00footnotetext: ${*}$Corresponding author.

# Extending structures of Rota-Baxter Lie algebras

Xiao-song Peng  School of Mathematics and Statistics, Jiangsu Normal
University, Xuzhou, Jiangsu 221116, P. R. China<EMAIL_ADDRESS>and  Yi
Zhang∗ School of Mathematics and Statistics, Nanjing University of Information
Science & Technology, Nanjing, Jiangsu 210044, P. R. China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first introduce the notion of an extending datum of a Rota-
Baxter Lie algebra through a vector space. We then construct a unified product
for the Rota-Baxter Lie algebra with a vector space as a main ingredient in
our approach. Finally, we solve the extending structures problem of Rota-
Baxter Lie algebras, which generalizes and unifies two problems in the study
of Rota-Baxter Lie algebras: the extension problem studied by Mishra-Das-Hazra
and the factorization problem investigated by Lang-Sheng.

###### 2010 Mathematics Subject Classification:

17B38, 17B05, 17B60

###### Contents

  1. 1 Introduction
    1. 1.1 Rota-Baxter (Lie) algebras
    2. 1.2  Extending structures problem
  2. 2 Unified products for Rota-Baxter Lie algebras
  3. 3 Special cases of unified products
    1. 3.1 Crossed products and the extension problem
    2. 3.2 Matched pairs and the factorization problem
  4. 4 Flag extending structures

##  1\. Introduction

The aim of this paper is to solve the extending structure problem for Rota-
Baxter Lie algebras which asks for the classification of all Rota-Baxter Lie
algebraic structures on a given vector space by the approach of a general
unified product.

###  1.1. Rota-Baxter (Lie) algebras

Let $\mathbf{k}$ be a field and $\lambda\in\mathbf{k}$. A Rota-Baxter algebra
of weight $\lambda$ is a pair $(R,P)$, where $R$ is an associative algebra
over $\mathbf{k}$, and $P:R\rightarrow R$ is a linear operator which satisfies
the Rota-Baxter identity

(1) $\displaystyle P(x)P(y)=P(xP(y)+P(x)y+\lambda xy)\,\text{ for }\,x,y\in
R.$

Then $P$ is called a Rota-Baxter operator of weight $\lambda$. The notion of a
Rota-Baxter algebra was probably introduced in the first time by G. Baxter
[12] in 1960 based on the study of Spitzer’s identity in fluctuation theory.
It is a natural algebraic interpretation of the formula of partial integration
in analysis, as in the case of a differential algebra could be treated as an
algebraic abstraction of differential equations. More precisely, let $R$ be
the $\mathbb{R}$-algebra of continuous functions on $\mathbb{R}$. Define
$P:R\rightarrow R$ to be the integration

$\displaystyle P(f)(x)=\int_{0}^{x}f(t)dt.$

Then the integration by parts formula

$\displaystyle\int_{0}^{x}P(f)^{\prime}(t)P(g)(t)dt=P(f)(x)P(g)(x)-\int_{0}^{x}P(f)(t)P(g)^{\prime}(t)dt$

is precise the Eq. (1) with $\lambda=0$.

After 60 years developments, the study of Rota-Baxter algebras has become a
vibrant research field and plays an important role in mathematics and
mathematical physics, such as multiple zeta values [20], pre-Lie algebras [1,
6], Hopf algebras [21, 30, 40, 42, 44], (antisymmetric) infinitesimal
bialgebras [5, 38], dendriform algebras [14, 15, 43], operads [34],
$\mathcal{O}$-operators [7], quantum field theory [13], classical
(associative) Yang-Baxter equations [7, 8, 10, 16]. For more details about
Rota-Baxter algebras, see the monograph [17].

Completely independent of the above developments, the Rota-Baxter operator in
the context of Lie algebras has its own motivations and developing history. In
fact, it is an operator form of the classical Yang-Baxter equation (CYBE),
named after the physicists C. N. Yang [39] and R. Baxter [11]. For a given Lie
algebra $\mathfrak{g}$, the classical Yang-Baxter equation (CYBE) was defined
by the following tensor form

$[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0,$

where $r\in\mathfrak{g}\otimes\mathfrak{g}$. In [36], Semonov-Tian-Shansky
showed that if there exists a nondegenerate symmetric invariant bilinear form
on $\mathfrak{g}$, then a skew-symmetric solution $r$ of the CYBE can be
equivalently expressed as a linear operator $R:\mathfrak{g}\to\mathfrak{g}$
satisfying the following identity

(2) $[R(x),R(y)]=R([R(x),y])+R([x,R(y)]),\;\;\forall x,y\in\mathfrak{g},$

which is precisely the Rota-Baxter relation (of weight zero) in Eq. (1) in the
context of Lie algebras. In this way, Eq. (2) is called an operator form of
the CYBE. We would like to emphasize that this result was generalized by
Kupershmidt [25] by the concept of an $\mathcal{O}$-operator, which was also
called a relative Rota-Baxter operator.

Quite recently the discovery of some new algebraic and geometric structures in
mathematics and physics [19] has led to a renewed interest in the study of
Rota-Baxter Lie algebras and their connections with deformations and homotopy
theory [27, 28, 37], cf. [37] for recent surveys. One of the main motivation
comes from the emergence of Rota-Baxter Lie groups. In [19], Lang-Sheng-Guo
showed that the differentiation of a Rota-Baxter Lie group is a Rota-Baxter
Lie algebra of weight 1, generalizing the fact that the differentiation of a
Lie group is a Lie algebra. Let $G$ be a Lie group. A Rota-Baxter Lie group is
a Lie group $G$ together with a smooth map $\mathcal{B}:G\rightarrow G$ on $G$
satisfying

$\displaystyle\mathcal{B}(g_{1})\mathcal{B}(g_{2})=\mathcal{B}(g_{1}\mathrm{Ad}_{\mathcal{B}(g_{1})}g_{2}),\
\ \forall g_{1},g_{2}\in G,$

where $\mathrm{Ad}_{g},g\in G$ is an adjoint action. Then the following
formula

$\displaystyle[u,v]_{\mathfrak{g}}=\frac{d^{2}}{dtds}\mid_{t,s=0}\mathrm{exp}^{tu}\mathrm{exp}^{sv}\mathrm{exp}^{-tu},\
\ \forall u,v\in\mathfrak{g}$

captures the relationship between the Lie bracket $[,]_{\mathfrak{g}}$ and the
Lie group multiplication. Let $e$ be an identity of $G$ and
$\mathfrak{g}=T_{e}G$ the Lie algebra of $G$. Then the tangent map
$B=\mathcal{B}_{\star e}:\mathfrak{g}\rightarrow\mathfrak{g}$ of $\mathcal{B}$
at the identity $e$ is a Rota-Baxter operator of weight $1$ and so
$(\mathfrak{g},[,]_{\mathfrak{g}},B)$ is a Rota-Baxter Lie algebra of weight
1.

Viewing Rota-Baxter Lie algebras in the framework of bialgebras, Lang and
Sheng [26] introduced the notion of a quadratic Rota-Baxter Lie algebra, which
has a one-one correspondence to the factorizable Lie bialgebras. This leads to
the introduction of a Rota-Baxter Lie bialgebra [26]. The study of Rota-Baxter
Lie bialgebras was partly motivated by the theory of Manin triple approach.
Except for the construction of a one-to-one correspondence between Manin
triples of Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras [26], Bai-
Guo-Liu-Ma [9] independently established a general bialgebra structure on
Rota-Baxter Lie algebras following the approach of Manin triples to Lie
bialgebras and antisymmetric infinitesimal bialgebras. A cohomology theory of
a relative Rota-Baxter Lie algebra with coefficients in a representation was
studied in [24].

###  1.2.  Extending structures problem

The extending structures(ES) problem arose in the study of group theory by
Agore and Militaru [2] around 2014. It can be regarded as a uniform of two
well-known problems in group theory– the extension problem of Hölder [22] and
the factorization problem of Ore [33], which served as baby models for the ES-
problem approach [2]. Later, Agore and Militaru argued in favor of the study
of ES-problems on several occasions. For example, the extending structures
problem for Lie algebras and unital associative algebras are solved in [3] and
[4] respectively. Apparently, the ES-problem becomes more popular when some
new motivations were found, coming on one hand from interesting applications
to conformal algebras [23] and on the other from close relation to the theory
of 3-Lie algebras [41] and superalgebras [45].

Considering the fact that the Rota-Baxter Lie algebra is a generalization of
the Lie algebra, it is almost a natural question to consider the following
extending structures problem.

######  Problem 1.1.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra. Let $E$ be
a vector space containing $\mathfrak{g}$ as a subspace. Up to an isomorphism
of Rota-Baxter Lie algebras which stabilizes $(\mathfrak{g},P_{\mathfrak{g}})$
, describe and classify all the Rota-Baxter Lie algebraic structures
$([-,-],P_{E})$ on $E$ such that $(\mathfrak{g},P_{\mathfrak{g}})$ is a Rota-
Baxter Lie subalgebra of $(E,[-,-],P_{E})$.

In order to solve the above problem, we follow the steps of Agore and Militaru
[3] on the construction of the Lie algebra case, we give the concept of an
extending datum of a Rota-Baxter Lie algebra through a vector space and a
Rota-Baxter Lie extending structure. Generalizing the results of Agore and
Militaru [3] of the Lie algebra case, the description part and the
classification part of the extending structure problem of Rota-Baxter Lie
algebras are answered by a unified product of Rota-Baxter Lie algebras and a
relative cohomology group. In particular, the non-abelian extension of Rota-
Baxter Lie algebras in [32] and the factorization problem of Rota-Baxter Lie
algebras in [26] appear to be special cases of the Rota-Baxter Lie extending
structures problem.

This paper is organized as follows. In Section 2, we give the concept of an
extending datum of Rota-Baxter Lie algebras with vector spaces and a unified
product from an extending datum. Then the description part of the extending
structure problem of Rota-Baxter Lie algebras is answered in Theorem 2.11. The
classification part of the extending structure problem of Rota-Baxter Lie
algebras is answered in Theorem 2.15 using a relative cohomology group. In
Section 3, we show that the cross products and bicrossed products of two Rota-
Baxter Lie algebras are special cases of the unified product. In Section 4, we
consider the flag extending structure of Rota-Baxter Lie algebras and give an
answer to the calculation of the classifying object of the extending structure
of the flag extending structure of Rota-Baxter Lie algebras.

Notation. Throughout this paper, let k be a field unless the contrary is
specified, which will be the base ring of all modules, algebras, coalgebras,
bialgebras, tensor products, as well as linear maps.

##  2\. Unified products for Rota-Baxter Lie algebras

In this section, we first introduce an extending datum of a Rota-Baxter Lie
algebra. We then give a construction of the unified product of a Rota-Baxter
Lie algebra with a vector space. We prove that any Rota-Baxter Lie algebra on
a vector space $E$ containing a given Rota-Baxter Lie algebra as a Rota-Baxter
Lie subalgebra is isomorphic to a unified product. Finally, the answer to the
classification part of the extending structure problem is given.

######  Definition 2.1.

Let $\lambda$ be a fixed element of ${\bf k}$. A Rota-Baxter Lie algebra of
weight $\lambda$ is a Lie algebra $\mathfrak{g}$ together with a linear map
$P_{\mathfrak{g}}$ such that

$\displaystyle P[g,h]=P([P(g),h]+[g,P(h)]+\lambda[g,h]),\,\,\forall
g,h\in\mathfrak{g}.$

Denote it by $(\mathfrak{g},[\cdot,\cdot],P_{\mathfrak{g}})$ or
$(\mathfrak{g},P_{\mathfrak{g}})$.

Throughout the rest of this paper, we say $(\mathfrak{g},P_{\mathfrak{g}})$ is
a Rota-Baxter Lie algebra to mean that $(\mathfrak{g},P_{\mathfrak{g}})$ is a
Rota-Baxter Lie algebra of weight $\lambda$.

######  Definition 2.2.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ and
$(\mathfrak{g}^{\prime},P_{\mathfrak{g}^{\prime}})$ be two Rota-Baxter Lie
algebras. A morphism of Rota-Baxter Lie algebras
$f:(\mathfrak{g},P_{\mathfrak{g}})\rightarrow(\mathfrak{g}^{\prime},P_{\mathfrak{g}^{\prime}})$
is a morphism of Lie algebras $f:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$
such that $f\circ P_{\mathfrak{g}}=P_{\mathfrak{g}^{\prime}}\circ f$. It is
said to be an isomorphism if $f$ is a linear isomorphism.

Let $E$ be a vector space and $\mathfrak{g}\subset E$ a subspace of $E$. A
subspace $V$ of $E$ is a complement of $\mathfrak{g}$ in $E$ if
$E=V+\mathfrak{g}$ and $V\cap\mathfrak{g}=0$. Note that such a complement is
unique up to isomorphism and its dimension is called the codimension of
$\mathfrak{g}$ in $E$.

For later use, we recall the notion of a Rota-Baxter module over a Rota-Baxter
Lie algebra given in [26].

######  Definition 2.3.

Let $\mathfrak{g}$ be a Lie algebra. A left $\mathfrak{g}$-module is a vector
space $V$ together with a linear map $\triangleright:\mathfrak{g}\otimes
V\rightarrow V$, called the left action of $\mathfrak{g}$ on $V$, such that

$\displaystyle[g,h]\triangleright x=g\triangleright(h\triangleright
x)-h\triangleright(g\triangleright x),$

for all $x\in V$ and $g,h\in\mathfrak{g}$.

The notion of right $\mathfrak{g}$-modules can be defined similarly. Note that
any right $\mathfrak{g}$-module is a left $\mathfrak{g}$-module via
$g\triangleright x:=-x\triangleleft g$ and viceversa. Hence the category of
right $\mathfrak{g}$-modules is isomorphic to the category of left
$\mathfrak{g}$-modules.

######  Definition 2.4.

[26, Definition 3.6] Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter
Lie algebra. A left $(\mathfrak{g},P_{\mathfrak{g}})$-module is a pair
$(V,T)$, where $V$ is a vector space and $T:V\rightarrow V$ is a linear map,
together with a linear map $\triangleright:\mathfrak{g}\otimes V\rightarrow V$
such that $V$ is a left $\mathfrak{g}$-module under the map $\triangleright$
and for $g\in\mathfrak{g}$ and $v\in V$,

$\displaystyle P_{\mathfrak{g}}(g)\triangleright
T(v)=T(P_{\mathfrak{g}}(g)\triangleright v+g\triangleright T(v)+\lambda
g\triangleright v).$

The notion of right $(\mathfrak{g},P_{\mathfrak{g}})$-modules can be defined
similarly.

