \left(G2\right)& & l_A(a·b)z=\left(l_A\left(b\right)z\right)◃a,\hfill \\ \hfill \left(G3\right)& & (y▹a)\circ c+l_V(y◃a)c=a·\left(l_V\left(y\right)c\right)+\left(r_A\left(c\right)y\right)▹a,\hfill \\ \hfill \left(G4\right)& & r_A\left(c\right)(y◃a)=\left(r_A\left(c\right)y\right)◃a,\hfill \\ \hfill \left(G5\right)& & (x▹b)\circ c+l_V(x◃b)c=x▹(b\circ c),\hfill \\ \hfill \left(G6\right)& & r_A\left(c\right)(x◃b)=x◃(b\circ c),\hfill \\ \hfill \left(G7\right)& & r_V\left(z\right)(y▹a)+f(y◃a,z)=a·f(y,z)+(y*z)▹a,\hfill \\ \hfill \left(G8\right)& & (y◃a)*z+l_A(y▹a)z=(y*z)◃a,\hfill \\ \hfill \left(G9\right)& & r_V\left(z\right)(x▹b)+f(x◃b,z)=x▹\left(r_V\left(z\right)b\right)+g(x,l_A\left(b\right)z),\hfill \\ \hfill \left(G10\right)& & (x◃b)*z+l_A(x▹b)z=x◃\left(r_V\left(z\right)b\right)+x•\left(l_A\left(b\right)z\right),\hfill \\ \hfill \left(G11\right)& & g(x,y)\circ c+l_V(x•y)c=x▹\left(l_V\left(y\right)c\right)+g(x,r_A\left(c\right)y),\hfill \\ \hfill \left(G12\right)& & r_A\left(c\right)(x•y)=x◃\left(l_V\left(y\right)c\right)+x•\left(r_A\left(c\right)y\right),\hfill \\ \hfill \left(G13\right)& & r_V\left(z\right)g(x,y)+f(x•y,z)=x▹f(y,z)+g(x,y*z),\hfill \\ \hfill \left(G14\right)& & (x•y)*z+l_A\left(g(x,y)\right)z=x◃f(y,z)+x•(y*z),\hfill \\ \hfill \left(G15\right)& & z▹(a\circ b)-a\circ (z▹b)-r_V(z◃b)a=z▹(b\circ a)-b\circ (z▹a)-r_V(z◃a)b,\hfill \\ \hfill \left(G16\right)& & z◃(a\circ b)-l_A\left(a\right)(z◃b)=z◃(b\circ a)-l_A\left(b\right)(z◃a),\hfill \\ \hfill \left(G17\right)& & \left(r_V\left(y\right)a\right)·c+\left(l_A\left(a\right)y\right)▹c-a\circ (y▹c)-r_V(y◃c)a\hfill \\ & & =\left(l_V\left(y\right)a\right)·c+\left(r_A\left(a\right)y\right)▹c-l_V\left(y\right)(a·c),\hfill \\ \hfill \left(G18\right)& & \left(l_A\left(a\right)y\right)◃c-l_A\left(a\right)(y◃c)=\left(r_A\left(a\right)y\right)◃c-r_A(a·c)y,\hfill \\ \hfill \left(G19\right)& & z▹\left(r_V\left(y\right)a\right)+g(l_A\left(a\right)y,z)-a\circ g(y,z)-r_V(y•z)a\hfill \\ & & =z▹\left(l_V\left(y\right)a\right)+g(r_A\left(a\right)y,z)-l_V\left(y\right)(z▹a)-f(y,z◃a),\hfill \\ \hfill \left(G20\right)& & z◃\left(r_V\left(y\right)a\right)+\left(l_A\left(a\right)y\right)•z-l_A\left(a\right)(y•z)\hfill \\ & & =z◃\left(l_V\left(y\right)a\right)+\left(r_A\left(a\right)y\right)•z-r_A(z▹a)y-y*(z◃a),\hfill \\ \hfill \left(G21\right)& & f(x,y)·c+(x*y)▹c-l_V\left(x\right)(y▹c)-f(x,y◃c)\hfill \\ & & =f(y,x)·c+(y*x)▹c-l_V\left(y\right)(x▹c)-f(y,x◃c),\hfill \\ \hfill \left(G22\right)& & (x*y)◃c-x*(y◃c)-r_A(y▹c)x\hfill \\ & & =(y*x)◃c-y*(x◃c)-r_A(x▹c)y,\hfill \\ \hfill \left(G23\right)& & z▹f(x,y)+g(x*y,z)-l_V\left(x\right)g(y,z)-f(x,y•z)\hfill \\ & & =z▹f(y,x)+g(y*x,z)-l_V\left(y\right)g(x,z)-f(y,x•z),\hfill \\ \hfill \left(G24\right)& & z◃f(x,y)+(x*y)•z-x*(y•z)-r_A\left(g(y,z)\right)x\hfill \\ & & =z◃f(y,x)+(y*x)•z-y*(x•z)-r_A\left(g(x,z)\right)y\hfill \end{array}$$

for all a, b, $c\in A$, x, y, $z\in V$, we say that $\mathsf{\Omega }(A,V)$ is aNovikov–Poisson extending datum of$(A,\circ ,·)$throughV. We denote the set of all Novikov–Poisson extending data of $(A,\circ ,·)$ through V by $\mathcal{GD}(A,V)$.

Theorem 1.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra, V be a vector space and $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ be an extending datum of $(A,\circ ,·)$ through V. Then, $A♮V$ is a unified product with the operations defined by (14) and (15) if and only if $\mathsf{\Omega }(A,V)$ is a Novikov–Poisson extending datum of $(A,\circ ,·)$ through V.

Proof. 

According to [20] (Theorem 3.2) , [18] (Theorem 2.2) and [18] (Definition 3.11), $(A♮V,\circ )$ is a Novikov algebra if and only if $(l_A,r_A,l_V,r_V,f,*)$ is a Novikov extending datum of $(A,\circ )$ through V, and $(A♮V,·)$ is a commutative associative algebra if and only if $(◃,▹,g,•)$ is a commutative associative extending datum of $(A,·)$ through V. Then, we only need to check that

$$\begin{array}{ccc}\hfill & & \left(\right(a+x)·(b+y\left)\right)\circ (c+z)=(a+x)·\left(\right(b+y)\circ (c+z\left)\right),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill & & \left(\right(a+x)\circ (b+y\left)\right)·(c+z)-(a+x)\circ \left(\right(b+y)·(c+z\left)\right)\hfill \\ & & =\left(\right(b+y)\circ (a+x\left)\right)·(c+z)-(b+y)\circ \left(\right(a+x)·(c+z\left)\right),\hfill \end{array}$$

(17)

for all a, b, $c\in A$, x, y, $z\in V$ if and only if $\left(G1\right)$-$\left(G24\right)$ hold.

Note that (16) holds for all a, b, $c\in A$, x, y, $z\in V$ if and only if it holds for all triples: $(a,b,c)$, $(a,b,z)$, $(a,y,c)$,$(x,b,c)$, $(a,y,z)$, $(x,b,z)$, $(x,y,c)$ and $(x,y,z)$, where a, b, $c\in A$, x, y, $z\in V$. Since $(A,\circ ,·)$ is a Novikov–Poisson algebra, (16) holds for the triple $(a,b,c)$. Since

$$\begin{array}{ccc}& & (a·b)\circ z=r_V\left(z\right)(a·b)+l_A(a·b)z,\hfill \\ & & a·(b\circ z)=(a·\left(r_V\left(z\right)b\right)+\left(l_A\left(b\right)z\right)⊳a)+\left(l_A\left(b\right)z\right)⊲a,\hfill \end{array}$$

we have that (16) holds for the triple $(a,b,z)$⇔$\left(G1\right)$ and $\left(G2\right)$ hold. Similarly, we can prove the following: (16) holds for the triple $(a,y,c)$⇔$\left(G3\right)$ and $\left(G4\right)$ hold; (16) holds for the triple $(x,b,c)$⇔$\left(G5\right)$ and $\left(G6\right)$ hold; (16) holds for the triple $(a,y,z)$⇔$\left(G7\right)$ and $\left(G8\right)$ hold; (16) holds for the triple $(x,b,z)$⇔$\left(G9\right)$ and $\left(G10\right)$ hold; (16) holds for the triple $(x,y,c)$⇔$\left(G11\right)$ and $\left(G12\right)$ hold; (16) holds for the triple $(x,y,z)$⇔$\left(G13\right)$ and $\left(G14\right)$ hold.