######  Definition 2.5.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra and let
$(E,P_{E})$ be a pair where $E$ is a vector space and $P_{E}:E\rightarrow E$
is a linear map. Suppose $\mathfrak{g}$ is a subspace of $E$ and $V$ is a
complement of $\mathfrak{g}$ in $E$. For a linear map $\phi:E\rightarrow E$
such that $\phi\circ P_{E}=P_{E}\circ\phi$, we consider the diagram:

where $\pi:E\rightarrow V$ is the canonical projection of $E=\mathfrak{g}+V$
on $V$ and $i:\mathfrak{g}\rightarrow E$ is the inclusion map such that
$i\circ P_{\mathfrak{g}}=P_{E}\circ i$. We say that $\phi:E\rightarrow E$
stabilizes $(\mathfrak{g},P_{\mathfrak{g}})$ (resp. co-stabilizes $V$ ) if the
left square (resp. the right square) of the above diagram is commutative.

######  Definition 2.6.

Suppose that $(E,[\cdot,\cdot],P)$ and $(E,[\cdot,\cdot]^{\prime},P^{\prime})$
are two Rota-Baxter Lie algebras on $E$ both containing
$(\mathfrak{g},P_{\mathfrak{g}})$ as a Rota-Baxter Lie subalgebra.

  1. (a)

$(E,[\cdot,\cdot],P)$ and $(E,[\cdot,\cdot]^{\prime},P^{\prime})$ are called
equivalent, denoted by
$(E,[\cdot,\cdot],P)\equiv(E,[\cdot,\cdot]^{\prime},P^{\prime})$, if there
exists a Rota-Baxter Lie algebra isomorphism
$\phi:(E,[\cdot,\cdot],P)\rightarrow(E,[\cdot,\cdot]^{\prime},P^{\prime})$
which stabilizes $(\mathfrak{g},P_{\mathfrak{g}})$.

  2. (b)

$(E,[\cdot,\cdot],P)$ and $(E,[\cdot,\cdot]^{\prime},P^{\prime})$ are called
cohomologous, denoted by
$(E,[\cdot,\cdot],P)\approx(E,[\cdot,\cdot]^{\prime},\\\ P^{\prime})$, if
there exists a Rota-Baxter Lie algebra isomorphism
$\phi:(E,[\cdot,\cdot],P)\rightarrow(E,[\cdot,\cdot]^{\prime},P^{\prime})$
which stabilizes $(\mathfrak{g},P_{\mathfrak{g}})$ and co-stabilizes $V$.

######  Remark 2.7.

  1. (a)

Note that $\equiv$ and $\approx$ are equivalence relations on the set of all
Rota-Baxter Lie algebras structures on $E$ containing
$(\mathfrak{g},P_{\mathfrak{g}})$ as a Rota-Baxter Lie subalgebra and we
denote by $Extd(E,(\mathfrak{g},P_{\mathfrak{g}}))$ (resp.
$Extd^{\prime}(E,(\mathfrak{g},P_{\mathfrak{g}}))$) the set of all equivalence
classes via $\equiv$ (resp. $\approx$).

  2. (b)

By Definition 2.6, $Extd(E,(\mathfrak{g},P_{\mathfrak{g}}))$ gives the set of
all isomorphism classes of Rota-Baxter Lie algebras on $E$ that stabilizes
$(\mathfrak{g},P_{\mathfrak{g}})$ and
$Extd^{\prime}(E,(\mathfrak{g},P_{\mathfrak{g}}))$ gives the set of all
isomorphism classes of Rota-Baxter Lie algebras on $E$ from the point of
extension theory.

  3. (c)

Since any two cohomologous Rota-Baxter Lie algebras structures on $E$ are
equivalent, there is a canonical projection

$\displaystyle Extd^{\prime}(E,(\mathfrak{g},P_{\mathfrak{g}}))\rightarrow
Extd(E,(\mathfrak{g},P_{\mathfrak{g}})).$

Generalizing the extending datum of Lie algebras through vector spaces [3], we
give the following notion.

######  Definition 2.8.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra and let $V$
be a vector space.

  1. (a)

An extending datum of $(\mathfrak{g},P_{\mathfrak{g}})$ through $V$ is a
system

$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$

consisting of four bilinear maps

$\displaystyle\triangleleft:V\times\mathfrak{g}\rightarrow
V,\,\triangleright:V\times\mathfrak{g}\rightarrow\mathfrak{g},\,f:V\times
V\rightarrow\mathfrak{g},\,\\{-,-\\}:V\times V\rightarrow V$

and two linear maps

$\displaystyle P_{1}:V\rightarrow\mathfrak{g},\,P_{2}:V\rightarrow V.$

  2. (b)

Let
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
be an extending datum of $(\mathfrak{g},P_{\mathfrak{g}})$ through $V$. Denote
by $\mathfrak{g}\natural V$ the vector space $\mathfrak{g}\times V$ with a
bilinear map $[-,-]:(\mathfrak{g}\times V)\times(\mathfrak{g}\times
V)\rightarrow\mathfrak{g}\times V$ defined by

(3) $\displaystyle[(g,x),(h,y)]:=([g,h]+x\triangleright h-y\triangleright
g+f(x,y),\\{x,y\\}+x\triangleleft h-y\triangleleft g)$

and a linear map $\mathcal{P}:\mathfrak{g}\times
V\rightarrow\mathfrak{g}\times V$ defined by

(4)
$\displaystyle\mathcal{P}((g,x)):=(P_{\mathfrak{g}}(g)+P_{1}(x),P_{2}(x)),$

for all $g,h\in\mathfrak{g}$ and $x,y\in V$. The object $\mathfrak{g}\natural
V$ is called a unified product of $(\mathfrak{g},P_{\mathfrak{g}})$ and $V$ if
it is a Rota-Baxter Lie algebra with the maps given by Eqs. (3) and (4).

  3. (c)

The extending datum
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\
\\},P_{1},P_{2})$ is called a Rota-Baxter Lie extending structure of
$(\mathfrak{g},P_{\mathfrak{g}})$ through $V$ if $\mathfrak{g}\natural V$ is a
unified product of $(\mathfrak{g},P_{\mathfrak{g}})$ and $V$.

  4. (d)

Denote by $\mathcal{L}((\mathfrak{g},P_{\mathfrak{g}}),V)$ the set of all
Rota-Baxter Lie extending structures of $(\mathfrak{g},P_{\mathfrak{g}})$
through $V$.

Now we provide an equivalent characterization of the unified product
$\mathfrak{g}\natural V$ of $(\mathfrak{g},P_{\mathfrak{g}})$ and $V$ as
follows.

######  Theorem 2.9.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra. Let $V$ be
a vector space and $\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)$ an extending
datum of $(\mathfrak{g},P_{\mathfrak{g}})$ through $V$. Then the following
statements are equivalent:

  1. (a)

$\mathfrak{g}\natural V$ is a unified product of
$(\mathfrak{g},P_{\mathfrak{g}})$ and $V$;

  2. (b)

For any $g,h\in\mathfrak{g}$ and $x,y,z\in V$,

    1. (i)

$f(x,x)=0,\,\\{x,x\\}=0$;

    2. (ii)

$(V,\triangleleft)$ is a right $\mathfrak{g}$-module;

    3. (iii)

$x\triangleright[g,h]=[x\triangleright g,h]+[g,x\triangleright
h]+(x\triangleleft g)\triangleright h-(x\triangleleft h)\triangleright g$;

    4. (iv)

$\\{x,y\\}\triangleleft g=\\{x,y\triangleleft g\\}+\\{x\triangleleft
g,y\\}+x\triangleleft(y\triangleright g)-y\triangleleft(x\triangleright g)$;

    5. (v)

$\\{x,y\\}\triangleright g=x\triangleright(y\triangleright
g)-y\triangleright(x\triangleright g)+[g,f(x,y)]+f(x,y\triangleleft
g)+f(x\triangleleft g,y)$;

    6. (vi)

$f(x,\\{y,z\\})+f(y,\\{z,x\\})+f(z,\\{x,y\\})+x\triangleright
f(y,z)+y\triangleright f(z,x)+z\triangleright f(x,y)=0$;

    7. (vii)

$\\{x,\\{y,z\\}\\}+\\{y,\\{z,x\\}\\}+\\{z,\\{x,y\\}\\}+x\triangleleft
f(y,z)+y\triangleleft f(z,x)+z\triangleleft f(x,y)=0$;

    8. (viii)

$[P_{\mathfrak{g}}(g),P_{1}(y)]-P_{2}(y)\triangleright
P_{\mathfrak{g}}(g)+P_{\mathfrak{g}}(y\triangleright
P_{\mathfrak{g}}(g))+P_{1}(y\triangleleft P_{\mathfrak{g}}(g))\\\
-P_{\mathfrak{g}}([\mathfrak{g},P_{1}(y)])+P_{\mathfrak{g}}(P_{2}(y)\triangleright
g)+P_{1}(P_{2}(y)\triangleleft g)+\lambda P_{\mathfrak{g}}(y\triangleright
g)\\\ +\lambda P_{1}(y\triangleleft g)=0$;

    9. (ix)

$P_{2}(y)\triangleleft P_{\mathfrak{g}}(g)-P_{2}(y\triangleleft
P_{\mathfrak{g}}(g))-P_{2}(P_{2}(y)\triangleleft g)-\lambda
P_{2}(y\triangleleft g)=0$;

    10. (x)

$[P_{1}(x),P_{1}(y)]+P_{2}(x)\triangleright P_{1}(y)-P_{2}(y)\triangleright
P_{1}(x)+f(P_{2}(x),P_{2}(y))\\\ +P_{\mathfrak{g}}(y\triangleright
P_{1}(x))-P_{\mathfrak{g}}(f(P_{2}(x),y))-P_{1}(\\{P_{2}(x),y\\})+P_{1}(y\triangleleft
P_{1}(x))\\\ -P_{\mathfrak{g}}(x\triangleright
P_{1}(y))-P_{\mathfrak{g}}(f(x,P_{2}(y)))-P_{1}(\\{x,P_{2}(y)\\})-P_{1}(x\triangleleft
P_{1}(y))\\\ -\lambda P_{\mathfrak{g}}(f(x,y))-\lambda P_{1}(\\{x,y\\})=0$;

    11. (xi)

$\\{P_{2}(x),P_{2}(y)\\}+P_{2}(x)\triangleleft P_{1}(y)-P_{2}(y)\triangleleft
P_{1}(x)-P_{2}(\\{P_{2}(x),y\\})\\\ +P_{2}(y\triangleleft
P_{1}(x))-P_{2}(\\{x,P_{2}(y)\\})-P_{2}(x\triangleleft P_{1}(y))-\lambda
P_{2}(\\{x,y\\})=0$.

###### Proof.

By Theorem 3.2 of [3], $\mathfrak{g}\natural V$ is a Lie algebra under the
product given by (3) if and only if Eqs. (b)(i)-(b)(vii) hold. Next we show
that the operator $\mathcal{P}$ given by Eq. (4) is a Rota-Baxter operator if
and only if Eqs. (b)(viii)-(b)(xi) hold. Since $(g,x)=(g,0)+(0,x)$ in
$\mathfrak{g}\natural V$ and $\mathcal{P}$ given by Eq. (4) is a linear map,
we only need to consider the action of $\mathcal{P}$ on all generators of
$\mathfrak{g}\natural V$, i.e. the set
$\\{(g,0)|g\in\mathfrak{g}\\}\cup\\{(0,x)|x\in V\\}$. As $[-,-]$ is
antisymmetric, there are three steps to consider. First, $\mathcal{P}$ is a
Rota-Baxter operator for any $(g,0),(h,0)$:

$\displaystyle\
[\mathcal{P}((g,0)),\mathcal{P}((h,0))]-\mathcal{P}\Big{(}[\mathcal{P}(g,0),(h,0)]+[(g,0),\mathcal{P}(h,0)]+\lambda[(g,0),(h,0)]\Big{)}$
$\displaystyle=$ $\displaystyle\
[(P_{\mathfrak{g}}(g),0),(P_{\mathfrak{g}}(h),0)]-\mathcal{P}\Big{(}[(P_{\mathfrak{g}}(g),0),(h,0)]+[(g,0),(P_{\mathfrak{g}}(h),0)]+\lambda[(g,0),(h,0)]\Big{)}$
$\displaystyle=$ $\displaystyle\
([P_{\mathfrak{g}}(g),P_{\mathfrak{g}}(h)],0)-\Big{(}P_{\mathfrak{g}}([P_{\mathfrak{g}}(g),h]+[g,P_{\mathfrak{g}}(h)]+\lambda[g,h]),0\Big{)}\quad(\text{by
Eqs.~{}(\ref{eq:uniproduct})-(\ref{eq:unilie})})$ $\displaystyle=$
$\displaystyle\ 0\quad\text{(As $P_{\mathfrak{g}}$ being a Rota-Baxter
operator of $\mathfrak{g}$)}.$

Next we prove that $\mathcal{P}$ is a Rota-Baxter operator for any
$(g,0),(0,y)$ if and only if Eqs. ((b)(viii))-((b)(ix)) hold. Indeed,

$\displaystyle\
[\mathcal{P}((g,0)),\mathcal{P}((0,y))]-\mathcal{P}\Big{(}[\mathcal{P}(g,0),(0,y)]+[(g,0),\mathcal{P}(0,y)]+\lambda[(g,0),(0,y)]\Big{)}$
$\displaystyle=$ $\displaystyle\
[(P_{\mathfrak{g}}(g),0),(P_{1}(y),P_{2}(y))]-\mathcal{P}\Big{(}[(P_{\mathfrak{g}}(g),0),(0,y)]+[(g,0),(P_{1}(y),P_{2}(y))]+\lambda[(g,0),(0,y)]\Big{)}$
$\displaystyle\hskip 312.9803pt(\text{by Eq.~{}(\ref{eq:unilie})})$
$\displaystyle=$ $\displaystyle\
\bigg{(}[P_{\mathfrak{g}}(g),P_{1}(y)]-P_{2}(y)\triangleright
P_{\mathfrak{g}}(y),-P_{2}(y)\triangleleft
P_{\mathfrak{g}}(g)\bigg{)}-\mathcal{P}\bigg{(}\Big{(}-y\triangleright
P_{\mathfrak{g}}(g),-y\triangleleft P_{\mathfrak{g}}(g)\Big{)}$
$\displaystyle\ +\Big{(}[g,P_{1}(y)]-P_{2}(y)\triangleright
g,-P_{2}(y)\triangleleft g\Big{)}+\lambda\Big{(}-y\triangleright
g,-y\triangleleft g\Big{)}\bigg{)}\quad(\text{by
Eq.~{}(\ref{eq:uniproduct})})$ $\displaystyle=$ $\displaystyle\
\bigg{(}[P_{\mathfrak{g}}(g),P_{1}(y)]-P_{2}(y)\triangleright
P_{\mathfrak{g}}(y),-P_{2}(y)\triangleleft P_{\mathfrak{g}}(g)\bigg{)}$
$\displaystyle\ -\bigg{(}P_{\mathfrak{g}}(-y\triangleright
P_{\mathfrak{g}}(g))+P_{1}(-y\triangleleft
P_{\mathfrak{g}}(g)),P_{2}(-y\triangleleft P_{\mathfrak{g}}(g))\bigg{)}$
$\displaystyle\
-\bigg{(}P_{\mathfrak{g}}([g,P_{1}(y)])-P_{\mathfrak{g}}(P_{2}(y)\triangleright
g)+P_{1}(-P_{2}(y)\triangleleft g),P_{2}(-P_{2}(y)\triangleleft g)\bigg{)}$
$\displaystyle\ -\lambda\Big{(}P_{\mathfrak{g}}(-y\triangleright
g)+P_{1}(-y\triangleleft g),P_{2}(-y\triangleleft g)\Big{)}\quad(\text{by
Eq.~{}(\ref{eq:unilie})})$ $\displaystyle=$ $\displaystyle\
\bigg{(}[P_{\mathfrak{g}}(g),P_{1}(y)]-P_{2}(y)\triangleright
P_{\mathfrak{g}}(g)+P_{\mathfrak{g}}(y\triangleright
P_{\mathfrak{g}}(g))+P_{1}(y\triangleleft
P_{\mathfrak{g}}(g))-P_{\mathfrak{g}}([g,P_{1}(y)])$ $\displaystyle\
+P_{\mathfrak{g}}(P_{2}(y)\triangleright g)+P_{1}(P_{2}(y)\triangleleft
g)+\lambda P_{\mathfrak{g}}(y\triangleright g)+\lambda P_{1}(y\triangleleft
g),$ $\displaystyle\ P_{2}(y)\triangleleft
P_{\mathfrak{g}}(g)-P_{2}(y\triangleleft
P_{\mathfrak{g}}(g))-P_{2}(P_{2}(y)\triangleleft g)-\lambda
P_{2}(y\triangleleft g)\bigg{)}.$

Hence $\mathcal{P}$ is a Rota-Baxter operator for $(g,0),(0,y)$ if and only if
Eqs. (b)(viii)-(b)(ix) hold. Finally, we prove that $\mathcal{P}$ is a Rota-
Baxter operator for $(0,x),(0,y)$ if and only if Eqs. (b)(x)-(b)(xi) hold.
Indeed,

$\displaystyle\
[\mathcal{P}((0,x)),\mathcal{P}((0,y))]-\mathcal{P}\bigg{(}[\mathcal{P}((0,x)),(0,y)]+[(0,x),\mathcal{P}((0,y))]+\lambda[(0,x),(0,y)]\bigg{)}$
$\displaystyle=$ $\displaystyle\
\left[\Big{(}P_{1}(x),P_{2}(x)\Big{)},\Big{(}P_{1}(y),P_{2}(y)\Big{)}\right]-\mathcal{P}\bigg{(}\left[\Big{(}P_{1}(x),P_{2}(x)\Big{)},(0,y)\right]+\left[(0,x),\Big{(}P_{1}(y),P_{2}(y)\Big{)}\right]$
$\displaystyle\ +\lambda[(0,x),(0,y)]\bigg{)}$ $\displaystyle=$
$\displaystyle\ \bigg{(}[P_{1}(x),P_{1}(y)]+P_{2}(x)\triangleright
P_{1}(y)-P_{2}(y)\triangleright
P_{1}(x)+f\Big{(}P_{2}(x),P_{2}(y)\Big{)},\\{P_{2}(x),P_{2}(y)\\}$
$\displaystyle\ +P_{2}(x)\triangleleft P_{1}(y)-P_{2}(y)\triangleleft
P_{1}(x)\bigg{)}-\mathcal{P}\bigg{(}\Big{(}-y\triangleright
P_{1}(x)+f(P_{2}(x),y),\\{P_{2}(x),y\\}-y\triangleleft P_{1}(x)\Big{)}$
$\displaystyle\ +\Big{(}x\triangleright
P_{1}(y)+f(x,P_{2}(y)),\\{x,P_{2}(y)\\}+x\triangleleft
P_{1}(y)\Big{)}+\lambda\Big{(}f(x,y),\\{x,y\\}\Big{)}\bigg{)}$
$\displaystyle=$ $\displaystyle\
\bigg{(}[P_{1}(x),P_{1}(y)]+P_{2}(x)\triangleright
P_{1}(y)-P_{2}(y)\triangleright
P_{1}(x)+f(P_{2}(x),P_{2}(y)),\\{P_{2}(x),P_{2}(y)\\}$ $\displaystyle\
+P_{2}(x)\triangleleft P_{1}(y)-P_{2}(y)\triangleleft
P_{1}(x)\bigg{)}-\bigg{(}P_{\mathfrak{g}}(-y\triangleright
P_{1}(x))+P_{\mathfrak{g}}(f(P_{2}(x),y))+P_{1}(\\{P_{2}(x),y\\})$
$\displaystyle\ -P_{1}(y\triangleleft
P_{1}(x)),P_{2}(\\{P_{2}(x),y\\})-P_{2}(y\triangleleft
P_{1}(x))\bigg{)}-\bigg{(}P_{\mathfrak{g}}(x\triangleright
P_{1}(y))+P_{\mathfrak{g}}(f(x,P_{2}(y)))$ $\displaystyle\
+P_{1}(\\{x,P_{2}(y)\\})+P_{1}(x\triangleleft
P_{1}(y)),P_{2}(\\{x,P_{2}(y)\\})+P_{2}(x\triangleleft P_{1}(y))\bigg{)}$
$\displaystyle\
-\lambda\bigg{(}P_{\mathfrak{g}}(f(x,y))+P_{1}(\\{x,y\\}),P_{2}(\\{x,y\\})\bigg{)}$
$\displaystyle=$ $\displaystyle\
\bigg{(}[P_{1}(x),P_{1}(y)]+P_{2}(x)\triangleright
P_{1}(y)-P_{2}(y)\triangleright
P_{1}(x)+f(P_{2}(x),P_{2}(y))+P_{\mathfrak{g}}(y\triangleright P_{1}(x))$
$\displaystyle\
-P_{\mathfrak{g}}(f(P_{2}(x),y))-P_{1}(\\{P_{2}(x),y\\})+P_{1}(y\triangleleft
P_{1}(x))-P_{\mathfrak{g}}(x\triangleright
P_{1}(y))-P_{\mathfrak{g}}(f(x,P_{2}(y)))$ $\displaystyle\
-P_{1}(\\{x,P_{2}(y)\\})-P_{1}(x\triangleleft P_{1}(y))-\lambda
P_{\mathfrak{g}}(f(x,y))-\lambda
P_{1}(\\{x,y\\}),\\{P_{2}(x),P_{2}(y)\\}+P_{2}(x)\triangleleft P_{1}(y)$
$\displaystyle\ -P_{2}(y)\triangleleft
P_{1}(x)-P_{2}(\\{P_{2}(x),y\\})+P_{2}(y\triangleleft
P_{1}(x))-P_{2}(\\{x,P_{2}(y)\\})-P_{2}(x\triangleleft P_{1}(y))-\lambda
P_{2}(\\{x,y\\})\bigg{)}.$

Hence $\mathcal{P}$ is a Rota-Baxter operator for $(0,x),(0,y)$ if and only if
Eqs. (b)(x) and (b)(xi) hold. This completes the proof. ∎

######  Remark 2.10.

Note that Eqs. (b)(ii) and (b)(xi) show that $(V,P_{2})$ is a right
$(\mathfrak{g},P_{\mathfrak{g}})$-module under the action $\triangleleft$.

Let
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
be a Rota-Baxter Lie extending structure and $\mathfrak{g}\natural V$ the
associated unified product. Then there is a canonical inclusion

$\displaystyle i_{\mathfrak{g}}:\mathfrak{g}\rightarrow\mathfrak{g}\natural
V,\quad i_{\mathfrak{g}}(g)=(g,0),\text{ \ for any \ }g\in\mathfrak{g}.$

Note that $i_{\mathfrak{g}}\circ P_{\mathfrak{g}}=\mathcal{P}\circ
i_{\mathfrak{g}}$. Then $i_{\mathfrak{g}}$ is an injective morphism of Rota-
Baxter Lie algebras. Hence $(\mathfrak{g},P_{\mathfrak{g}})$ can be viewed as
a Rota-Baxter Lie subalgebra of $\mathfrak{g}\natural V$ through the
identification $\mathfrak{g}\cong
i_{\mathfrak{g}}(\mathfrak{g})\cong\mathfrak{g}\times\\{0\\}$. Next we will
prove that any Rota-Baxter Lie algebra structure on a vector space $E$
containing $(\mathfrak{g},P_{\mathfrak{g}})$ as a Rota-Baxter Lie subalgebra
is isomorphic to a unified product of $(\mathfrak{g},P_{\mathfrak{g}})$
through a vector space.

Now we arrive at our first main result in this section.

######  Theorem 2.11.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra. Let $E$ be
a vector space containing $\mathfrak{g}$ as a subspace and $(E,[-,-],P_{E})$ a
Rota-Baxter Lie algebra such that $(\mathfrak{g},P_{\mathfrak{g}})$ is a Rota-
Baxter Lie subalgebra of $(E,[-,-],P_{E})$. Then there exists a Rota-Baxter
Lie extending structure

$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$

of $(\mathfrak{g},P_{\mathfrak{g}})$ through a subspace $V$ of $E$ and an
isomorphism of Rota-Baxter Lie algebras

$(E,P_{E})\cong(\mathfrak{g}\natural V,\mathcal{P})$

that stabilizes $(\mathfrak{g},P_{\mathfrak{g}})$ and co-stabilizes $V$.

###### Proof.

Since ${\bf k}$ is a field, we have a linear map $p:E\rightarrow\mathfrak{g}$
such that $p(g)=g$, for all $g\in\mathfrak{g}$. Define $V:=\mathrm{ker}(p)$.
Then $V$ is a complement of $\mathfrak{g}$ in $E$, $p\circ p=p$ and
$p\circ(id_{E}-p)=0$. Now define the extending datum of
$(\mathfrak{g},P_{\mathfrak{g}})$ through $V$ as follows:

$\displaystyle\triangleright:$ $\displaystyle\
V\times\mathfrak{g}\rightarrow\mathfrak{g},\quad x\triangleright g:=p([x,g]),$
$\displaystyle\triangleleft:$ $\displaystyle\ V\times\mathfrak{g}\rightarrow
V,\quad x\triangleleft g:=[x,g]-p([x,g]),$ $\displaystyle f:$ $\displaystyle\
V\times V\rightarrow\mathfrak{g},\quad f(x,y):=p([x,y]),$
$\displaystyle\\{,\\}:$ $\displaystyle\ V\times V\rightarrow
V,\quad\\{x,y\\}:=[x,y]-p([x,y]),$ $\displaystyle P_{1}:$ $\displaystyle\
V\rightarrow\mathfrak{g},\quad P_{1}(x)=p(P_{E}(x)),$ $\displaystyle P_{2}:$
$\displaystyle\ V\rightarrow V,\quad P_{2}(x)=P_{E}(x)-p(P_{E}(x)),$

for any $g\in\mathfrak{g}$ and $x,y\in V$. Note that by definition of the map
$p$, for any $g\in\mathfrak{g}$ and $x,y\in V$, $x\triangleleft g\in V$,
$\\{x,y\\}\in V$ and $P_{2}(x)\in V$. Hence the above maps are all well-
defined linear maps or bilinear maps. Now we prove that
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
is a Rota-Baxter Lie extending structure of $(\mathfrak{g},P_{\mathfrak{g}})$
through $V$ and

$\displaystyle\phi:\mathfrak{g}\natural V\rightarrow E,\quad\phi(g,x):=g+x$

is an isomorphism of Rota-Baxter Lie algebras that stabilizes $\mathfrak{g}$
and co-stabilizes $V$. Analogue to the proof of Theorem 3.4 in [3],

$\phi:\mathfrak{g}\times V\rightarrow E,\quad\phi(g,x):=g+x$

is a linear isomorphism between the vector space $\mathfrak{g}\times V$ and
the Lie algebra $E$ with the inverse given by

$\displaystyle\phi^{-1}(y):=(p(y),y-p(y)),\text{ \ for all \ }y\in E.$

As $\phi$ is a linear isomorphism, there exists a unique Rota-Baxter Lie
algebra structure on $\mathfrak{g}\times V$ with the Rota-Baxter operator and
the product on $\mathfrak{g}\times V$ given by

(5)
$\displaystyle\mathcal{P}((g,x)):=\phi^{-1}(P_{E}(\phi(g,x))),\quad[(g,x),(h,y)]:=\phi^{-1}([\phi(g,x),\phi(h,y)]),$

for all $g,h\in\mathfrak{g}$ and $x,y\in V$, such that $\phi$ is an
isomorphism of Rota-Baxter Lie algebras. In fact, by the proof of Theorem 3.4
in [3], the product in Eq. (5) coincides with the one defined by Eq. (3).
Hence, it remains to prove that the Rota-Baxter operator in Eq. (5) coincides
with the one defined by Eq. (4). Indeed, for any $g\in\mathfrak{g}$ and $x\in
V$, we have

$\displaystyle\
\mathcal{P}((g,x))=\phi^{-1}(P_{E}(\phi(g,x)))=\phi^{-1}(P_{E}(g+x))$
$\displaystyle=$ $\displaystyle\
\phi^{-1}(P_{E}(g)+P_{E}(x))=\phi^{-1}(P_{\mathfrak{g}}(g)+P_{E}(x))$ (by
$(\mathfrak{g},P_{\mathfrak{g}})$ being a Rota-Baxter Lie subalgebra of
$(E,P_{E})$) $\displaystyle=$ $\displaystyle\
\phi^{-1}(P_{\mathfrak{g}}(g)+p(P_{E}(x))+P_{E}(x)-p(P_{E}(x)))$
$\displaystyle=$ $\displaystyle\
(P_{\mathfrak{g}}(g)+p(P_{E}(x)),P_{E}(x)-p(P_{E}(x)))\quad\text{(by $p\circ
p=p$ and $p\circ(id_{E}-p)=0$)}$ $\displaystyle=$ $\displaystyle\
(P_{\mathfrak{g}}(x)+P_{1}(x),P_{2}(x)).$

Finally, we can check the following diagram is commutative

where $\pi:E\rightarrow V$ is the projection of $E=\mathfrak{g}+V$ on the
vector space $V$ and $q:\mathfrak{g}\natural V\rightarrow V,q(g,x):=x$ is the
canonical projection. Hence $\phi$ stabilizes
$(\mathfrak{g},P_{\mathfrak{g}})$ and $V$. This completes the proof. ∎

######  Remark 2.12.

By Theorem 2.11, any Rota-Baxter Lie algebra structure on a vector space $E$
containing $(\mathfrak{g},P_{\mathfrak{g}})$ as a Rota-Baxter Lie subalgebra
is isomorphic to a unified product of $(\mathfrak{g},P_{\mathfrak{g}})$
through a given complement $V$ of $\mathfrak{g}$ in $E$. Hence the
classification of all Rota-Baxter Lie algebra structures on $E$ that contains
$(\mathfrak{g},P_{\mathfrak{g}})$ as a Rota-Baxter Lie subalgebras is
equivalent to the classification of all unified products $\mathfrak{g}\natural
V$, associated to all Rota-Baxter Lie extending structures
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$,
for a given complement $V$ of $\mathfrak{g}$ in $E$.