As for (17), it also holds if $(a+x,b+y,c+z)$ is changed to $(b+y,a+x,c+z)$. Thus, (17) holds for all a, b, $c\in A$, x, y, $z\in V$ if and only if it holds for all triples $(a,b,c)$, $(a,b,z)$, $(a,y,c)$, $(a,y,z)$, $(x,b,z)$, and $(x,y,z)$, where a, b, $c\in A$, x, y, $z\in V$. Obviously, (17) holds for the triple $(a,b,c)$, since $(A,\circ ,·)$ is a Novikov–Poisson algebra. Similarly, we can obtain: (17) holds for the triple $(a,b,z)$⇔$\left(G15\right)$ and $\left(G16\right)$ hold; (17) holds for the triple $(a,y,c)$⇔$\left(G17\right)$ and $\left(G18\right)$ hold; (17) holds for the triple $(a,y,z)$⇔$\left(G19\right)$ and $\left(G20\right)$ hold; (17) holds for the triple $(x,y,c)$⇔$\left(G21\right)$ and $\left(G22\right)$ hold; (17) holds for the triple $(x,y,z)$⇔$\left(G23\right)$ and $\left(G24\right)$ hold.

Then, the proof is finished. □

Remark 2.$\left(G0\right)$, $\left(G2\right)$, $\left(G4\right)$, $\left(G6\right)$, $\left(G16\right)$ and $\left(G18\right)$ mean that $(V,l_A,r_A,⊲)$ is a right module of $(A,\circ ,·)$.

Example 1.

Let $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ be an extending datum of $(A,\circ ,·)$ through a vector space V, where $l_A$, $r_A$ and ◃ are trivial maps. We can simplify the extending datum into $\mathsf{\Omega }(A,V)=(l_V,r_V,f,*,▹,g,•)$. Then, $\mathsf{\Omega }(A,V)$ is a Novikov–Poisson extending datum of $(A,\circ ,·)$ through V if and only if $(l_V,r_V,f,*)$ is a Novikov extending datum of $(A,\circ )$ through V, $(▹,g,•)$ is a commutative associative extending datum of $(A,·)$ through V, $(V,*,•)$ is a Novikov–Poisson algebra, and the following conditions hold:

$$\begin{array}{ccc}\hfill \left(H1\right)& & r_V\left(z\right)(a·b)=a·\left(r_V\left(z\right)b\right),\hfill \\ \hfill \left(H2\right)& & (y▹a)\circ b=a·\left(l_V\left(y\right)b\right),\hfill \\ \hfill \left(H3\right)& & (x▹a)\circ b=x▹(a\circ b),\hfill \\ \hfill \left(H4\right)& & r_V\left(z\right)(y▹a)=a·f(y,z)+(y*z)▹a,\hfill \\ \hfill \left(H5\right)& & r_V\left(z\right)(x▹a)=x▹\left(r_V\left(z\right)a\right),\hfill \\ \hfill \left(H6\right)& & g(x,y)\circ a+l_V(x•y)a=x▹\left(l_V\left(y\right)a\right),\hfill \\ \hfill \left(H7\right)& & r_V\left(z\right)g(x,y)+f(x•y,z)=x▹f(y,z)+g(x,y*z),\hfill \\ \hfill \left(H8\right)& & z▹(a\circ b)-a\circ (z▹b)=z▹(b\circ a)-b\circ (z▹a),\hfill \\ \hfill \left(H9\right)& & \left(r_V\left(y\right)a\right)·b-a\circ (y▹b)=\left(l_V\left(y\right)a\right)·b-l_V\left(y\right)(a·b),\hfill \\ \hfill \left(H10\right)& & z▹\left(r_V\left(y\right)a\right)-a\circ g(y,z)-r_V(y•z)a=z▹\left(l_V\left(y\right)a\right)-l_V\left(y\right)(z▹a),\hfill \\ \hfill \left(H11\right)& & f(x,y)·a+(x*y)▹a-l_V\left(x\right)(y▹a)\hfill \\ & & =f(y,x)·a+(y*x)▹a-l_V\left(y\right)(x▹a),\hfill \\ \hfill \left(H12\right)& & z▹f(x,y)+g(x*y,z)-l_V\left(x\right)g(y,z)-f(x,y•z)\hfill \\ & & =z▹f(y,x)+g(y*x,z)-l_V\left(y\right)g(x,z)-f(y,x•z),\hfill \end{array}$$

for all a, $b\in A$, x, y, $z\in V$. We denote the associated unified product $A♮V$ in this case by $A\diamond V$, which is called thecrossed productof $(A,\circ ,·)$ and $(V,*,•)$. The crossed product associated with $(A,V,l_V,r_V,f,▹,g)$ satisfying the compatibility conditions above, is the Novikov–Poisson algebra, defined as follows:

$$\begin{array}{ccc}& & (a+x)\circ (b+y)=(a\circ b+l_V\left(x\right)b+r_V\left(y\right)a+f(x,y))+(x*y),\hfill \\ & & (a+x)·(b+y)=(a·b+y▹a+x▹b+g(x,y\left)\right)+(x•y),\hfill \end{array}$$

for all a, $b\in A$, x, $y\in V$. Note that $(A,\circ ,·)$ is an ideal of $(A\diamond V,\circ ,·)$.

Example 2.

Let $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ be an extending datum of $(A,\circ ,·)$ through a vector space V, where f and g are trivial maps. We also simplify the extending datum into $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,*,◃,▹,•)$. Then, $\mathsf{\Omega }(A,V)$ is a Novikov–Poisson extending datum of $(A,\circ ,·)$ through V if and only if $(l_A,r_A,l_V,r_V,*)$ is a Novikov extending datum of $(A,\circ )$ through V, $(◃,▹,•)$ is a commutative associative extending datum of $(A,·)$ through V, $(V,*,•)$ is a Novikov–Poisson algebra, $(V,◃,l_A,r_A)$ is a right module of $(A,\circ ,·)$, $(A,▹,l_V,r_V)$ is a left module of $(V,*,•)$, and they satisfy $\left(G1\right)$, $\left(G3\right)$, $\left(G5\right)$, $\left(G8\right)$, $\left(G10\right)$, $\left(G12\right)$, $\left(G15\right)$, $\left(G17\right)$, $\left(G20\right)$ and $\left(G22\right)$. In this case, the associated unified product $A♮V$ denoted by $A\bowtie V$ is called thebicrossed productof Novikov–Poisson algebras $(A,\circ ,·)$ and $(V,*,•)$ associated with the matched pair $(l_A,r_A,l_V,r_V,◃,▹)$. The bicrossed product associated with $(l_A,r_A,l_V,r_V,◃,▹)$ satisfying the compatibility conditions above is the Novikov–Poisson algebra defined as follows:

$$\begin{array}{ccc}& & (a+x)\circ (b+y)=(a\circ b+l_V\left(x\right)b+r_V\left(y\right)a)+(x*y+l_A\left(a\right)y+r_A\left(b\right)x),\hfill \\ & & (a+x)·(b+y)=(a·b+y▹a+x▹b)+(y◃a+x◃b+x•y),\hfill \end{array}$$

for all a, $b\in A$, x, $y\in V$. Note that $(A,\circ ,·)$ and $(V,*,•)$ are both subalgebras of $A\bowtie V$.

Theorem 2.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra and $(E,\circ ,·)$ be a Novikov–Poisson algebra containing $(A,\circ ,·)$ as a subalgebra. Then, there exists a Novikov–Poisson extending datum $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ of $(A,\circ ,·)$, such that $(E,\circ ,·)\cong A♮V$ as Novikov–Poisson algebras whose restriction on A is the identity map.

Proof. 