To classify all unified products $\mathfrak{g}\natural V$, we need the
following lemma.

######  Lemma 2.13.

Let
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$,
$\Omega^{\prime}((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft^{\prime},\triangleright^{\prime},f^{\prime},\\{-,-\\}^{\prime},P^{\prime}_{1},P^{\prime}_{2})$
be two Rota-Baxter Lie extending structures of
$(\mathfrak{g},P_{\mathfrak{g}})$ through $V$ and let $\mathfrak{g}\natural
V,\mathfrak{g}\natural^{\prime}V$ be their corresponding associated unified
products. Then there exists a bijection between the set of all morphisms of
Rota-Baxter Lie algebras $\psi:\mathfrak{g}\natural
V\rightarrow\mathfrak{g}\natural^{\prime}V$ which stabilizes
$(\mathfrak{g},P_{\mathfrak{g}})$, and the set of pairs $(r,v)$, where
$r:V\rightarrow\mathfrak{g}$, $v:V\rightarrow V$ are two linear maps
satisfying the following conditions: for any $g\in\mathfrak{g}$, $x,y\in V$,

  1. (a)

$v(x)\triangleleft^{\prime}g=v(x\triangleleft g)$;

  2. (b)

$r(x\triangleleft g)=[r(x),g]-x\triangleright g+v(x)\triangleright^{\prime}g$;

  3. (c)

$v(\\{x,y\\})=\\{v(x),v(y)\\}^{\prime}+v(x)\triangleleft^{\prime}r(y)-v(y)\triangleleft^{\prime}r(x)$;

  4. (d)

$r(\\{x,y\\})=[r(x),r(y)]+v(x)\triangleright^{\prime}r(y)-v(y)\triangleright^{\prime}r(x)+f^{\prime}(v(x),v(y))-f(x,y)$;

  5. (e)

$P_{1}(x)=P_{\mathfrak{g}}(r(x))+P^{\prime}_{1}(v(x))-r(P_{2}(x))$;

  6. (f)

$v(P_{2}(x))=P^{\prime}_{2}(v(x))$.

Under the above bijection, the morphism of Rota-Baxter Lie algebras
$\psi=\psi_{(r,v)}:\mathfrak{g}\natural
V\rightarrow\mathfrak{g}\natural^{\prime}V$ corresponding to $(r,v)$ is given
by

$\displaystyle\psi(g,x)=(g+r(x),v(x)),$

for any $g\in\mathfrak{g}$ and $x\in V$. Moreover, $\psi=\psi_{(r,v)}$ is an
isomorphism of Rota-Baxter Lie algebras if and only if $v:V\rightarrow V$ is a
bijection and $\psi=\psi_{(r,v)}$ co-stabilizes $V$ if and only if $v=id_{V}$.

###### Proof.

By the proof of Lemma 3.5 in [3], a linear map $\psi:\mathfrak{g}\natural
V\rightarrow\mathfrak{g}\natural^{\prime}V$ which stabilizes $\mathfrak{g}$ is
uniquely determined by two linear maps $r:V\rightarrow\mathfrak{g}$ and
$v:V\rightarrow V$ such that $\psi(g,x)=(g+r(x),v(x))$ for all
$g\in\mathfrak{g}$ and $x\in V$. Moreover, $\psi$ is a morphism of Lie
algebras if and only if the conditions (a)-(d) hold. Denote by $\mathcal{P}$
and $\mathcal{P}^{\prime}$ the Rota-Baxter operators associated to
$\mathfrak{g}\natural V$ and $\mathfrak{g}\natural^{\prime}V$. Next, we show
that $\psi$ is compatible with the Rota-Baxter operators, i.e.
$\psi\circ\mathcal{P}=\mathcal{P}^{\prime}\circ\psi$, if and only if the
conditions (e)-(f) hold. For $g,h\in\mathfrak{g}$, we have

$\displaystyle\psi\circ\mathcal{P}((g,0))=\psi((P_{\mathfrak{g}}(g),0))=(P_{\mathfrak{g}}(g),0)=\mathcal{P}^{\prime}((g,0))=\mathcal{P}^{\prime}\circ\psi((g,0)).$

$\psi$ is compatible with the Rota-Baxter operators under this case. For $x\in
V$, we have

$\displaystyle\
\psi\circ\mathcal{P}((0,x))-\mathcal{P}^{\prime}\circ\psi((0,x))=\psi((P_{1}(x),P_{2}(x)))-\mathcal{P}^{\prime}((r(x),v(x)))$
$\displaystyle=$ $\displaystyle\
(P_{1}(x)+r(P_{2}(x)),v(P_{2}(x)))-(P_{\mathfrak{g}}(r(x))+P^{\prime}_{1}(v(x)),P^{\prime}_{2}(v(x))).$

Hence $\psi\circ\mathcal{P}((0,x))=\mathcal{P}^{\prime}\circ\psi((0,x))$ if
and only if the conditions (e)-(f) hold.

Moreover, by the proof of Lemma 3.5 in [3], $\psi=\psi_{(r,v)}$ is an
isomorphism if and only if $v:V\rightarrow V$ is a bijection and
$\psi=\psi_{(r,v)}$ co-stabilizes $V$ if and only if $v=id_{V}$. ∎

By Lemma 2.13, we give the following definition.

######  Definition 2.14.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra and let $V$
be a vector space. Two Rota-Baxter Lie extending structures of
$(\mathfrak{g},P_{\mathfrak{g}})$ through $V$,
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
and
$\Omega^{\prime}((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft^{\prime},\triangleright^{\prime},f^{\prime},\\{-.-\\}^{\prime},P^{\prime}_{1},P^{\prime}_{2})$
are called equivalent, denoted by

$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)\equiv\Omega^{\prime}((\mathfrak{g},P_{\mathfrak{g}}),V),$

if there exists a pair $(r,v)$ of linear maps, where
$r:V\rightarrow\mathfrak{g}$ and $v\in Aut_{{\bf k}}(V)$ such that

$(\triangleleft^{\prime},\triangleright^{\prime},f^{\prime},\\{-,-\\}^{\prime},P^{\prime}_{1},P^{\prime}_{2})$

is obtained from $(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
using $(r,v)$ via

$\displaystyle x\triangleleft^{\prime}g=$ $\displaystyle\
v(v^{-1}(x)\triangleleft g),$ $\displaystyle x\triangleright^{\prime}g=$
$\displaystyle\ r(v^{-1}(x)\triangleleft g)+v^{-1}(x)\triangleright
g+[g,r(v^{-1}(x))],$ $\displaystyle f^{\prime}(x,y)=$ $\displaystyle\
f(v^{-1}(x),v^{-1}(y))+r(\\{v^{-1}(x),v^{-1}(y)\\})+[r(v^{-1}(x)),r(v^{-1}(y))]-r(v^{-1}(x)\triangleleft
r(v^{-1}(y)))$ $\displaystyle\ -v^{-1}(x)\triangleright
r(v^{-1}(y))+r(v^{-1}(y)\triangleleft r(v^{-1}(x)))+v^{-1}(y)\triangleright
r(v^{-1}(x)),$ $\displaystyle\\{x,y\\}^{\prime}=$ $\displaystyle\
v(\\{v^{-1}(x),v^{-1}(y)\\})-v(v^{-1}(x)\triangleleft
r(v^{-1}(y)))+v(v^{-1}(y)\triangleleft r(v^{-1}(x))),$ $\displaystyle
P^{\prime}_{1}(x)=$ $\displaystyle\
P_{1}(v^{-1}(x))-P_{\mathfrak{g}}(r(v^{-1}(x)))+r(P_{2}(v^{-1}(x))),$
$\displaystyle P^{\prime}_{2}(x)=$ $\displaystyle\ v(P_{2}(v^{-1}(x))).$

Recall that ${Extd}(E,(\mathfrak{g},P_{\mathfrak{g}}))$ is the set of all
equivalence classes of Rota-Baxter Lie algebra structures on $E$ which
stabilizes $(\mathfrak{g},P_{\mathfrak{g}})$. Now we give a classification of
${Extd}(E,(\mathfrak{g},P_{\mathfrak{g}}))$ as our second main results.

######  Theorem 2.15.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra, $E$ a
vector space that contains $\mathfrak{g}$ as a subspace and $V$ a complement
of $\mathfrak{g}$ in $E$. Then:

  1. (a)

$\equiv$ is an equivalence relation on the set
$\mathcal{L}((\mathfrak{g},P_{\mathfrak{g}}),V)$ of all Rota-Baxter Lie
extending structures of $(\mathfrak{g},P_{\mathfrak{g}})$ through $V$. Denote
by
$H^{2}_{(\mathfrak{g},P_{\mathfrak{g}})}(V,(\mathfrak{g},P_{\mathfrak{g}})):=\mathcal{L}((\mathfrak{g},P_{\mathfrak{g}}),V)/\equiv$,
the quotient set.

  2. (b)

The map

$\displaystyle\Phi:H^{2}_{(\mathfrak{g},P_{\mathfrak{g}})}(V,(\mathfrak{g},P_{\mathfrak{g}}))\rightarrow
Extd(E,(\mathfrak{g},P_{\mathfrak{g}})),\quad\overline{(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})}\mapsto(\mathfrak{g}\natural
V,[-,-],\mathcal{P})$

is a bijection, where
$\overline{(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})}$ is the
equivalence class of $(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
via $\equiv$.

###### Proof.

(a) By Theorems 2.9, 2.11 and Lemma 2.13,
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)\equiv\Omega^{\prime}((\mathfrak{g},P_{\mathfrak{g}}),V)$
in the sense of Definition 2.14 if and only if there exists an isomorphism of
Rota-Baxter Lie algebras $\psi:\mathfrak{g}\natural
V\rightarrow\mathfrak{g}\natural^{\prime}V$ which stabilizes
$(\mathfrak{g},P_{\mathfrak{g}})$. Therefore, $\equiv$ is an equivalence
relation on the set $\mathcal{L}((\mathfrak{g},P_{\mathfrak{g}}),V)$ of all
Rota-Baxter Lie extending structures.

(b) By Lemma 2.13, the map $\Phi$ is a well-defined injection. By Theorem
2.11, $\Phi$ is a surjection. Hence $\Phi$ is a bijection. ∎

######  Remark 2.16.

Let
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
and
$\Omega^{\prime}((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft^{\prime},\triangleright^{\prime},f^{\prime},\\{-.-\\}^{\prime},P^{\prime}_{1},\\\
P^{\prime}_{2})$ be two Rota-Baxter Lie extending structures of
$(\mathfrak{g},P_{\mathfrak{g}})$ through $V$. They are called cohomologous,
denoted by
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)\approx\Omega^{\prime}((\mathfrak{g},P_{\mathfrak{g}}),V)$,
if $\triangleleft^{\prime}=\triangleleft$, $P^{\prime}_{2}=P_{2}$ and there
exists a linear map $r:V\rightarrow\mathfrak{g}$ such that

$\displaystyle x\triangleright^{\prime}g$ $\displaystyle\ =x\triangleright
g+r(x\triangleleft g)-[r(x),g],$ $\displaystyle f^{\prime}(x,y)$
$\displaystyle\ =f(x,y)+r(\\{x,y\\})+[r(x),r(y)]+y\triangleright
r(x)-x\triangleright r(y)+r(y\triangleleft r(x))-r(x\triangleleft r(y)),$
$\displaystyle\\{x,y\\}^{\prime}$ $\displaystyle\ =\\{x,y\\}-x\triangleleft
r(y)+y\triangleleft r(x),$ $\displaystyle P^{\prime}_{1}(x)$ $\displaystyle\
=P_{1}(x)-P_{\mathfrak{g}}(r(x))+r(P_{2}(x)),$

for all $g\in\mathfrak{g}$ and $x,y\in V$.

Similar to the proof of Theorem 2.15, we can conclude that
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)\approx\Omega^{\prime}((\mathfrak{g},P_{\mathfrak{g}}),V)$
if and only if there exists an isomorphism of Rota-Baxter Lie alebras
$\phi:\mathfrak{g}\natural V\rightarrow\mathfrak{g}\natural^{\prime}V$ which
stabilizes $(\mathfrak{g},P_{\mathfrak{g}})$ and co-stabilizes $V$. Then
$\approx$ is also an equivalent relation on the set
$\mathcal{L}((\mathfrak{g},P_{\mathfrak{g}}),V)$ of all Rota-Baxter Lie
extending structures of $(\mathfrak{g},P_{\mathfrak{g}})$ through $V$. Define

$H^{2}(V,(\mathfrak{g},P_{\mathfrak{g}})):=\mathcal{L}((\mathfrak{g},P_{\mathfrak{g}}),V)/\approx,$

then the map

$\displaystyle H^{2}(V,(\mathfrak{g},P_{\mathfrak{g}}))\rightarrow
Extd^{\prime}(E,\mathfrak{g}),\quad\overline{(\triangleleft,\triangleright,f,\\{-,-\\})}\mapsto(\mathfrak{g}\natural
V,[-,-],\mathcal{P})$

is a bijection between $H^{2}(V,(\mathfrak{g},P_{\mathfrak{g}}))$ and the
isomorphism classes of all Rota-Baxter Lie algebras structures on $E$ which
stabilizes $(\mathfrak{g},P_{\mathfrak{g}})$ and co-stabilizes $V$.

##  3\. Special cases of unified products

In the section, we first introduce the notion of crossed products and
bicrossed products of Rota-Baxter Lie algebras. Then we prove that both
products are two special cases of the unified products of Rota-Baxter Lie
algebras.

Throughout the rest of this paper, we shall omit the map if it is one of the
maps $\triangleleft,\triangleright,f,\\{-,-\\},P_{1}$ or $P_{2}$ of an
extending datum
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
and it is trivial.

###  3.1. Crossed products and the extension problem

In this subsection, we introduce the concept of a crossed product of Rota-
Baxter Lie algebras and show that the extension problem studied by Mishra-Das-
Hazra [32] is a special case of the Rota-Baxter Lie extending structures
problem.

Let
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\})$
be an extending datum of $(\mathfrak{g},P_{\mathfrak{g}})$ through $V$ such
that $\triangleleft$ is a trivial map, i.e. $x\triangleleft g=0$, for all
$x\in V$ and $g\in\mathfrak{g}$. It follows from Theorem 2.9 that
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleright,f,\\{-,-\\})$ is a
Rota-Baxter Lie extending structure of $(\mathfrak{g},P_{\mathfrak{g}})$
through $V$ if and only if $(V,\\{-,-\\},P_{2})$ is a Lie algebra that satisfy
the following relations:

$\displaystyle\ f(x,x)=0,$ $\displaystyle\
x\triangleright[g,h]=[x\triangleright g,h]+[g,x\triangleright h],$
$\displaystyle\ \\{x,y\\}\triangleright g=x\triangleright(y\triangleright
g)-y\triangleright(x\triangleright g)+[g,f(x,y)],$ $\displaystyle\
f(x,\\{y,z\\}+f(y,\\{z,x\\})+f(z,\\{x,y\\})+x\triangleright
f(y,z)+y\triangleright f(z,x)+z\triangleright f(x,y)=0,$ $\displaystyle\
[P_{\mathfrak{g}}(g),P_{1}(y)]-P_{2}(y)\triangleright
P_{\mathfrak{g}}(g)+P_{\mathfrak{g}}(y\triangleright
P_{\mathfrak{g}}(g))-P_{\mathfrak{g}}([g,P_{1}(y)])+P_{\mathfrak{g}}(P_{2}(y)\triangleright
g)$ $\displaystyle\ +\lambda P_{\mathfrak{g}}(y\triangleright g)=0,$
$\displaystyle\ [P_{1}(x),P_{1}(y)]+P_{2}(x)\triangleright
P_{1}(y)-P_{2}(y)\triangleright
P_{1}(x)+f(P_{2}(x),P_{2}(y))+P_{\mathfrak{g}}(y\triangleright P_{1}(x))$
$\displaystyle\
-P_{\mathfrak{g}}(f(P_{2}(x),y))-P_{1}(\\{P_{2}(x),y\\})-P_{\mathfrak{g}}(x\triangleright
P_{1}(y))-P_{\mathfrak{g}}(f(x,P_{2}(y)))-P_{1}(\\{x,P_{2}(y)\\})$
$\displaystyle\ -\lambda P_{\mathfrak{g}}(f(x,y))-\lambda P_{1}(\\{x,y\\})=0.$

for any $g,h\in\mathfrak{g}$ and $x,y,z\in V$.

In this case, the associated unified product $\mathfrak{g}\natural V$ is the
crossed product of the Rota-Baxter Lie algebras
$(\mathfrak{g},P_{\mathfrak{g}})$ and $(V,P_{2})$. A system
$((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleright,f,P_{1})$
consisting of two Rota-Baxter Lie algebras
$(\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2})$, two bilinear maps
$\triangleright:V\times\mathfrak{g}\rightarrow\mathfrak{g},f:V\times
V\rightarrow\mathfrak{g}$ and a linear map $P_{1}:V\rightarrow\mathfrak{g}$
satisfying the above conditions is called a crossed system of Rota-Baxter Lie
algebras. The Rota-Baxter Lie algebra $(\mathfrak{g}\natural
V,[-,-],\mathcal{P})$ associated to the crossed system
$((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleright,f,P_{1})$ is
defined as follows: for any $g,h\in\mathfrak{g}$ and $x,y\in V$,

$\displaystyle[(g,x),(h,y)]:=([g,h]+x\triangleright h-y\triangleright
g+f(x,y),\\{x,y\\}),\quad\mathcal{P}((g,x)):=(P_{\mathfrak{g}}(g)+P_{1}(x),P_{2}(x)).$

Note that the crossed system provides the answer to the following restricted
version of the extending structures problem:

######  Problem 3.1.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra, $E$ a
vector space containing $\mathfrak{g}$ as a subspace. Describe and classify
all Rota-Baxter Lie algebra structures on $E$ such that
$(\mathfrak{g},P_{\mathfrak{g}})$ is a Rota-Baxter ideal of $E$.

Let $((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleright,f,P_{1})$ be a
crossed system of two Rota-Baxter Lie algebras. Since

$\displaystyle[(g,0),(h,y)]=([g,h]-y\triangleright g,0)\ \text{ and }\
\mathcal{P}((g,0))=(P_{\mathfrak{g}}(g),0),$

we have that $\mathfrak{g}\cong\mathfrak{g}\times\\{0\\}$ is a Rota-Baxter
ideal of the Rota-Baxter Lie algebra $\mathfrak{g}\natural V$. Next we will
prove that any Rota-Baxter Lie algebra structure on a vector space $E$
containing $(\mathfrak{g},P_{\mathfrak{g}})$ as a Rota-Baxter ideal can be
described by a crossed product of Rota-Baxter Lie algebras.

######  Corollary 3.2.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra and let $E$
be a vector space containing $\mathfrak{g}$ as a subspace. Then any Rota-
Baxter Lie algebra structure on $E$ that contains
$(\mathfrak{g},P_{\mathfrak{g}})$ as a Rota-Baxter ideal is isomorphic to a
Rota-Baxter Lie algebra associated to a crossed system
$((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleright,f,P_{1})$.

###### Proof.

Suppose $(E,P_{E})$ is a Rota-Baxter Lie algebra such that
$(\mathfrak{g},P_{\mathfrak{g}})$ is a Rota-Baxter ideal in $(E,P_{E})$. Then
$(\mathfrak{g},P_{\mathfrak{g}})$ is also a Rota-Baxter Lie subalgebra of
$(E,P_{E})$. Hence by Theorem 2.11, there is a Rota-Baxter Lie extending
structure
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\})$
of $(\mathfrak{g},P_{\mathfrak{g}})$ through a subspace $V$ of $E$ and an
isomorphism of Rota-Baxter Lie algebras $E\cong\mathfrak{g}\natural V$. As
$(\mathfrak{g},P_{\mathfrak{g}})$ is a Rota-Baxter ideal of $(E,P_{E})$, for
any $x\in V$ and $g\in\mathfrak{g}$, we have that $[x,g]\in\mathfrak{g}$ and
hence $p([x,g])=[x,g]$, where $p$ is the linear map in the proof of Theorem
2.11. Thus, $x\triangleleft_{p}g=[x,g]-p([x,g])=0$ and $(E,P_{E})$ is
isomorphic to the Rota-Baxter Lie algebra associated to the crossed system
$((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleright,f,P_{1})$, where
$V=\mathrm{ker}(p)$. ∎

######  Definition 3.3.

[32, Definition 3.1] Let $(\mathfrak{g},P_{\mathfrak{g}})$ and
$(\mathfrak{h},P_{\mathfrak{h}})$ be two Rota-Baxter Lie algebras. A non-
abelian extension of $(\mathfrak{g},P_{\mathfrak{g}})$ by
$(\mathfrak{h},P_{\mathfrak{h}})$ is a Rota-Baxter Lie algebra
$(\mathfrak{k},P_{\mathfrak{k}})$ equipped with a short exact sequence of
Rota-Baxter Lie algebras

$\displaystyle
0\rightarrow\mathfrak{h}\rightarrow\mathfrak{k}\rightarrow\mathfrak{g}\rightarrow
0.$

The equivalent classes of non-abelian extension of
$(\mathfrak{g},P_{\mathfrak{g}})$ by $(\mathfrak{h},P_{\mathfrak{h}})$ is
characterized in [32]. Actually, the restricted version of the extending
structures problem is in face an equivalent reformulation of the non-abelian
extension problem. Indeed, the Rota-Baxter Lie algebra $\mathfrak{g}\natural
V$ associated to a crossed system
$((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleright,f,P_{1})$ is an
extension of $(V,P_{2})$ by $(\mathfrak{g},P_{\mathfrak{g}})$ via the
following sequence

$\displaystyle 0\rightarrow\mathfrak{g}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\mathfrak{g}\natural
V\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}V\rightarrow 0,$

where $i:\mathfrak{g}\rightarrow\mathfrak{g}\natural V$, $i(g)=(g,0)$ and
$\pi:\mathfrak{g}\natural V\rightarrow V$, $\pi(g,x)=x$. Conversely, let
$(\mathfrak{k},P_{\mathfrak{k}})$ be an extension of $(V,P_{2})$ by
$(\mathfrak{g},P_{\mathfrak{g}})$. Then there is an exact sequence of Rota-
Baxter Lie algebras

$\displaystyle 0\rightarrow\mathfrak{g}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\mathfrak{k}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}V\rightarrow
0.$

As $\mathfrak{g}\cong Im(i)=Ker(pi)$, $(\mathfrak{g},P_{\mathfrak{g}})$ can be
viewed as a Rota-Baxter ideal of $(\mathfrak{k},P_{\mathfrak{k}})$. By
Corollary 3.2, there exists a crossed system
$((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleright,f)$ of Rota-Baxter
Lie algebras such that $\mathfrak{g}\natural V\cong\mathfrak{k}$ as Rota-
Baxter Lie algebras.

###  3.2. Matched pairs and the factorization problem

In this subsection, we first recall the concept of a matched pair of Lie
algebras [29, 31] and then show that how the factorization problem studied by
Lang and Sheng [26] is a special case of Theorem 2.15.

######  Definition 3.4.

A matched pair of Lie algebras consists of two Lie algebras
$(\mathfrak{g},[-,-])$ and $(V,\\{-,-\\})$, which satisfy that $\mathfrak{g}$
is a left $V$-module under
$\triangleright:V\otimes\mathfrak{g}\rightarrow\mathfrak{g}$, $V$ is a right
$\mathfrak{g}$-module under $\triangleleft:V\otimes\mathfrak{g}\rightarrow V$
and for all $g,h\in\mathfrak{g}$ and $u,v\in V$,

$\displaystyle u\triangleright[g,h]=$ $\displaystyle\ [u\triangleright
g,h]+[g,u\triangleright h]+(u\triangleleft g)\triangleright h-(u\triangleleft
h)\triangleright g,$ $\displaystyle\\{u,v\\}\triangleleft g=$ $\displaystyle\
\\{u,v\triangleleft g\\}+\\{u\triangleleft
g,v\\}+u\triangleleft(v\triangleright g)-v\triangleleft(u\triangleright g).$

The notion of representations of Rota-Baxter associative algebras was
introduced in [18] and further studied in [35]. The concept of representations
of Rota-Baxter Lie algebras of weight 0 was given in [24] and the general
notion of representations of Rota-Baxter Lie algebras was given in [26] in the
study of matched pairs of Rota-Baxter Lie algebras.

######  Definition 3.5.

[26, Definition 3.12] A matched pair of Rota-Baxter Lie algebras consists of
two Rota-Baxter Lie algebras $(\mathfrak{g},P_{\mathfrak{g}})$ and
$(\mathfrak{h},P_{\mathfrak{h}})$, which satisfy that
$(\mathfrak{g},P_{\mathfrak{g}})$ is a left
$(\mathfrak{h},P_{\mathfrak{h}})$-module under
$\triangleright:\mathfrak{h}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$,
$(\mathfrak{h},P_{\mathfrak{h}})$ is a right
$(\mathfrak{g},P_{\mathfrak{g}})$-module under
$\triangleleft:V\otimes\mathfrak{g}\rightarrow V$ and
$(\mathfrak{g},\mathfrak{h},\triangleleft,\triangleright)$ is a matched pair
of Lie algebras.

Let
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
be an extending datum of $(\mathfrak{g},P_{\mathfrak{g}})$ through $V$ such
that $f$ and $P_{1}$ are the trivial maps, i.e. $f(x,y)=0$ and $P_{1}(x)=0$,
for all $x,y\in V$. Then by Theorem 2.9,
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,\\{-,-\\},P_{2})$
is a Rota-Baxter Lie extending structure of $(\mathfrak{g},P_{\mathfrak{g}})$
through $V$ if and only $(V,\\{-,-\\},P_{2})$ is a Rota-Baxter Lie algebra and
$((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleleft,\triangleright)$ is
a matched pair of Rota-Baxter Lie algebras. In this case, we denote the
associated unified product $\mathfrak{g}\natural V$ by $\mathfrak{g}\bowtie V$
and call it the bicrossed product of the matched pair
$((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleleft,\triangleright)$ of
Rota-Baxter Lie algebras.

The bicrossed product of two Rota-Baxter Lie algebras is the construction
which provides the answer for the following factorization problem.

######  Problem 3.6.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ and $(\mathfrak{h},P_{\mathfrak{h}})$ be
two Rota-Baxter Lie algebras. Describe and classify all Rota-Baxter Lie
algebras $(\mathfrak{k},P_{\mathfrak{k}})$ that factorize through
$(\mathfrak{g},P_{\mathfrak{g}})$ and $(\mathfrak{h},P_{\mathfrak{h}})$, i.e.
$(\mathfrak{k},P_{\mathfrak{k}})$ contains $(\mathfrak{g},P_{\mathfrak{g}})$
and $(\mathfrak{h},P_{\mathfrak{h}})$ as Rota-Baxter Lie subalgebras such that
$\mathfrak{k}=\mathfrak{g}+\mathfrak{h}$ and
$\mathfrak{g}\cap\mathfrak{h}=\\{0\\}$.