Let $\pi :E\to A$ be the natural linear map such that $\pi \left(a\right)=a$ for all $a\in A$. Let $V=\mathrm{Ker}\left(\pi \right)\subset E$, which is a complement of A in E as a vector space. Then, we define the extending datum of $(A,\circ ,·)$ through V as follows:

$$\begin{array}{ccc}& & l_A:A\to gl\left(V\right),l_A\left(a\right)x:=a\circ x-\pi (a\circ x),\hfill \\ & & r_A:A\to gl\left(V\right),r_A\left(a\right)x:=x\circ a-\pi (x\circ a),\hfill \\ & & l_V:V\to gl\left(A\right),l_V\left(x\right)a:=\pi (x\circ a),\hfill \\ & & r_V:V\to gl\left(A\right),r_V\left(x\right)a:=\pi (a\circ x),\hfill \\ & & f:V\times V\to A,f(x,y):=\pi (x\circ y),\hfill \\ & & *:V\times V\to V,x*y:=x\circ y-\pi (x\circ y),\hfill \\ & & ◃:V\times A\to V,x◃a:=x·a-\pi (x·a),\hfill \\ & & ▹:V\times A\to A,x▹a:=\pi (x·a),\hfill \\ & & g:V\times V\to A,g(x,y):=\pi (x·y),\hfill \\ & & •:V\times V\to V,•:=x·y-\pi (x·y),\hfill \end{array}$$

for all $a\in A$, $x,y\in V$. According to the proof of Theorem 3.9 in [20] and the proof of Theorem 2.4 in [18], it is easy to prove that $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ is a Novikov–Poisson extending datum of $(A,\circ ,·)$ through V and the linear map $\phi :A♮V\to (E,\circ ,[·,·\left]\right)$, where $\phi (a+x):=a+x$ is an isomorphism of Novikov–Poisson algebras whose restriction on A is the identity map. □

Remark 3.

According to Theorem 2 and Example 1, any Novikov–Poisson algebra structure on the vector space $E=A\oplus V$, such that $(A,\circ ,·)$ is an ideal of E is isomorphic to a crossed product $A\diamond V$ for some Novikov–Poisson algebra $(V,*,•)$.

Let $(A,\circ ,·)$ and $(V,*,•)$ be two Novikov–Poisson algebras. By Theorem 2 and Example 2, any Novikov–Poisson algebra structure on the vector space $E=A\oplus V$ such that $(A,\circ ,·)$ and $(V,*,•)$ are two subalgebras is isomorphic to a bicrossed product $A\bowtie V$.

Definition 9.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra and V a vector space. If there is a pair of linear maps $(\lambda ,\mu )$ in which $\lambda :V\to A$ is a linear map, $\mu \in {Aut}_{\mathbf{k}}\left(V\right)$ such that the Novikov–Poisson extending datum $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ can be obtained from another Novikov–Poisson extending datum ${\mathsf{\Omega }}^{{}^{\prime }}(A,V)=(l_A^{{}^{\prime }},r_A^{{}^{\prime }},l_V^{{}^{\prime }},r_V^{{}^{\prime }},f^{{}^{\prime }},{*}^{{}^{\prime }},{◃}^{{}^{\prime }},{▹}^{{}^{\prime }},g^{{}^{\prime }},{•}^{{}^{\prime }})$ through $(\lambda ,\mu )$ as follows:

$$\begin{array}{ccc}\hfill \left(D1\right)& & r_A\left(a\right)x={\mu }^{-1}\left(r_A^{{}^{\prime }}\left(a\right)\mu \left(x\right)\right),\hfill \\ \hfill \left(D2\right)& & l_A\left(a\right)x={\mu }^{-1}\left(l_A^{{}^{\prime }}\left(a\right)\mu \left(x\right)\right),\hfill \\ \hfill \left(D3\right)& & l_V\left(x\right)a=\lambda \left(x\right)\circ a+l_V^{{}^{\prime }}\left(\mu \left(x\right)\right)a-\lambda \left({\mu }^{-1}\left(r_A^{{}^{\prime }}\left(a\right)\mu \left(x\right)\right)\right),\hfill \\ \hfill \left(D4\right)& & r_V\left(x\right)a=a\circ \lambda \left(x\right)+r_V^{{}^{\prime }}\left(\mu \left(x\right)\right)a-\lambda \left({\mu }^{-1}\left(l_A^{{}^{\prime }}\left(a\right)\mu \left(x\right)\right)\right),\hfill \\ \hfill \left(D5\right)& & x*y={\mu }^{-1}\left(\mu \left(x\right){*}^{{}^{\prime }}\mu \left(y\right)\right)+{\mu }^{-1}\left(l_A^{{}^{\prime }}\left(\lambda \left(x\right)\right)\mu \left(y\right)\right)+{\mu }^{-1}\left(r_A^{{}^{\prime }}\left(\lambda \left(y\right)\right)\mu \left(x\right)\right),\hfill \\ \hfill \left(D6\right)& & f(x,y)=\lambda \left(x\right)\circ \lambda \left(y\right)+l_V^{{}^{\prime }}\left(\mu \left(x\right)\right)\lambda \left(y\right)+r_V^{{}^{\prime }}\left(\mu \left(y\right)\right)\lambda \left(x\right)+f^{{}^{\prime }}(\mu \left(x\right),\mu \left(y\right))\hfill \\ & & -\lambda \left({\mu }^{-1}\left(\mu \left(x\right){*}^{{}^{\prime }}\mu \left(y\right)\right)\right)-\lambda \left({\mu }^{-1}\left(l_A^{{}^{\prime }}\left(\lambda \left(x\right)\right)\mu \left(y\right)\right)\right)-\lambda \left({\mu }^{-1}\left(r_A^{{}^{\prime }}\left(\lambda \left(y\right)\right)\mu \left(x\right)\right)\right),\hfill \\ \hfill \left(D7\right)& & x◃a={\mu }^{-1}\left(\mu \left(x\right){◃}^{{}^{\prime }}a\right),\hfill \\ \hfill \left(D8\right)& & x▹a=\lambda \left(x\right)·a+\mu \left(x\right){▹}^{{}^{\prime }}a-\lambda \left({\mu }^{-1}\left(\mu \left(x\right){◃}^{{}^{\prime }}a\right)\right),\hfill \\ \hfill \left(D9\right)& & g(x,y)=\lambda \left(x\right)·\lambda \left(y\right)+\mu \left(x\right){▹}^{{}^{\prime }}\lambda \left(y\right)+\mu \left(y\right){▹}^{{}^{\prime }}\lambda \left(x\right)+g^{{}^{\prime }}(\mu \left(x\right),\mu \left(y\right))\hfill \\ & & -\lambda \left({\mu }^{-1}\left(\mu \left(x\right){•}^{{}^{\prime }}\mu \left(y\right)\right)\right)-\lambda \left({\mu }^{-1}\left(\mu \left(x\right){◃}^{{}^{\prime }}\lambda \left(y\right)\right)\right)-\lambda \left({\mu }^{-1}\left(\mu \left(y\right){◃}^{{}^{\prime }}\lambda \left(x\right)\right)\right),\hfill \\ \hfill \left(D10\right)& & x•y={\mu }^{-1}\left(\mu \left(x\right){◃}^{{}^{\prime }}\lambda \left(y\right)\right)+{\mu }^{-1}\left(\mu \left(y\right){◃}^{{}^{\prime }}\lambda \left(x\right)\right)+{\mu }^{-1}\left(\mu \left(x\right){•}^{{}^{\prime }}\mu \left(y\right)\right),\hfill \end{array}$$

for all $a\in A$, x, $y\in V$, then $\mathsf{\Omega }(A,V)$ and ${\mathsf{\Omega }}^{{}^{\prime }}(A,V)$ are calledequivalent, which is denoted by $\mathsf{\Omega }(A,V)\equiv {\mathsf{\Omega }}^{{}^{\prime }}(A,V)$.

Lemma 1.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra, V a vector space, $\mathsf{\Omega }(A,V)=$$(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ and ${\mathsf{\Omega }}^{{}^{\prime }}(A,V)=(l_A^{{}^{\prime }},r_A^{{}^{\prime }},l_V^{{}^{\prime }},r_V^{{}^{\prime }},f^{{}^{\prime }},{*}^{{}^{\prime }}$, ${◃}^{{}^{\prime }},{▹}^{{}^{\prime }},g^{{}^{\prime }},{•}^{{}^{\prime }})$ be two Novikov–Poisson extending datums of $(A,\circ ,·)$ through V. Let $A♮V$ and $A{♮}^{{}^{\prime }}V$ be the corresponding unified products, respectively. Then, $A♮V\equiv A{♮}^{{}^{\prime }}V$ if and only if $\mathsf{\Omega }(A,V)\equiv {\mathsf{\Omega }}^{{}^{\prime }}(A,V)$.

Proof. 