By Proposition 3.14 in [26], a Rota-Baxter Lie algebra
$(\mathfrak{k},P_{\mathfrak{k}})$ factorizes through
$(\mathfrak{g},P_{\mathfrak{g}})$ and $(\mathfrak{h},P_{\mathfrak{h}})$ if and
only if there is a matched pair of Rota-Baxter Lie algebras
$((\mathfrak{g},P_{\mathfrak{g}}),(\mathfrak{h},P_{\mathfrak{h}}),\triangleleft,\triangleright)$
such that
$\mathfrak{g}\bowtie\mathfrak{h}\cong(\mathfrak{k},P_{\mathfrak{k}})$. Hence
the factorization problem can be restated as:

######  Problem 3.7.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ and $(\mathfrak{h},P_{\mathfrak{h}})$ be
two Rota-Baxter Lie algebras. Describe the set of all matched pairs
$((\mathfrak{g},P_{\mathfrak{g}}),(\mathfrak{h},P_{\mathfrak{h}}),\triangleleft,\triangleright)$
and classify up to an isomorphism all bicrossed product
$\mathfrak{g}\bowtie\mathfrak{h}$.

As a bicrossed product of a matched pair
$((\mathfrak{g},P_{\mathfrak{g}}),(V,P_{2}),\triangleleft,\triangleright)$ of
Rota-Baxter Lie algebras is a special case of the unified product of an
extending datum
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
of $(\mathfrak{g},P_{\mathfrak{g}})$ through $V$ such that $f$ and $P_{1}$ are
the trivial maps, Problem 3.7 can be characterized by Theorem 2.15 as a
special case.

##  4\. Flag extending structures

In this section, we apply our main theorem to a flag extending structure of
Rota-Baxter Lie algebras as a special case.

Given a vector space $E$ containing a Rota-Baxter Lie algebra
$(\mathfrak{g},P_{\mathfrak{g}})$ as a subspace, there is a complement $V$ of
$\mathfrak{g}$ in $E$. Although Theorem 2.15 gives an classification of the
extending structure problem of Rota-Baxter Lie algebras, there is a
computational problem.

######  Problem 4.1.

How to compute the object
$H^{2}_{(\mathfrak{g},P_{\mathfrak{g}})}(V,(\mathfrak{g},P_{\mathfrak{g}}))$
explicitly and how to obtain all Rota-Baxter Lie algebra structures on $E$
which contains $(\mathfrak{g},P_{\mathfrak{g}})$ as a Rota-Baxter Lie
subalgebra?

Now we provide a way of answering the problem for a special case.

######  Definition 4.2.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra and let $E$
be a vector space containing $\mathfrak{g}$ as a subspace. A Rota-Baxter Lie
algebra on $E$ such that $(\mathfrak{g},P_{\mathfrak{g}})$ is a Rota-Baxter
Lie subalgebra is called a flag extending structure of
$(\mathfrak{g},P_{\mathfrak{g}})$ to $E$ if there is a finite chain of Rota-
Baxter Lie subalgebras of $E$

$\displaystyle(\mathfrak{g},P_{\mathfrak{g}})=(E_{0},P_{0})\subset(E_{1},P_{1})\subset\cdots\subset(E_{m},P_{m})=(E,P_{E})$

such that $E_{i}$ has codimension 1 in $E_{i+1}$, for all $i=0,\ldots,m-1.$

All flag extending structures of $(\mathfrak{g},P_{\mathfrak{g}})$ to $E$ can
be described recursively: the first step is to describe and classify all
unified products $\mathfrak{g}\natural V_{1}$, for a 1-dimensional vector
space $V_{1}$; then replace $(\mathfrak{g},P_{\mathfrak{g}})$ with
$\mathfrak{g}\natural V_{1}$, describe and classify all unified products of
$(\mathfrak{g}\natural V_{1})\natural V_{2}$, where $V_{1}$ and $V_{2}$ are
vector spaces of dimension 1. After $m$ steps, we obtain all flag extending
structure of $(\mathfrak{g},P_{\mathfrak{g}})$ to $E$.

Now we show that all unified products of $\mathfrak{g}\natural V$ with $V$ a
1-dimensional vector space are parameterized by the space
$ExDer((\mathfrak{g},P_{\mathfrak{g}}))$ of all extended derivations of
$(\mathfrak{g},P_{\mathfrak{g}})$. Now we recall the notion of twisted
derivation of a Lie algebra [3].

######  Definition 4.3.

[3, Definition 5.2] Let $\mathfrak{g}$ be a Lie algebra. A twisted derivation
of $\mathfrak{g}$ is a pair $(\varepsilon,D)$, where
$\varepsilon:\mathfrak{g}\rightarrow{\bf k}$ and
$D:\mathfrak{g}\rightarrow\mathfrak{g}$ are two linear maps such that for all
$g,h\in\mathfrak{g}$,

(6) $\displaystyle\varepsilon([g,h])=$ $\displaystyle\ 0,$ (7) $\displaystyle
D([g,h])=$ $\displaystyle\
[D(g),h]+[g,D(h)]+\varepsilon(g)D(h)-\varepsilon(h)D(g).$

######  Definition 4.4.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra. A extended
derivation of $(\mathfrak{g},P_{\mathfrak{g}})$ is a quadruple
$(\varepsilon,D,g_{0},k_{0})$, where $\varepsilon:\mathfrak{g}\rightarrow{\bf
k}$ and $D:\mathfrak{g}\rightarrow\mathfrak{g}$ are two linear maps,
$g_{0}\in\mathfrak{g}$ and $k_{0}\in{\bf k}$ such that $(\varepsilon,D)$ is a
twisted derivation of $\mathfrak{g}$ and for all $g\in\mathfrak{g}$,

$\displaystyle[P_{\mathfrak{g}}(g),g_{0}]$ $\displaystyle\
-k_{0}D(P_{\mathfrak{g}}(g))+P_{\mathfrak{g}}(D(P_{\mathfrak{g}}(g)))+\varepsilon(P_{\mathfrak{g}}(g))g_{0}-P_{\mathfrak{g}}([g,g_{0}])$
(8) $\displaystyle\
+P_{\mathfrak{g}}(k_{0}D(g))+k_{0}\varepsilon(g)g_{0}+\lambda
P_{\mathfrak{g}}(D(g))+\lambda\varepsilon(g)g_{0}=0,$ (9) $\displaystyle\
k_{0}^{2}\varepsilon(g)+\lambda k_{0}\varepsilon(g)=0.$

Denote the set of all extended derivations of
$(\mathfrak{g},P_{\mathfrak{g}})$ by $ExDer((\mathfrak{g},P_{\mathfrak{g}}))$.
We shall now prove that the set of all Rota-Baxter Lie extending structures
$\mathcal{L}((\mathfrak{g},P_{\mathfrak{g}}),V)$ of a Rota-Baxter Lie algebra
$(\mathfrak{g},P_{\mathfrak{g}})$ through a 1-dimensional vector space $V$ is
parameterized by $ExDer((\mathfrak{g},P_{\mathfrak{g}}))$.

######  Proposition 4.5.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra and let $V$
be a vector space of dimension 1 with a basis $\\{x\\}$. Then there exists a
bijection between the set $\mathcal{L}((\mathfrak{g},P_{\mathfrak{g}}),V)$ of
all Rota-Baxter Lie extending structures of $(\mathfrak{g},P_{\mathfrak{g}})$
through $V$ and the space $ExDer((\mathfrak{g},P_{\mathfrak{g}}))$ of all
extended derivations of $(\mathfrak{g},P_{\mathfrak{g}})$. Through the above
bijection, the Rota-Baxter Lie extending structure
$\Omega((\mathfrak{g},P_{\mathfrak{g}}),V)=(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})$
associated to $(\varepsilon,D,g_{0},k_{0})\in
ExDer((\mathfrak{g},P_{\mathfrak{g}}))$ is given by

$\displaystyle x\triangleleft g=\varepsilon(g)x,\,x\triangleright
g=D(g),\,f=0,\,\\{-,-\\}=0,P_{1}(x)=g_{0},\,P_{2}(x)=k_{0}x,$

for all $g\in\mathfrak{g}$ and $x\in V$. Denote the corresponding unified
product $\mathfrak{g}\natural V$ by
$\mathfrak{g}\natural_{(\varepsilon,D,g_{0},k_{0})}V$.

###### Proof.

Note that the set $\mathcal{L}((\mathfrak{g},P_{\mathfrak{g}}),V)$ of all
Rota-Baxter Lie extending structures of $(\mathfrak{g},P_{\mathfrak{g}})$
through $V$ is equivalent to the set of all the maps
$\\{(\triangleleft,\triangleright,f,\\{-,-\\},P_{1},P_{2})\\}$, where
$\triangleleft:V\times\mathfrak{g}\rightarrow\mathfrak{g},\triangleright:V\times\mathfrak{g}\rightarrow
V,f:V\times V\rightarrow\mathfrak{g},\\{-,-\\}:V\times V\rightarrow
V,P_{1}:V\rightarrow\mathfrak{g}$ and $P_{2}:V\rightarrow V$ satisfying the
conditions (b)(i)-(b)(xi) of Theorem 2.9. Since $V$ has dimension 1, the
condition (b)(i) is equivalent to that $f$ and $\\{-,-\\}$ are trivial maps.
Again by the dimension of $V$ is 1, there is a bijection between the set of
all bilinear maps $\triangleleft:V\times\mathfrak{g}\rightarrow V$ and the set
of all linear maps $\varepsilon:\mathfrak{g}\rightarrow{\bf k}$. The bijection
is given such that the action $\triangleleft:V\times\mathfrak{g}\rightarrow V$
associated to $\varepsilon$ is given by the formula: $x\triangleleft
g=\varepsilon(g)x$, for all $g\in\mathfrak{g}$. Similarly, any bilinear map
$\triangleright:V\times\mathfrak{g}\rightarrow\mathfrak{g}$ is uniquely
determined by a linear map $D:\mathfrak{g}\rightarrow\mathfrak{g}$ via the
formula: $x\triangleright g=D(g)$, for all $g\in\mathfrak{g}$. Moreover, the
linear map $P_{1}:V\rightarrow\mathfrak{g}$ is uniquely determined by an
element $g_{0}\in\mathfrak{g}$ via the formula: $P_{1}(x)=g_{0}$ and the
linear map $P_{2}:V\rightarrow V$ is uniquely determined by an element
$k_{0}\in{\bf k}$ via the formula: $P_{2}(x)=k_{0}x$.

Since $dim_{{\bf k}}(V)=1,f=0,\\{-,-\\}=0$, we find that the conditions
(b)(iv)-(b)(vii), (b)(x)-(b)(xi) hold automatically. Moreover, the condition
(b)(ii) is equivalent to Eq. (6) holds, the condition (b)(iii) is equivalent
to Eq. (7) holds, the condition (b)(viii) is equivalent to Eq. (8) holds and
the condition (b)(ix) is equivalent to Eq. (9) holds. This completes the
proof. ∎

######  Definition 4.6.

Let $(\varepsilon,D,g_{0},k_{0})$ and
$(\varepsilon^{\prime},D^{\prime},g^{\prime}_{0},k^{\prime}_{0})$ be two
extended derivations of a Rota-Baxter Lie algebra
$(\mathfrak{g},P_{\mathfrak{g}})$. $(\varepsilon,D,g_{0},k_{0})$ and
$(\varepsilon^{\prime},D^{\prime},g^{\prime}_{0},k^{\prime}_{0})$ are
equivalent, denoted by
$(\varepsilon,D,g_{0},k_{0})\equiv(\varepsilon^{\prime},D^{\prime},g^{\prime}_{0},k^{\prime}_{0})$,
if $\varepsilon=\varepsilon^{\prime},k_{0}=k^{\prime}_{0}$ and there exists an
element $g_{1}\in\mathfrak{g}$ and a nonzero element $k_{1}\in{\bf k}$ such
that for any $g\in g$,

(10) $\displaystyle D(g)=$ $\displaystyle\
k_{1}D^{\prime}(g)+[g_{1},g]-\varepsilon(g)g_{1},$ (11) $\displaystyle g_{0}=$
$\displaystyle\ P_{\mathfrak{g}}(g_{1})+k_{1}g^{\prime}_{0}-k_{0}g_{1}.$

######  Theorem 4.7.

Let $(\mathfrak{g},P_{\mathfrak{g}})$ be a Rota-Baxter Lie algebra of
codimension 1 in a vector space $E$ and let $V$ be a complement of
$\mathfrak{g}$ in $E$ with basis $\\{x\\}$. Then $\equiv$ is an equivalent
relation of the set $ExDer((\mathfrak{g},P_{\mathfrak{g}}))$ of all extended
derivations of $(\mathfrak{g},P_{\mathfrak{g}})$ and
$Extd(E,\mathfrak{g})\cong
H^{2}_{(\mathfrak{g},P_{\mathfrak{g}})}(V,(\mathfrak{g},P_{\mathfrak{g}}))\cong
ExDer((\mathfrak{g},P_{\mathfrak{g}}))/\equiv$.

###### Proof.

By Lemma 2.13, there is an isomorphism of Rota-Baxter Lie algebras between
$\mathfrak{g}\natural_{(\varepsilon,D,g_{0},k_{0})}V$ and
$\mathfrak{g}\natural_{(\varepsilon^{\prime},D^{\prime},g^{\prime}_{0},k^{\prime}_{0})}$
which stabilizes $(\mathfrak{g},P_{\mathfrak{g}})$ if and only if there are
linear maps $r:V\rightarrow\mathfrak{g}$ and $v:V\rightarrow V$ satisfying the
conditions (a)-(f) and $v$ is a bijection. Since $V={\bf k}\\{x\\}$, the map
$r:V\rightarrow\mathfrak{g}$ is uniquely determined by an element
$g_{1}\in\mathfrak{g}$ such that $r(x)=g_{1}$ and the bijective map
$v:V\rightarrow V$ is uniquely determined by a nonzero element $k_{1}\in{\bf
k}$ such that $v(x)=k_{1}x$. Note that $f=f^{\prime}=0$ and
$\\{-,-\\}=\\{-,-\\}^{\prime}=0$ in the corresponding Rota-Baxter Lie
extending structures, the conditions (c) and (d) hold trivially. The condition
(a) is equivalent to $k_{1}\varepsilon^{\prime}(g)x=k_{1}\varepsilon(g)x$, for
all $g\in\mathfrak{g}$. As $k_{1}\neq 0$, $\varepsilon=\varepsilon^{\prime}$.
The condition (b) is equivalent to Eq. (10) and the condition (e) is
equivalent to Eq. (11). Moreover, the condition (f) is equivalent to
$k_{0}k_{1}x=k^{\prime}_{0}k_{1}x$ and hence $k_{0}=k^{\prime}_{0}$ by
$k_{1}\neq 0$. Thus
$\mathfrak{g}\natural_{(\varepsilon,D,g_{0},k_{0})}V\cong\mathfrak{g}\natural_{(\varepsilon^{\prime},D^{\prime},g^{\prime}_{0},k^{\prime}_{0})}V$
as Rota-Baxter Lie algebras if and only if
$(\varepsilon,D,g_{0},k_{0})\equiv(\varepsilon^{\prime},D^{\prime},g^{\prime}_{0},k^{\prime}_{0})$.
This completes the proof. ∎

Acknowledgments: This research is supported by the National Natural Science
Foundation of China (Grant No.12101316).