Let $\phi :A♮V\to A{♮}^{{}^{\prime }}V$ be an isomorphism of Novikov–Poisson algebras whose restriction on A is the identity map. Since ${\phi |}_A$ is the identity map, we assume that ${\phi }_{\lambda ,\mu }(a+x)=(a+\lambda \left(x\right))+\mu \left(x\right)$, where $\lambda :V\to A$ and $\mu :V\to V$ are two linear maps. Note that a homomorphism of Novikov–Poisson algebras is also a homomorphism of Novikov algebras and a homomorphism of commutative associative algebras. By Lemma 3.12 in [20] and Lemma 2.5 in [18], we get that $\phi$ is an isomorphism of Novikov–Poisson algebras whose restriction on A is the identity map if and only if $\mu \in {\mathrm{Aut}}_{\mathbf{k}}\left(V\right)$ and $\left(D1\right)$-$\left(D10\right)$ hold. Then, this conclusion follows via Definition 9. □

Theorem 3.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra and E a vector space which contains A as a subspace and V a complement of A in E as a vector space. Denote ${\mathcal{GH}}^2(V,A):=\mathcal{GD}(A,V)/\equiv$. Then, the map

$$\begin{array}{c}\hfill {\mathcal{GH}}^2(V,A)\mapsto Extd(E,A),\overline{\mathsf{\Omega }(A,V)}\to (A♮V,\circ ,·)\end{array}$$

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is bijective, where $\overline{\mathsf{\Omega }(A,V)}$ is the equivalence class of $\mathsf{\Omega }(A,V)$ via ≡.

Proof. 

This conclusion follows directly from Theorem 1, Theorem 2 and Lemma 1. □

Remark 4.

By Theorem 3, ${\mathcal{GH}}^2(V,A)$ describes and classifies all Novikov–Poisson algebra structures on $E=A\oplus V$ such that $(A,\circ ,·)$ is a subalgebra up to isomorphism whose restriction on A is the identity map, i.e., it gives a theoretical answer to the extending structures problem for Novikov–Poisson algebras.

Remark 5.

Note that unified product is a very general construction which includes bicrossed product, crossed product and so on. Therefore, from Theorem 3, we can also obtain many interesting results in some special cases. We also give an example which corresponds to a crossed product as follows.

According to Theorem 3, Example 1 and Remark 3, ${\mathcal{GH}}^2(V,A):=\mathcal{GD}(A,V)/\equiv$, where $l_A$, $r_A$ and ⊲ are trivial in these Novikov–Poisson extending datums, describes and classifies all Novikov–Poisson algebra structures on $E=A\oplus V$, such that $(A,\circ ,·)$ is an ideal up to isomorphism whose restriction on A is the identity map.

4. Unified Products $\mathbf{A}♮\mathbf{V}$ When $\dim \left(\mathbf{V}\right)=\mathbf{1}$

In this section, we consider the unified product when $\dim \left(V\right)=1$ in detail.

Definition 10

([20], Definition 4.3). Let $(A,\circ )$ be a Novikov algebra. A Novikov flag datum of $(A,\circ )$ is a 6-tuple $(h,k,D,T,a_0,\alpha )$ consisting of four linear maps: h, $k:A\to \mathbf{k}$, D, $T:A\to A$ and two elements $a_0\in A$, $\alpha \in \mathbf{k}$ satisfying the following conditions:

$$\begin{array}{ccc}\hfill \left(FN1\right)& & h(a\circ b)=h(b\circ a),k(a\circ b)=k\left(a\right)k\left(b\right),\hfill \\ \hfill \left(FN2\right)& & D(a\circ b)=D\left(a\right)\circ b+a\circ D\left(b\right)+\left(k\right(a)-h(a\left)\right)D\left(b\right)+k\left(b\right)T\left(a\right)-T\left(a\right)\circ b,\hfill \\ \hfill \left(FN3\right)& & T(a\circ b)-T(b\circ a)=h\left(b\right)T\left(a\right)-h\left(a\right)T\left(b\right)+a\circ T\left(b\right)-b\circ T\left(a\right),\hfill \\ \hfill \left(FN4\right)& & T^2\left(a\right)=T\left(D\left(a\right)\right)-D\left(T\left(a\right)\right)+a\circ a_0+(k\left(a\right)-2h\left(a\right))a_0+\alpha T\left(a\right),\hfill \\ \hfill \left(FN5\right)& & h\left(D\right(a\left)\right)-h\left(T\right(a\left)\right)=k\left(T\right(a\left)\right)+\alpha \left(h\right(a)-k(a\left)\right),\hfill \\ \hfill \left(FN6\right)& & D\left(a\right)\circ b+k\left(a\right)D\left(b\right)=D\left(b\right)\circ a+k\left(b\right)D\left(a\right),\hfill \\ \hfill \left(FN7\right)& & T(a\circ b)=T\left(a\right)\circ b+h\left(a\right)D\left(b\right),\hfill \\ \hfill \left(FN8\right)& & h(a\circ b)=h\left(a\right)k\left(b\right),\hfill \\ \hfill \left(FN9\right)& & T\left(D\left(a\right)\right)=a_0\circ a+\alpha D\left(a\right)-k\left(a\right)a_0,\hfill \\ \hfill \left(FN10\right)& & h\left(D\right(a\left)\right)=0,\hfill \end{array}$$

for all a, $b\in A$.

Definition 11

([20], Definition 2.5). Let $(A,\circ )$ be a Novikov algebra. If a linear map $T:A\to A$ satisfies

$$\begin{array}{ccc}& & T(a\circ b)=T\left(a\right)\circ b,\hfill \end{array}$$

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$$\begin{array}{ccc}& & T(a\circ b)-T(b\circ a)=a\circ T\left(b\right)-b\circ T\left(a\right),\mathrm{for}\mathrm{all}a,b\in A,\hfill \end{array}$$

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then the linear map T is called a quasicentroid.

Remark 6.

For any $b\in A$, there is a quasicentroid $T_b$ associated to it defined by $T_b\left(a\right)=a\circ b$ for all $a\in A$. We call $T_b$ aninner quasicentroidof $(A,\circ )$.

Definition 12

([18], Definition 4.2). Let $(A,·)$ be a commutative associative algebra. A commutative associative flag datum of $(A,·)$ is a 4-tuple $(\wedge ,P,a_1,\beta )$ consisting of two linear maps: $\wedge :A\to \mathbf{k}$, $P:A\to A$ and two elements: $a_1\in A$, $\beta \in \mathbf{k}$ satisfying the following conditions:

$$\begin{array}{ccc}\hfill \left(CN1\right)& & \wedge \left(P\right(a\left)\right)=0,\hfill \\ \hfill \left(CN2\right)& & \wedge (a·b)=\wedge \left(a\right)\wedge \left(b\right),\hfill \\ \hfill \left(CN3\right)& & P(a·b)=\wedge \left(a\right)P\left(b\right)+P\left(a\right)·b,\hfill \\ \hfill \left(CN4\right)& & P^2\left(a\right)=\beta P\left(a\right)+a_1·a-\wedge \left(a\right)a_1,\hfill \end{array}$$

for all a, $b\in A$.

Definition 13.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra. ANovikov–Poisson flag datumof $(A,\circ ,·)$ is a 10-tuple $(h,k,D,T,a_0,\alpha ,\wedge ,P,a_1,\beta )$ consisting of six linear maps: h, k, $\wedge :A\to \mathbf{k}$, D, T, $P:A\to A$ and four elements $a_0,a_1\in A$, $\alpha ,\beta \in \mathbf{k}$ satisfying the following conditions:

$$\begin{array}{ccc}\hfill \left(GF0\right)& & (h,k,D,T,a_0,\alpha )\mathit{is}a\mathit{Novikov}\mathit{flag}\mathit{datum}\mathit{of}(A,\circ )\hfill \\ & & \mathit{and}(\wedge ,P,a_1,\beta )\mathit{is}a\mathit{commutative}\mathit{associative}\mathit{flag}\mathit{datum}\mathit{of}(A,·),\hfill \\ \hfill \left(GF1\right)& & h\left(P\left(a\right)\right)=0,\wedge \left(D\left(c\right)\right)=0,h\left(a_1\right)=\wedge \left(a_0\right),\hfill \\ \hfill \left(GF2\right)& & h(a·b)=\wedge \left(a\right)h\left(b\right),\wedge (a\circ b)=\wedge \left(a\right)k\left(b\right),k(a·b)=\wedge \left(b\right)k\left(a\right),\hfill \\ \hfill \left(GF3\right)& & T(a·b)=a·T\left(b\right)+h\left(b\right)P\left(a\right),\hfill \\ \hfill \left(GF4\right)& & P\left(a\right)\circ c+\wedge \left(a\right)D\left(c\right)=a·D\left(c\right)+k\left(c\right)P\left(a\right),\hfill \\ \hfill \left(GF5\right)& & T\left(P\left(a\right)\right)+\wedge \left(a\right)a_0=a·a_0+\alpha P\left(a\right),\hfill \\ \hfill \left(GF6\right)& & T\left(P\left(b\right)\right)+\wedge \left(b\right)a_0=P\left(T\left(b\right)\right)+h\left(b\right)a_1,\hfill \\ \hfill \left(GF7\right)& & \wedge \left(b\right)\alpha +h\left(P\right(b\left)\right)=\wedge \left(T\right(b\left)\right)+\beta h\left(b\right),\hfill \\ \hfill \left(GF8\right)& & a_1\circ c+\beta D\left(c\right)=P\left(D\left(c\right)\right)+k\left(c\right)a_1,\hfill \\ \hfill \left(GF9\right)& & T\left(a_1\right)+\beta a_0=P\left(a_0\right)+\alpha a_1,\hfill \\ \hfill \left(GF10\right)& & P\left(b\right)\circ c+\wedge \left(b\right)D\left(c\right)=P(b\circ c),\hfill \\ \hfill \left(GF11\right)& & P(a\circ b)-P(b\circ a)=a\circ P\left(b\right)-b\circ P\left(a\right)+\wedge \left(b\right)T\left(a\right)-\wedge \left(a\right)T\left(b\right),\hfill \\ \hfill \left(GF12\right)& & \wedge (a\circ b)-\wedge (b\circ a)+\wedge \left(a\right)h\left(b\right)-\wedge \left(b\right)h\left(a\right)=0,\hfill \\ \hfill \left(GF13\right)& & \left(T\right(a)-D(a\left)\right)·c+\left(h\right(a)-k(a\left)\right)P\left(c\right)-a\circ P\left(c\right)\hfill \\ & & -\wedge \left(c\right)T\left(a\right)+D(a·c)=0,\hfill \\ \hfill \left(GF14\right)& & P\left(T\left(a\right)\right)-P\left(D\left(a\right)\right)+(h\left(a\right)-k\left(a\right))a_1-a\circ a_1\hfill \\ & & -\beta T\left(a\right)+D\left(P\left(a\right)\right)+\wedge \left(a\right)a_0=0,\hfill \\ \hfill \left(GF15\right)& & \wedge \left(T\right(a\left)\right)-\wedge \left(D\right(a\left)\right)=\beta k\left(a\right)-\alpha \wedge \left(a\right)-k\left(P\right(a\left)\right),\hfill \end{array}$$

for all a, $b\in A$. We denote the set of all flag datums of $(A,\circ ,·)$ by $\mathcal{F}\left(A\right)$.

Proposition 1.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra and V a vector space of dimension 1 with a basis $\left\{x\right\}$. Then, there exists a bijection between the set $\mathcal{GD}(A,V)$ of all Novikov–Poisson extending datums of $(A,\circ ,·)$ through V and $\mathcal{F}\left(A\right)$ of all Novikov–Poisson flag datums of $(A,\circ ,·)$.

Proof. 

Let $\mathsf{\Omega }(A,V)=(l_A,r_A,l_V,r_V,f,*,◃,▹,g,•)$ be a Novikov–Poisson extending datum of $(A,\circ ,·)$ through V. Since $\dim \left(V\right)=1$, we can set

$$\begin{array}{ccc}& & l_A\left(a\right)x=h\left(a\right)x,r_A\left(a\right)x=k\left(a\right)x,l_V\left(x\right)a=D\left(a\right),r_V\left(x\right)a=T\left(a\right),f(x,x)=a_0,\hfill \\ & & x*x=\alpha x,x◃a=\wedge \left(a\right)x,x▹a=P\left(a\right),g(x,x)=a_1,x•x=\beta x,\hfill \end{array}$$

where $a_0,a_1\in A$, $\alpha ,\beta \in \mathbf{k}$, and h, k, $\wedge :A\to \mathbf{k}$, D, T, $P:A\to A$ are six linear maps. Take them into $\left(G0\right)$–$\left(G24\right)$. [20] (Proposition 4.4) and [18] (Proposition 4.3) we get that $\left(G0\right)$ is equivalent to $\left(GF0\right)$. Let $y=x$ in $\left(G3\right)$. Since $x⊳a=P\left(a\right)$, $x⊲a=\wedge \left(a\right)x$, $l_V\left(x\right)c=D\left(c\right)$ and $r_A\left(c\right)x=k\left(c\right)x$, one can directly obtain that $\left(G3\right)$ is equivalent to $\left(GF4\right)$. Similarly, we can get that $\left(G1\right)$–$\left(G24\right)$ are equivalent to $\left(GF1\right)$–$\left(GF15\right)$. Then, the proof is finished. □

In the sequel, we denote the unified product associated with the Novikov–Poisson extending datum corresponding to a Novikov–Poisson flag datum $(h,k,D,T,a_0,$$\alpha ,\wedge ,P,a_1,\beta )$ by $GD(A,x\mid h,k,D,T,a_0,\alpha ,\wedge ,P,a_1,\beta )$.

Theorem 4.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra of codimension 1 in the vector space E. Then, there exists a bijection

$$\begin{array}{ccc}& & Extd(E,A)\cong {\mathcal{GH}}^2(\mathbf{k},A)\cong \mathcal{F}\left(A\right)/\equiv ,\hfill \end{array}$$

where ≡ is the equivalence relation on the set $\mathcal{F}\left(A\right)$ of all Novikov–Poisson flag data of $(A,\circ ,·)$ defined as follows: $(h,k,D,T,a_0,\alpha ,\wedge ,P,a_1,\beta )\equiv (h^{{}^{\prime }},k^{{}^{\prime }},D^{{}^{\prime }},T^{{}^{\prime }},a_0^{{}^{\prime }},{\alpha }^{{}^{\prime }},{\wedge }^{{}^{\prime }},P^{{}^{\prime }},a_1^{{}^{\prime }},{\beta }^{{}^{\prime }})$ if and only if $h=h^{{}^{\prime }}$, $k=k^{{}^{\prime }}$, $\wedge ={\wedge }^{{}^{\prime }}$ and there exists a pair $(b_0,\gamma )\in A\times {\mathbf{k}}^{*}$, such that the following conditions hold:

$$\begin{array}{ccc}\hfill \left(FD1\right)& & D\left(a\right)=b_0\circ a+\gamma D^{{}^{\prime }}\left(a\right)-k\left(a\right)b_0,\hfill \\ \hfill \left(FD2\right)& & T\left(a\right)=a\circ b_0+\gamma T^{{}^{\prime }}\left(a\right)-h\left(a\right)b_0,\hfill \\ \hfill \left(FD3\right)& & \alpha =\gamma {\alpha }^{{}^{\prime }}+h\left(b_0\right)+k\left(b_0\right),\hfill \\ \hfill \left(FD4\right)& & a_0=b_0\circ b_0+\gamma D^{{}^{\prime }}\left(b_0\right)+\gamma T^{{}^{\prime }}\left(b_0\right)+{\gamma }^2{a}_0^{{}^{\prime }}-\alpha b_0,\hfill \\ \hfill \left(FD5\right)& & P\left(a\right)=b_0·a+\gamma P^{{}^{\prime }}\left(a\right)-\wedge \left(a\right)b_0,\hfill \\ \hfill \left(FD6\right)& & a_1=b_0·b_0+2\gamma P^{{}^{\prime }}\left(b_0\right)+{\gamma }^2{a}_1^{{}^{\prime }}-\gamma {\beta }^{{}^{\prime }}{b}_0-2\wedge \left(b_0\right)b_0,\hfill \\ \hfill \left(FD7\right)& & \beta =2\wedge \left(b_0\right)+\gamma {\beta }^{{}^{\prime }},\hfill \end{array}$$

for all $a\in A$. The bijection between $\mathcal{F}\left(A\right)/\equiv$ and Extd $(E,A)$ is given by

$$\begin{array}{c}\hfill \overline{(h,k,D,T,a_0,\alpha ,\wedge ,P,a_1,\beta )}\to GD(h,k,D,T,a_0,\alpha ,\wedge ,P,a_1,\beta ),\end{array}$$

where $\overline{(h,k,D,T,a_0,\alpha ,\wedge ,P,a_1,\beta )}$ is the equivalence class of $(h,k,D,T,a_0,\alpha ,\wedge ,P,$$a_1,\beta )$ by ≡.

Proof. 