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# A note on the order of the Tate–Shafarevich group modulo squares

Alexandros Konstantinou  University College London, London WC1H 0AY, UK
<EMAIL_ADDRESS>
###### Abstract.

We investigate the order of the Tate–Shafarevich group of abelian varieties
modulo rational squares. Our main result shows that every square-free natural
number appears as the non square-free part of the Tate–Shafarevich group of
some abelian variety, thereby validating a conjecture of W. Stein.

###### 2010 Mathematics Subject Classification:

11G10 (11G40)

###### Contents

  1. 1 Introduction
  2. 2 Expression for mod squares
  3. 3  of non-square order

##  1\. Introduction

In the following note, we study the order of the Tate–Shafarevich group
$\Sha(A/K)$ of an abelian variety $A$ defined over a number field $K$. Its
non-zero elements correspond to principal homogeneous spaces for $A/K$ with a
rational point over all completions of $K$ though lacking a point in $K$
itself. Despite its central role in the study of rational points, several
foundational questions regarding $\Sha(A/K)$ remain open. These include the
Tate–Shafarevich conjecture which stipulates its finiteness. In addition, a
conjecture of Stein focuses on the possible values that the order of the
Tate–Shafarevich group can take (if finite) modulo rational squares. In the
following note, we validate Stein’s conjecture, which can be phrased as
follows.

######  Theorem 1.1 (=Corollary 3.6).

For every positive square-free integer $n$, there exists an abelian variety
$A/\mathbb{Q}$ with finite Tate–Shafarevich group of order $nm^{2}$ for some
integer $m\geq 1$.

###  1.1. Previous work

In the case of principally polarised abelian varieties, the Cassels–Tate
pairing,

$\langle,\rangle:\Sha(A/K)\times\Sha(A/K)\to\mathbb{Q}/\mathbb{Z},$

lays down significant constraints on the structure of the Tate–Shafarevich
group. In particular, for elliptic curves, this is known to be alternating,
and therefore the order of the Tate–Shafarevich group of an elliptic curve
must be a square, if finite.

The work of Flach [4] establishes the (slightly weaker) skew symmetric
condition $\langle x,y\rangle=-\langle y,x\rangle$ for principally polarised
abelian varieties. From this, one can only deduce that the Tate–Shafarevich
group is either a square or twice a square. Building upon this, Poonen and
Stoll present a criterion for deducing if the Tate–Shafarevich group (provided
it’s finite) is of square order purely based on the vanishing of $\langle
c,c\rangle$ for an element $c\in\Sha(A/K)[2]$ canonically associated to $A$.
They then use this criterion to present the first example of an abelian
variety for which the order of its Tate–Shafarevich group is twice a square.

Subsequently, Stein [13] presents the first examples of abelian varieties
defined over $\mathbb{Q}$ in which the order of their Tate–Shafarevich groups
satisfy $\left|\Sha(A/\mathbb{Q})\right|=pm^{2}$ for $p<25000$ an odd prime.
His construction relies on the existence of an abelian variety defined over
$\mathbb{Q}$ which becomes isomorphic to a product $E^{p-1}$ of elliptic
curves over a cyclic extension. Following this, Keil’s work [6] presents
examples of abelian surfaces, given as quotients of a product of two elliptic
curves, for which $\left|\Sha\right|=pm^{2}$ for $p\leq 13$ an odd prime.

###  1.2. Acknowledgements

I would like to extend my thanks to Vladimir Dokchitser for his guidance and
many constructive discussions during the course of this work. In addition, I
would like to thank Adam Morgan for many helpful suggestions on a draft
version of this paper and for many useful discussions that came before that.

##  2\. Expression for mod squares

Our proof relies on breaking down the Weil restriction of scalars of an
abelian variety up to isogeny followed by an application of a formula of
Cassels–Tate. This allows us to express the size of the Tate–Shafarevich group
modulo squares in terms of Birch–Swinnerton-Dyer constants.

###  2.1. Cassels–Tate formula

In the following subsection, we briefly recall a result of Cassels and Tate
which proves invariance of the Birch–Swinnerton-Dyer conjecture under isogeny.
We first fix some notation.

######  Notation 2.1.

We let $K$ be a number field. Given a $K$-rational isogeny $\phi:A_{1}\to
A_{2}$, we write

$\begin{split}Q(\phi)=&|\textup{coker}(\phi:A_{1}(K)/A_{1}(K)_{\textup{tors}}\to
A_{2}(K)/A_{2}(K)_{\textup{tors}})|\\\
\times&|\textup{ker}(\phi:\Sha(A_{1}/K)_{\textup{div}}\to\Sha(A_{2}/K)_{\textup{div}})|,\end{split}$

where ${}_{\textup{div}}$ denotes the divisible part of .

For an abelian variety $A/K$ with a fixed non-zero global exterior form
$\omega$, we denote the Birch–Swinnerton-Dyer periods by

$\Omega_{\mathbb{C}}(A,\omega)=2^{\textup{dim}A}\int_{A(\mathbb{C})}\left|\omega\wedge\overline{\omega}\right|,\
\ \ \ \
\Omega_{\mathbb{R}}(A,\omega)=\int_{A(\mathbb{R})}\left|\omega\right|.$

We write $\Omega(A/K,\omega)=\prod_{v\mid\infty}\Omega_{K_{v}}(A,\omega)$
where $K_{v}=\mathbb{R}$ (resp. $\mathbb{C}$) if $v$ is real (resp. complex).
In addition, for a non-archimedean place $v$, we write
$w_{A,v}^{\textup{min}}$ for a Néron minimal exterior form on $A$,
$c_{v}(A/K)$ for its Tamagawa number and
$C(A/K,\omega)=\prod_{v\nmid\infty}c_{v}(A/K)|{\omega}/{\omega_{A,v}^{\textup{min}}}|_{v}$.
We write ${}_{0}=\Sha/{}_{\textup{div}}$ and $\square$ to denote the square of
a rational number.

The following theorem is a version of the Cassels–Tate formula as found in [3,
Theorem 4.3].

######  Theorem 2.2 (Cassels–Tate).

Let $\phi:A_{1}\to A_{2}$ be a $K$-rational isogeny. Then, for any non-zero
global exterior forms $\omega_{1}$ and $\omega_{2}$ on $A_{1}$ and $A_{2}$,

$\frac{Q(\phi^{\vee})}{Q(\phi)}=\frac{|A_{2}(K)_{\textup{tors}}|}{|A_{1}(K)_{\textup{tors}}|}\frac{|A_{2}^{\vee}(K)_{\textup{tors}}|}{|A_{1}^{\vee}(K)_{\textup{tors}}|}\frac{C(A_{1}/K,\omega_{1})}{C(A_{2}/K,\omega_{2})}\frac{\Omega(A_{1}/K,\omega_{1})}{\Omega(A_{2}/K,\omega_{2})}\prod_{p\mid\textup{deg}(\phi)}\frac{|{}_{0}(A_{1}/K)[p^{\infty}]|}{|{}_{0}(A_{2}/K)[p^{\infty}]|}.$

Given a field extension $L/K$, we write $\phi_{L}$ for the base change of
$\phi$ to $L$. The following proposition shows that for carefully chosen $L$
and working modulo squares, the Cassels–Tate formula simplifies.

######  Lemma 2.3.

Let $L/K$ be a quadratic extension such that all bad primes of $A_{1}/K$ and
$A_{2}/K$ and all places $v\mid\textup{deg}(\phi)$ split in $L$. In addition,
suppose that $\textup{deg}(\phi)$ is a square. Then,

$\frac{Q(\phi_{L}^{\vee})}{Q(\phi_{L})}=\frac{|A_{2}(L)_{\textup{tors}}|}{|A_{1}(L)_{\textup{tors}}|}\frac{|A_{2}^{\vee}(L)_{\textup{tors}}|}{|A_{1}^{\vee}(L)_{\textup{tors}}|}\prod_{p\mid\textup{deg}(\phi)}\frac{|{}_{0}(A_{1}/K)[p^{\infty}]|}{|{}_{0}(A_{2}/K)[p^{\infty}]|}\
\mathrm{mod}\ \square.$

###### Proof.

We note that $C(A_{1}/L,\omega_{1})\Omega(A_{1}/L,\omega_{1})$ is independent
of $\omega_{1}$. By our assumptions on $L/K$, the only terms in this product
we have to consider are complex places of $L$ which lie above real places of
$K$ and non-archimedean places $w\nmid\textup{deg}(\phi)$. As in the proof of
[9, Theorem 7.3], when $w$ is complex, we have
${\Omega_{\mathbb{C}}(A_{1},\phi^{*}\omega_{2})}/{\Omega_{\mathbb{C}}(A_{2},\omega_{2})}=\textup{deg}(\phi)=\square.$
For a non-archimedean place $w\nmid\textup{deg}(\phi)$,
$|{(\phi^{*}\omega_{2}/\omega_{A_{1},w}^{\textup{min}})}\big{/}{(\omega_{2}/\omega_{A_{2},w}^{\textup{min}})}|_{w}=1$
(see [12, pp. 12]). To conclude,
$\frac{C(A_{1}/L,\omega_{1})}{C(A_{2}/L,\omega_{2})}\frac{\Omega(A_{1}/L,\omega_{1})}{\Omega(A_{2}/L,\omega_{2})}=\square$
giving the required result. ∎

###  2.2. A decomposition for the Weil-restriction of scalars

In what follows, we let $A/K$ be a principally polarised abelian variety
defined over a number field $K$. We let $p$ be an odd prime, $F/K$ a cyclic
extension of degree $p$ and $\textup{Res}_{K}^{F}A$ the restriction of scalars
from $F$ to $K$. For brevity, we write $B=\textup{Res}_{K}^{F}A$. This is an
$F/K$-twist of $A^{p}$, and the product polarisation on $A^{p}$ descends to a
principal polarisation on $B/K$. Then, there are $K$-morphisms

$i:A\to B,\ \ \ \ \ \textup{Tr}:B\to A.$

These are the inclusion and the trace map respectively.

After identifying the base change $B_{F}=A^{p}$ as principally polarised
abelian varieties, the base changed morphisms $i_{F}$ and $\textup{Tr}_{F}$
coincide with the diagonal inclusion and the summation map respectively. In
addition, writing $i^{\vee}:B\to A$ for the dual of $i$ (having suppressed the
canonical principal polarisations on both sides), we have
$i^{\vee}=\textup{Tr.}$ Finally, we write $R$ to denote the kernel of the
trace map. This is an abelian variety defined over $K$ and is an $F/K$-twist
of $A^{p-1}$ (see [13, Proposition 2.4]).

######  Lemma 2.4.

Let $\phi:R\times A\to B$ be given by $(x,y)\mapsto x+i(y)$. Then, $\phi$ is a
$K$-rational isogeny with $\textup{ker}(\phi)\cong A[p]$.

###### Proof.

It suffices to show $\textup{ker}(\phi)\cong A[p]$. After base change to
$\overline{K}$, we identify $R_{\overline{K}}=A^{p-1}$ and
$B_{\overline{K}}=A^{p}$. Then, on $\overline{K}$-points $\phi$ is given by
$(x_{1},\dots,x_{p-1},y)\mapsto(y+x_{1},\dots,y+x_{p-1},y-\sum_{i=1}^{p-1}x_{i})$
. Thus, restricting the projection $R\times A\to A$ to $\textup{ker}(\phi)$
allows us to deduce that $\textup{ker}(\phi)\cong A[p]$. ∎

###  2.3. An expression for $\Sha(R/L)$ modulo squares

In this subsection, we relate $|\Sha(R/L)[p^{\infty}]|\ \mathrm{mod}\ \square$
to the torsion subgroup of $R$ and the rank of $A/L$.

######  Lemma 2.5.

Let $L/K$ be a number field extension such that $L\otimes_{K}F$ is a field.
Then, the following hold.

  1. (1)

For all odd primes $q\neq p$, $\left|{}_{0}(R/L)[q^{\infty}]\right|=\square$.

  2. (2)

If, in addition, $A$ is an elliptic curve,
$\left|{}_{0}(R/L)[2^{\infty}]\right|=\square$.

###### Proof.

By Lemma 2.4, $\textup{deg}(\phi)=p^{2\textup{dim}(A)}$. When $q\neq p$, this
isogeny induces an isomorphism
${}_{0}(A/L)[q^{\infty}]\times{}_{0}(R/L)[q^{\infty}]\stackrel{{\scriptstyle\sim}}{{\to}}{}_{0}(B/L)[q^{\infty}]$.
Since $A$ and $B$ are principally polarised, [10, Theorem 8] asserts that
${}_{0}(A/L)[q^{\infty}]$ and ${}_{0}(B/L)[q^{\infty}]$ are both of square
order for all odd $q$. This proves (1). Further, by [2, Proposition A.5.2],
$B_{L}\cong\textup{Res}_{L}^{L\otimes_{K}F}A$, and by combining [8, pp.
178(a)] with Shapiro’s Lemma,
${}_{0}(B/L)[2^{\infty}]\cong{}_{0}(A/L\otimes_{K}F)[2^{\infty}]$. Therefore,
if $A$ is an elliptic curve, both ${}_{0}(A/L)[2^{\infty}]$ and
${}_{0}(B/L)[2^{\infty}]$ are of square order by [10, Theorem 8, Corollary
12]. Claim (2) follows. ∎

In what follows, we write $\textup{rk}_{p}(A/K)$ for the
$\mathbb{Z}_{p}$-corank of the $p^{\infty}$-Selmer group
$\textup{Sel}_{p^{\infty}}(A/K):=\varinjlim_{n\geq
1}\textup{Sel}_{p^{n}}(A/K)$. This coincides with the Mordell–Weil rank of $A$
plus the multiplicity of $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ in its
Tate–Shafarevich group.

######  Proposition 2.6.

Let $L/K$ be a quadratic extension such that all bad primes of $A/K$ and all
primes dividing either $\textup{deg}(\phi)$ or the relative discriminant
$\Delta(F/K)$ split in $L$. Then, the following hold.