Let $(h,k,D,T,a_0,\alpha ,\wedge ,P,a_1,\beta )$, $(h^{{}^{\prime }},k^{{}^{\prime }},D^{{}^{\prime }},T^{{}^{\prime }},a_0^{{}^{\prime }},{\alpha }^{{}^{\prime }},{\wedge }^{{}^{\prime }},P^{{}^{\prime }},a_1^{{}^{\prime }},{\beta }^{{}^{\prime }})\in \mathcal{F}\left(A\right)$ and $\mathsf{\Omega }(A,V)$, ${\mathsf{\Omega }}^{\prime }(A,V)$ be the corresponding Novikov–Poisson-extending data, respectively. Assume that x is a basis of V. Then, we can set $\lambda$, $\mu$ in Lemma 1 as follows:

$$\begin{array}{c}\hfill \lambda \left(x\right)=b_0,\mu \left(x\right)=\gamma x,\end{array}$$

where $b_0\in A$ and $\gamma \in {\mathbf{k}}^{*}$. Then, the desired conclusion follows directly from Lemma 1, Theorem 3 and Proposition 1. □

If $(h,k,D,T,a_0,\alpha ,\wedge ,P,a_1,\beta )\in \mathcal{F}\left(A\right)$, where h, k, T and ∧ are trivial, then we denote the flag datum as $(D,a_0,\alpha ,P,a_1,\beta )$. We denote the set of all such flag data of $(A,\circ ,·)$ as ${\mathcal{F}}_1\left(A\right)$. According to Theorem 4, $(D,a_0,\alpha ,P,a_1,\beta )\equiv (D^{{}^{\prime }},a_0^{{}^{\prime }},{\alpha }^{{}^{\prime }},P^{{}^{\prime }},a_1^{{}^{\prime }},$${\beta }^{{}^{\prime }})$ if and only if there exists a pair $(b_0,\gamma )\in A\times {\mathbf{k}}^{*}$, such that the following conditions hold:

$$\begin{array}{ccc}& & D\left(a\right)=b_0\circ a+\gamma D^{{}^{\prime }}\left(a\right),\hfill \\ & & a\circ b_0=0,\alpha =\gamma {\alpha }^{{}^{\prime }},\beta =\gamma {\beta }^{{}^{\prime }},\hfill \\ & & a_0=b_0\circ b_0+\gamma D^{{}^{\prime }}\left(b_0\right)+{\gamma }^2{a}_0^{{}^{\prime }}-\alpha b_0,\hfill \\ & & P\left(a\right)=b_0·a+\gamma P^{{}^{\prime }}\left(a\right),\hfill \\ & & a_1=b_0·b_0+2\gamma P^{{}^{\prime }}\left(b_0\right)+{\gamma }^2{a}_1^{{}^{\prime }}-\gamma {\beta }^{{}^{\prime }}{b}_0,\hfill \end{array}$$

for all $a\in A$. Particularly, when $a_0=0$, we denote such flag datum of $(A,\circ ,·)$ by $(D,\alpha ,P,a_1,\beta )$. Denote the set of all such flag data of $(A,\circ ,·)$ by ${\mathcal{F}}_2\left(A\right)$. Define $(D,\alpha ,P,a_1,\beta )$$\approx (D^{{}^{\prime }},{\alpha }^{{}^{\prime }},P^{{}^{\prime }},a_1^{{}^{\prime }},{\beta }^{{}^{\prime }})$ if there exists $\gamma \in {\mathbf{k}}^{*}$, such that the following conditions hold:

$$\begin{array}{ccc}& & D\left(a\right)=\gamma D^{{}^{\prime }}\left(a\right),\alpha =\gamma {\alpha }^{{}^{\prime }},P\left(a\right)=\gamma P^{{}^{\prime }}\left(a\right),\hfill \\ & & a_1={\gamma }^2{a}_1^{{}^{\prime }},\beta =\gamma {\beta }^{{}^{\prime }},\hfill \end{array}$$

for all $a\in A$.

If $(h,k,D,T,a_0,\alpha ,\wedge ,P,a_1,\beta )\in \mathcal{F}\left(A\right)$, where h, T are trivial and $\alpha =0$, then we denote such flag data of $(A,\circ ,·)$ by $(k,D,a_0,\wedge ,P,a_1,\beta )$. Denote the set of all such flag data of $(A,\circ ,·)$, where $k\ne 0$ by ${\mathcal{F}}_3\left(A\right)$. Using Theorem 4, $(k,D,a_0,\wedge ,P,a_1,\beta )\equiv (k^{{}^{\prime }},D^{{}^{\prime }},a_0^{{}^{\prime }},{\wedge }^{{}^{\prime }},P^{{}^{\prime }},a_1^{{}^{\prime }},{\beta }^{{}^{\prime }})$ if and only if $k=k^{{}^{\prime }}$, $\wedge ={\wedge }^{{}^{\prime }}$ and there is a pair $(b_0,\gamma )\in A\times {\mathbf{k}}^{*}$, such that the following conditions hold:

$$\begin{array}{ccc}& & D\left(a\right)=b_0\circ a+\gamma D^{{}^{\prime }}\left(a\right)-k\left(a\right)b_0,\hfill \\ & & a\circ b_0=0,k\left(b_0\right)=0,\hfill \\ & & a_0=b_0\circ b_0+\gamma D^{{}^{\prime }}\left(b_0\right)+{\gamma }^2{a}_0^{{}^{\prime }},\hfill \\ & & P\left(a\right)=b_0·a+\gamma P^{{}^{\prime }}\left(a\right)-\wedge \left(a\right)b_0,\hfill \\ & & a_1=b_0·b_0+2\gamma P^{{}^{\prime }}\left(b_0\right)+{\gamma }^2{a}_1^{{}^{\prime }}-\gamma {\beta }^{{}^{\prime }}{b}_0-2\wedge \left(b_0\right)b_0,\hfill \\ & & \beta =2\wedge \left(b_0\right)+\gamma {\beta }^{{}^{\prime }},\hfill \end{array}$$

for all $a\in A$. Furthermore, we define that $(k,D,a_0,\wedge ,P,a_1,\beta )\approx (k^{{}^{\prime }},D^{{}^{\prime }},a_0^{{}^{\prime }},{\wedge }^{{}^{\prime }},P^{{}^{\prime }},$$a_1^{{}^{\prime }},{\beta }^{{}^{\prime }})$ if $k=k^{{}^{\prime }}$, $\wedge ={\wedge }^{{}^{\prime }}$ and there exists $\gamma \in A$, such that the following conditions hold:

$$\begin{array}{ccc}& & D\left(a\right)=\gamma D^{{}^{\prime }}\left(a\right),a_0={\gamma }^2{a}_0^{{}^{\prime }},\hfill \\ & & P\left(a\right)=\gamma P^{{}^{\prime }}\left(a\right),a_1={\gamma }^2{a}_1^{{}^{\prime }},\beta =\gamma {\beta }^{{}^{\prime }},\hfill \end{array}$$

for all $a\in A$.

Corollary 1.

Let $(A,\circ ,·)$ be a Novikov–Poisson algebra, its sub-adjacent Lie algebra $\left(g\right(A),[·,·\left]\right)$ is perfect, i.e., $\left[g\right(A),g(A\left)\right]=g\left(A\right)$, and all quasicentroids of $(A,\circ )$ are inner. Then,

$$\begin{array}{ccc}& & Extd(E,A)\cong {\mathcal{GH}}^2(\mathbf{k},A)\cong ({\mathcal{F}}_1\left(A\right)/\equiv )\cup ({\mathcal{F}}_3\left(A\right)/\equiv ).\hfill \end{array}$$

In particular, if $\{b\mid a\circ b=0foralla\in A\}=\left\{0\right\}$, then

$$\begin{array}{ccc}& & Extd(E,A)\cong {\mathcal{GH}}^2(\mathbf{k},A)\cong ({\mathcal{F}}_2\left(A\right)/\approx )\cup ({\mathcal{F}}_3\left(A\right)/\approx ).\hfill \end{array}$$

Proof. 

Since $\left[g\right(A),g(A\left)\right]=g\left(A\right)$, using the proof of Corollary 4.6 in [20], we obtain the following two cases:

(1)

h, k and T are trivial,

(2)

h, T are trivial, $\alpha =0$ and k is non-trivial.

In Case (1), by $\left(GF2\right)$, we get $\wedge (a\circ b)=0$ for all a, $b\in A$. Note that $A\circ A=A$ due to $\left[g\right(A),g(A\left)\right]=g\left(A\right)$. Therefore, $\wedge =0$. Thus, in this case, we only need to consider those flag data in ${\mathcal{F}}_1\left(A\right)$. Similarly, in Case (2), we only need to compute those flag data in ${\mathcal{F}}_3\left(A\right)$. Next, we consider both cases, when $\{b\mid a\circ b=0\mathrm{for}\mathrm{all}a\in A\}=\left\{0\right\}$. Using $\left(FN4\right)$, we get $a_0=0$. Moreover, using $\left(FD2\right)$, we obtain $a\circ b_0=0$ for all $a\in A$. Therefore, $b_0=0$. Then, this corollary follows directly from Theorem 4. □

Finally, we present an example to compute ${\mathcal{GH}}^2(\mathbf{k},A)$.

Example 3.

Let $\mathbf{k}=\mathbb{C}$ and $(A,\circ ,·)$ be a two-dimensional Novikov–Poisson algebra with a basis $\{e_1,e_2\}$ and the products given by:

$$\begin{array}{c}\hfill \begin{array}{ccc}& & e_1\circ e_1=0,e_1\circ e_2=e_1,e_2\circ e_1=-e_1,e_2\circ e_2=e_2,\\ & & e_1·e_1=0,e_1·e_2=e_1,e_2·e_1=e_1,e_2·e_2=e_2.\end{array}\end{array}$$

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This is one of the classifications of two-dimensional Novikov–Poisson algebras studied in [9].

Using a long but straightforward calculation, the following flag data $(h,k,D,T,a_0,$$\alpha ,\wedge ,P,a_1,\beta )$ satisfying $\left(GF0\right)$-$\left(GF15\right)$ are as follows.

There are five cases in total.

Case A1

$$\begin{array}{ccc}& & h=0,k=0,D=\left(\begin{array}{cc}-b_1& -b_2\\ 0& b_1\end{array}\right),T=\left(\begin{array}{cc}{b}_1& b_2\\ 0& b_1\end{array}\right),\alpha =b_3,\hfill \\ & & a_0=b_2{b}_3{e}_1+(b_1^2-b_1{b}_3)e_2,\wedge \left(e_1\right)=0,\wedge \left(e_2\right)=0,\hfill \\ & & P=\left(\begin{array}{cc}{b}_1& -b_2\\ 0& b_1\end{array}\right),\beta =b_4,a_1=(b_2{b}_4-2b_1{b}_2)e_1+(b_1^2-b_1{b}_4)e_2,\hfill \\ & & forall{b}_1,b_2,b_3,b_4\in \mathbb{C}.\hfill \end{array}$$

This tells us that any three-dimensional Novikov–Poisson algebra that contains $(A,\circ ,·)$ as a subalgebra is isomorphic to the following Novikov–Poisson algebra denoted by ${A1}_{b_1,b_2,b_3,b_4}$ with the basis $\{e_1,e_2,x\}$ and the products given by (21) and

$$\begin{array}{ccc}& & x\circ e_1=-b_1{e}_1,e_1\circ x=b_1{e}_1,x\circ e_2=-b_2{e}_1+b_1{e}_2,e_2\circ x=b_2{e}_1+b_1{e}_2,\hfill \\ & & x\circ x=b_2{b}_3{e}_1+(b_1^2-b_1{b}_3)e_2+b_{3}x,x·e_1=e_1·x=b_1{e}_1,\hfill \\ & & x·e_2=e_2·x=-b_2{e}_1+b_1{e}_2,x·x=(b_2{b}_4-2b_1{b}_2)e_1+(b_1^2-b_1{b}_4)e_2+b_{4}x.\hfill \end{array}$$

Two such Novikov–Poisson algebras ${A1}_{b_1,b_2,b_3,b_4}$ and ${A1}_{b_1^{{}^{\prime }},b_2^{{}^{\prime }},b_3^{{}^{\prime }},b_4^{{}^{\prime }}}$ are equivalent if and only if there is u, $v\in \mathbb{C}$ and $\gamma \in {\mathbb{C}}^{*}$ such that $b_1=v+\gamma b_1^{{}^{\prime }}$, $b_2=-u+\gamma b_2^{{}^{\prime }}$, $b_3=\gamma b_3^{{}^{\prime }}$ and $b_4=\gamma b_4^{{}^{\prime }}$. Then, ${A1}_{b_1,b_2,b_3,b_4}$ is equivalent to ${A1}_{0,0,b_3,b_4}$ by setting $\gamma =1$, $v=b_1$ and $u=-b_2$. Furthermore, ${A1}_{0,0,b_3,b_4}$ is equivalent to ${A1}_{0,0,b_3^{{}^{\prime }},b_4^{{}^{\prime }}}$ if and only if there exists $\gamma \in {\mathbb{C}}^{*}$, such that $b_3=\gamma b_3^{{}^{\prime }}$ and $b_4=\gamma b_4^{{}^{\prime }}$.

Case A2

$$\begin{array}{ccc}& & h\left(e_1\right)=0,h\left(e_2\right)=0,k\left(e_1\right)=0,k\left(e_2\right)=1,D=0,T=\left(\begin{array}{cc}0& b_1\\ 0& 0\end{array}\right),\hfill \\ & & \alpha =0,a_0=0,\wedge \left(e_1\right)=0,\wedge \left(e_2\right)=1,P=0,\beta =0,a_1=b_2{e}_1,\hfill \\ & & forall{b}_1,b_2\in \mathbb{C}.\hfill \end{array}$$

This tell us that any three-dimensional Novikov–Poisson algebra that contains $(A,\circ ,·)$ as a subalgebra is isomorphic to the following Novikov–Poisson algebra denoted by ${A2}_{b_1,b_2}$ with the basis $\{e_1,e_2,x\}$ and the products given by (21) and

$$\begin{array}{ccc}& & x\circ e_1=0,e_1\circ x=0,x\circ e_2=x,e_2\circ x=b_1{e}_1,x\circ x=0,\hfill \\ & & x·e_1=e_1·x=0,x·e_2=e_2·x=x,x·x=b_2{e}_1.\hfill \end{array}$$

Two such Novikov–Poisson algebras ${A2}_{b_1,b_2}$ and ${A2}_{b_1^{{}^{\prime }},b_2^{{}^{\prime }}}$ are equivalent if and only if $u\in \mathbb{C}$ and $\gamma \in {\mathbb{C}}^{*}$, such that $b_1=-u+\gamma b_1^{{}^{\prime }}$ and $b_2={\gamma }^2{b}_2^{{}^{\prime }}$. Then, ${A2}_{b_1,b_2}$ is equivalent to ${A2}_{0,0}$ and ${A2}_{0,1}$, setting $\gamma =1$ and $u=-b_2$. Therefore, in this case, there are only two equivalence classes, i.e., ${A2}_{0,0}$ and ${A2}_{0,1}$.

Case A3

$$\begin{array}{ccc}& & h\left(e_1\right)=0,h\left(e_2\right)=0,k\left(e_1\right)=0,k\left(e_2\right)=1,D=\left(\begin{array}{cc}-b_1& 0\\ 0& 0\end{array}\right),T=\left(\begin{array}{cc}{b}_1& b_2\\ 0& b_1\end{array}\right),\hfill \\ & & \alpha =b_1,a_0=b_1{b}_2{e}_1,\wedge \left(e_1\right)=0,\wedge \left(e_2\right)=1,P=\left(\begin{array}{cc}{b}_1& 0\\ 0& 0\end{array}\right),\beta =2b_1,\hfill \\ & & a_1=b_3{e}_1-b_1^2{e}_2,\hfill \\ & & forall{b}_1,b_2,b_3\in \mathbb{C}.\hfill \end{array}$$

This tells us that any three-dimensional Novikov–Poisson algebra that contains $(A,\circ ,·)$ as a subalgebra is isomorphic to the following Novikov–Poisson algebra denoted by ${A3}_{b_1,b_2,b_3}$, with the basis $\{e_1,e_2,x\}$ and the products given by (21) and

$$\begin{array}{ccc}& & x\circ e_1=-b_1{e}_1,e_1\circ x=b_1{e}_1,x\circ e_2=x,e_2\circ x=b_2{e}_1+b_1{e}_2,\hfill \\ & & x\circ x=b_1{b}_2{e}_1+b_{1}x,x·e_1=e_1·x=b_1{e}_1,\hfill \\ & & x·e_2=e_2·x=x,x·x=b_3{e}_1-b_1^2{e}_2+2b_{1}x.\hfill \end{array}$$

Two such Novikov–Poisson algebras ${A3}_{b_1,b_2,b_3}$ and $A{3}_{b_1^{\prime },b_2^{\prime },b_3^{\prime }}$ are equivalent if and only if there is u, $v\in \mathbb{C}$ and $\gamma \in {\mathbb{C}}^{*}$, such that $b_1=v+\gamma b_1^{{}^{\prime }}$, $b_2=-u+\gamma b_2^{{}^{\prime }}$ and $b_3={\gamma }^2{b}_3^{{}^{\prime }}$. Then, ${A3}_{b_1,b_2,b_3}$ is equivalent to either ${A3}_{0,0,0}$ or ${A3}_{0,0,1}$. Therefore, in this case, there are only two equivalence classes, i.e., ${A3}_{0,0,0}$ and ${A3}_{0,0,1}$.

Case A4

$$\begin{array}{ccc}& & h\left(e_1\right)=0,h\left(e_2\right)=1,k\left(e_1\right)=0,k\left(e_2\right)=1,D=\left(\begin{array}{cc}{b}_1& 0\\ 0& 0\end{array}\right),T=\left(\begin{array}{cc}-b_1& b_2\\ 0& 0\end{array}\right),\hfill \\ & & \alpha =b_3,a_0=\frac{b_2{b}_3}{2}{e}_1+(b_1^2+b_1{b}_3)e_2,\wedge \left(e_1\right)=0,\wedge \left(e_2\right)=1,P=\left(\begin{array}{cc}-b_1& 0\\ 0& 0\end{array}\right),\hfill \\ & & \beta =b_3,a_1=(\frac{b_2{b}_3}{2}+b_1{b}_2)e_1+(b_1^2+b_1{b}_3)e_2,\hfill \\ & & forall{b}_1,b_2,b_3\in \mathbb{C}.\hfill \end{array}$$

This tells us that any three-dimensional Novikov–Poisson algebra that contains $(A,\circ ,·)$ as a subalgebra is isomorphic to the following Novikov–Poisson algebra denoted by ${A4}_{b_1,b_2,b_3}$, with the basis $\{e_1,e_2,x\}$ and the products given by (21) and

$$\begin{array}{ccc}& & x\circ e_1=b_1{e}_1,e_1\circ x=-b_1{e}_1,x\circ e_2=x,e_2\circ x=b_2{e}_1+x,\hfill \\ & & x\circ x=\frac{b_2{b}_3}{2}{e}_1+(b_1^2+b_1{b}_3)e_2+b_{3}x,x·e_1=e_1·x=-b_1{e}_1,\hfill \\ & & x·e_2=e_2·x=x,x·x=(\frac{b_2{b}_3}{2}+b_1{b}_2)e_1+(b_1^2+b_1{b}_3)e_2+b_{3}x.\hfill \end{array}$$

Two such Novikov–Poisson algebras ${A4}_{b_1,b_2,b_3}$ and ${A4}_{b_1^{{}^{\prime }},b_2^{{}^{\prime }},b_3^{{}^{\prime }}}$ are equivalent if and only if u, $v\in \mathbb{C}$ and $\gamma \in {\mathbb{C}}^{*}$, such that $b_1=-v+\gamma b_1^{{}^{\prime }}$, $b_2=-2u+\gamma b_2^{{}^{\prime }}$, and $b_3=2v+\gamma b_3^{{}^{\prime }}$. Then, ${A4}_{b_1,b_2,b_3}$ is equivalent to ${A4}_{b_1,0,b_3}$ by letting $u=-\frac{b_2}{2}$, $v=0$ and $\gamma =1$. Moreover, ${A4}_{b_1,0,b_3}$ is equivalent to ${A4}_{b_1^{{}^{\prime }},0,b_3^{{}^{\prime }}}$ if and only if $v\in \mathbb{C}$ and $\gamma \in {\mathbb{C}}^{*}$, such that $b_1=-v+\gamma b_1^{{}^{\prime }}$ and $b_3=2v+\gamma b_3^{{}^{\prime }}$.

Case A5

$$\begin{array}{ccc}& & h\left(e_1\right)=0,h\left(e_2\right)=b_1,k\left(e_1\right)=0,k\left(e_2\right)=1,D=\left(\begin{array}{cc}{b}_2& 0\\ 0& 0\end{array}\right),\hfill \\ & & T=\left(\begin{array}{cc}-b_2& b_3\\ 0& (b_1-1)b_2\end{array}\right),\alpha =-(b_1+1)b_2,a_0=-b_2{b}_3{e}_1-b_1{b}_2^2{e}_2,\hfill \\ & & \wedge \left(e_1\right)=0,\wedge \left(e_2\right)=1,P=\left(\begin{array}{cc}-b_2& 0\\ 0& 0\end{array}\right),\beta =-2b_2,a_1=-b_2^2{e}_2,\hfill \\ & & forall{b}_1,b_2,b_3\in \mathbb{C},b_1\ne 0and{b}_1\ne 1.\hfill \end{array}$$

This tells us that any three-dimensional Novikov–Poisson algebra that contains $(A,\circ ,·)$ as a subalgebra is isomorphic to the following Novikov–Poisson algebra denoted by ${A5}_{b_1,b_2,b_3}$, with the basis $\{e_1,e_2,x\}$ and the products given by (21) and

$$\begin{array}{ccc}& & x\circ e_1=b_2{e}_1,e_1\circ x=-b_2{e}_1,x\circ e_2=x,e_2\circ x=b_3{e}_1+(b_1-1)b_2{e}_2+b_{1}x,\hfill \\ & & x\circ x=-b_2{b}_3{e}_1-b_1{b}_2^2{e}_2-(b_1+1)b_{2}x,x·e_1=e_1·x=-b_2{e}_1,\hfill \\ & & x·e_2=e_2·x=x,x·x=-b_2^2{e}_2-2b_{2}x.\hfill \end{array}$$

Two such Novikov–Poisson algebras ${A5}_{b_1,b_2,b_3}$ and ${A5}_{b_1^{{}^{\prime }},b_2^{{}^{\prime }},b_3^{{}^{\prime }}}$ are equivalent if and only if u, $v\in \mathbb{C}$ and $\gamma \in {\mathbb{C}}^{*}$, such that $b_1=b_1^{{}^{\prime }}$, $b_2=-v+\gamma b_2^{{}^{\prime }}$ and $b_3=-(b_1+1)u+\gamma b_3^{{}^{\prime }}$. Then, ${A5}_{b_1,b_2,b_3}$ is equivalent to ${A5}_{-1,0,0}$, ${A5}_{-1,0,1}$ or ${A5}_{b_1,0,0}$, where $b_1\ne 0$, $b_1\ne 1$ and $b_1\ne -1$. Therefore, in this case, there are three equivalence classes, i.e., ${A5}_{-1,0,0},{A5}_{-1,0,1}$ and ${A5}_{b_1,0,0}$, where $b_1\ne 0$, $b_1\ne 1$ and $b_1\ne -1$.

Therefore, according to the discussion above and Theorem 4, ${\mathcal{GH}}^2(\mathbb{C},A)$ can be described by the disjointed union of the equivalence classes of ${A1}_{0,0,b_3,b_4}$, ${A2}_{0,0}$, ${A2}_{0,1}$, ${A3}_{0,0,0}$ and ${A3}_{0,0,1}$, the equivalence classes of ${A4}_{b_1,0,b_3}$ and ${A5}_{-1,0,0},{A5}_{-1,0,1}$ and ${A5}_{b_1,0,0}$, where $b_1\ne 0$, $b_1\ne 1$ and $b_1\ne -1$.

5. Conclusions

Based on the study in the previous sections, we give the following conclusions and discussions.

By means of the definition of a unified product for Novikov–Poisson algebras, we have constructed an object ${\mathcal{GH}}^2(V,A)$ to give a theoretical answer to the extending structures problem for Novikov–Poisson algebras raised in the introduction. Note that unified product is a very general product, which includes crossed product, bicrossed product and so on. Therefore, the general theory developed in this paper can be also applied to these special cases; for example, replacing a unified product with a crossed product, we can naturally obtain an extension theory of Novikov–Poisson algebras. Note that for an algebra object, the extension theory is closely related to its cohomology theory and the bicrossed product is related to its bialgebra theory. Therefore, these results will enrich the structure theory of Novkov-Poisson algebras and can be applied to investigate the cohomology theory and bialgebra theory of Novikov–Poisson algebras. Moreover, we have shown that when $\dim \left(V\right)=1$, ${\mathcal{GH}}^2(V,A)$ can be computed by studying flag data. These results will be useful for classifying low-dimensional Novikov–Poisson algebras. In addition, as [6] shows, a Novikov–Poisson algebra corresponds to a Novikov conformal algebra. The theory in this paper will be helpful for investigating the extending structures problem for Novikov conformal algebras, which will be studied in our future research.