  1. (1)

Up to rational squares,

$\left|{}_{0}(R/L)[p^{\infty}]\right|\equiv\frac{Q(\phi_{L})}{Q{(}\phi^{\vee}_{L}{)}}\frac{\left|R(L)[p^{\infty}]\right|}{\left|R^{\vee}(L)[p^{\infty}]\right|}\
\mathrm{mod}\ \square.$

  2. (2)

If, in addition, $\textup{rk}_{p}(A/L)=\textup{rk}_{p}(A/L\otimes_{K}F)$, then

$\left|{}_{0}(R/L)[p^{\infty}]\right|\equiv
p^{\textup{rk}_{p}(A/L)}\frac{\left|R(L)[p^{\infty}]\right|}{\left|R^{\vee}(L)[p^{\infty}]\right|}\
\mathrm{mod}\ \square.$

###### Proof.

Let $v$ be a place of bad reduction for $B/K$. By [8, Proposition 1], this can
only happen when $A/K$ has bad reduction at $v$ or when $v\mid\Delta(F/K)$.
This shows that all bad primes of $B/K$ split in $L$. Since $A/K$ is
principally polarised,
$\left|A(L)[p^{\infty}]\right|=\left|A^{\vee}(L)[p^{\infty}]\right|$ and
$\left|{}_{0}(A/L)[p^{\infty}]\right|=\left|{}_{0}(A^{\vee}/L)[p^{\infty}]\right|$.
The same holds for $B$. Part (1) follows by combining Lemma 2.3 and Lemma 2.4.
For the second claim, it suffices to show that if
$\textup{rk}_{p}(A/L)=\textup{rk}_{p}(A/L\otimes_{K}F)$, then
${Q(\phi_{L})}/{Q{(}\phi^{\vee}_{L}{)}}=p^{\textup{rk}_{p}(A/L)}$. After
suppressing the principal polarisations $A=A^{\vee},B=B^{\vee}$, we view
$\phi^{\vee}$ as a morphism $B\to A\times R^{\vee}$. Therefore,
$\frac{Q(\phi_{L})}{Q(\phi^{\vee}_{L})}\equiv{Q(\phi^{\vee}_{L}\circ\phi_{L}:A\times
R\to A\times R^{\vee}})\ \mathrm{mod}\ \square.$ Since the degree of
$\phi^{\vee}\circ\phi$ is a power of $p$, it induces an isomorphism
$\Sha(A\times
R/L)_{\textup{div}}[q^{\infty}]\stackrel{{\scriptstyle\sim}}{{\to}}\Sha(A\times{R}^{\vee}/L)_{\textup{div}}[q^{\infty}]$
for $q\neq p$. From this, we deduce
$\textup{ker}(\phi_{L}^{\vee}\circ\phi_{L}\ |\
{}_{\textup{div}})=\textup{ker}(\phi_{L}^{\vee}\circ\phi_{L}\ |\
{}_{\textup{div}}[p^{\infty}])$. Since
$B_{L}\cong\textup{Res}_{L}^{L\otimes_{K}F}A$ (see [2, Proposition A.5.2]),
then by combining [8, pp. 178(a)] with Shapiro’s Lemma gives
$\textup{Sel}_{p^{\infty}}(B/L)\cong\textup{Sel}_{p^{\infty}}(A/L\otimes_{K}F)$.
By assumption, $\textup{rk}_{p}(A/L)=\textup{rk}_{p}(A/L\otimes_{K}F)$, from
which we deduce $\textup{rk}_{p}(R/L)=\textup{rk}_{p}(R^{\vee}/L)=0$. As a
result, ${Q(\phi^{\vee}_{L}\circ\phi_{L}})$ coincides with
$Q(i_{L}^{\vee}\circ i_{L})$, where we view $i_{L}^{\vee}\circ i_{L}$ as an
endomorphism of $A_{L}$. Using the description of $i$ and $i^{\vee}$ from
§2.2, $i^{\vee}\circ i=[p]_{A}$. This gives the required result. ∎

##  3\.  of non-square order

We retain all the notation from §2. In this section, we restrict to the case
when $A=E$ is an elliptic curve, and so $R$ is an abelian variety of dimension
$p-1$. In addition, we fix the following notation.

######  Notation 3.1.

We write $K_{\infty}$ to denote the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}$, and $K_{n}=\mathbb{Q}(\zeta_{p^{n}}+\zeta_{p^{n}}^{-1})$ for the
$n^{\textup{th}}$ layer of this extension. For $d$ a positive, square-free
integer, we let $L_{n}=K_{n}\cdot\mathbb{Q}(\sqrt{-d})$.

We first show that for $n$ sufficiently large, $\Sha(R/L_{n})$ is finite for
any choice of $d$. Further, having carefully selected $d$, we show that
$|\Sha(R/L_{n})|\equiv p\ \mathrm{mod}\ \square$. We begin by proving a lemma
involving torsion on $R$ and $R^{\vee}$.

######  Lemma 3.2.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Then, for any $n\geq
0$ and any prime $p\geq 11$ both $R(L_{n})[p^{\infty}]$ and
$R^{\vee}(L_{n})[p^{\infty}]$ are trivial.

###### Proof.

As abelian varieties over $L_{n+1}$, both $R$ and $R^{\vee}$ are isomorphic to
$E^{p-1}.$ It therefore suffices to show that $E(L_{n+1})[p^{\infty}]$ is
trivial for $p\geq 11$. Let $E_{-d}$ be the quadratic twist of $E$ by $-d$.
Then, there exists a map $E(L_{n+1})\to E(K_{n+1})\times E_{-d}(K_{n+1})$
whose kernel and cokernel are $2$-groups. Then, for any odd prime $p$, this
induces an isomorphism

$E(L_{n+1})[p^{\infty}]\stackrel{{\scriptstyle\sim}}{{\to}}E(K_{n+1})[p^{\infty}]\times
E_{-d}(K_{n+1})[p^{\infty}].$

By [1, Theorem 1.1], $E(K_{\infty})[p^{\infty}]=E(\mathbb{Q})[p^{\infty}]$
(similarly for $E_{-d})$ for any $p\geq 5$. If $p\geq 11$, Mazur’s theorem [7,
Theorem 2] asserts that $E(\mathbb{Q})[p^{\infty}]$ and
$E_{-d}(\mathbb{Q})[p^{\infty}]$ are both trivial. Therefore,
$E(L_{n+1})[p^{\infty}]$ is trivial. ∎

The following result is a formal consequence of a theorem by Kato and
Rohrlich.

######  Theorem 3.3.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Then, there exists a
positive integer $n$ such that

  1. (1)

$\textup{ord}_{s=1}L(E/L_{n},s)=\textup{ord}_{s=1}L(E/L_{n+1},s),$

  2. (2)

$\Sha(R/L_{n})$ is finite,

  3. (3)

$\textup{rk}_{p}(E/L_{n+1})=\textup{rk}_{p}(E/L_{n}).$

###### Proof.

As $E$ is defined over $\mathbb{Q}$, $\textup{Res}_{K_{n}}^{L_{n}}E$ is
isogenous to $E\times E_{-d}$ giving an equality of $L$-functions

$L(E/L_{n},s)=L(E/K_{n},s)L(E_{-d}/K_{n},s),$

By Rohrlich [11, pp. 409], the sequence
$\\{\textup{ord}_{s=1}L(E/K_{n},s)\\}_{n\geq 1}$ stabilises (similarly for
$E_{-d}$), and so
$\textup{ord}_{s=1}L(E/L_{n},s)=\textup{ord}_{s=1}L(E/L_{n+1},s)$ for $n$
sufficiently large. This proves (1). Part (2) is a consequence of a result by
Kato [5]. Following the notation introduced there, for $G$ an abelian group,
$\tau\in\widehat{G}$, and a $G$-module $M$, we let $M^{(\tau)}:=\\{x\in
M:I_{\tau}\cdot x=0\\}$ where $I_{\tau}\subseteq\mathbb{Z}[G]$ denotes the
kernel of the map $\mathbb{Z}[G]\to\mathbb{C}^{\times}$ induced by $\tau$. We
apply this to
$G=\textup{Gal}(L_{n+1}/\mathbb{Q})\cong\mathbb{Z}/2p^{n+1}\mathbb{Z}$. For
brevity, we write $H=\textup{Gal}(L_{n+1}/L_{n})\cong\mathbb{Z}/p\mathbb{Z}$.
Further, the composition $i\circ\textup{Tr}$ from §2.2 is equal to
$N_{H}:=\sum_{h\in H}h$ viewed as a $K$-endomorphism of $B$. It follows that
the composition $R\hookrightarrow{}B\stackrel{{\scriptstyle N_{H}}}{{\to}}B$
is the zero map, which in turn gives a homomorphism
$\Sha(R/L_{n})\to\textup{ker}(N_{H}\ |\ \Sha(E/L_{n+1}))$. We claim this has
finite kernel. To see this, we look at the short exact sequence of abelian
varieties

$0\to R\to B\xrightarrow{\textup{Tr}}E\to 0.$

Taking Galois cohomology gives an exact sequence

$0\to E(L_{n})/\textup{Norm}_{L_{n}}^{L_{n+1}}E(L_{n+1})\to H^{1}(L_{n},R)\to
H^{1}(L_{n},B).$

By the Mordell–Weil theorem $E(L_{n})/pE(L_{n})$ is finite, from which we
deduce that $E(L_{n})/\textup{Norm}_{L_{n}}^{L_{n+1}}E(L_{n+1})$ must be
finite. Therefore, the induced map $\Sha(R/L_{n})\to\textup{ker}(N_{H}\ |\
\Sha(E/L_{n+1}))$ has finite kernel. For a primitive character
$\tau\in\widehat{G}$, we have $N_{H}\in I_{\tau}$. Therefore, for such $\tau$,
$\textup{ker}(N_{H}\ |\ \Sha(E/L_{n+1}))\subseteq\Sha(E/L_{n+1})^{(\tau)}$. By
Artin formalism,

$L(E/L_{n+1},s)=L(E/L_{n},s)\prod_{\begin{subarray}{c}\chi\in\widehat{H}\\\
\chi\neq\textup{id}\end{subarray}}L(E/L_{n},\chi,s).$

We let $n$ be as in part (1) and $\chi$ a non-trivial character of $H$. Then,
$L(E/L_{n},\chi,s)=L(E/\mathbb{Q},\textup{Ind}_{H}^{G}\chi,1)\neq 0$.
Therefore, $L(E/\mathbb{Q},\tau,1)\neq 0$ for all primitive characters
$\tau\in\widehat{G}$. Then, by [5, Corollary 14.3], the $\tau$-part
$({}_{n+1})^{(\tau)}$ is finite for all primitive $\tau\in\widehat{G}$. This
completes the proof of (2). For (3), it suffices to show that $R(L_{n})$ is
finite. This follows in the same way as (2). ∎

######  Theorem 3.4.

Let $p\geq 11$ be a prime. Then, there exists an abelian extension
$L/\mathbb{Q}$ and an abelian variety $R/L$ such that
$\left|\Sha(R/L)\right|=p\cdot\square.$

###### Proof.

We let $E/\mathbb{Q}$ be an elliptic curve defined over $\mathbb{Q}$ and $d$ a
positive integer such that $p$ and all bad primes of $E/\mathbb{Q}$ split in
$\mathbb{Q}(\sqrt{-d})$. For the remainder of this proof, we fix $n$ as in
Theorem 3.3 so that $\textup{rk}_{p}(E/L_{n+1})=\textup{rk}_{p}(E/L_{n})$ and
$\Sha(R/L_{n})$ is finite. In view of Lemma 2.5(1)-(2), we deduce that
$\left|\Sha(R/L_{n})\right|\equiv\left|{}_{0}(R/L_{n})[p^{\infty}]\right|$.
Combining this with Proposition 2.6(2) (applied with
$K=K_{n},F=K_{n+1},L=L_{n}$) and Lemma 3.2, we get

$\left|\Sha(R/L_{n})\right|\equiv p^{\textup{rk}_{p}(E/L_{n})}\ \mathrm{mod}\
\square.$

Since $E$ is defined over $\mathbb{Q}$, its global root number
$\omega(E/\mathbb{Q}(\sqrt{-d}))=-1$. This is because the only contribution in
the root number computation comes from the unique archimedean place of
$\mathbb{Q}(\sqrt{-d})$. Since
$\textup{Res}_{\mathbb{Q}}^{\mathbb{Q}(\sqrt{-d})}E$ is isogenous to $E\times
E_{-d}$, we deduce that

$(-1)^{\textup{rk}_{p}(E/\mathbb{Q}(\sqrt{-d}))}=(-1)^{\textup{rk}_{p}(E/\mathbb{Q})}(-1)^{\textup{rk}_{p}(E_{-d}/\mathbb{Q})}\stackrel{{\scriptstyle\textup{\cite[cite]{[\@@bibref{}{BSD-
mod-squares}{}{}, Thm 1.4]}}}}{{=}}w(E/\mathbb{Q})w(E_{-d}/\mathbb{Q}).$

In view of the isogeny,
$w(E/\mathbb{Q}(\sqrt{-d}))=w(E/\mathbb{Q})w(E_{-d}/\mathbb{Q})$, from which
we deduce that $\textup{rk}_{p}(E/\mathbb{Q}(\sqrt{-d}))\equiv 1\
\mathrm{mod}\ 2$. Since the parity of $p^{\infty}$-Selmer ranks remains
unchanged in odd degree Galois extensions [3, Corollary 4.15], then
$\textup{rk}_{p}(E/L_{n})\equiv\textup{rk}_{p}(E/\mathbb{Q}(\sqrt{-d}))\equiv
1\ \mathrm{mod}\ 2$. Combining this with (3) gives the required result. ∎

######  Theorem 3.5.

Let $p$ be a rational prime. Then, there exists an abelian variety
$R^{\prime}/\mathbb{Q}$ such that
$\left|\Sha(R^{\prime}/\mathbb{Q})\right|=p\cdot\square.$

###### Proof.

For $p\geq 11$, we consider $R^{\prime}=\textup{Res}^{L}_{\mathbb{Q}}R$ where
$R$ is the abelian variety from Theorem 3.4. Since
$\Sha(R^{\prime}/\mathbb{Q})\cong\Sha(R/L)$ (combine Shapiro’s lemma with [8,
pp. 178(a)]), the result follows by Theorem 3.4. For $p\leq 7$, see [10] and
[13]. ∎

######  Corollary 3.6.

For every positive square-free integer $n$, there exists an abelian variety
defined over $\mathbb{Q}$ with finite Tate–Shafarevich group of order
$n\cdot\square$.

###### Proof.

Given abelian varieties $A_{1},A_{2}$, $\Sha(A_{1}\times
A_{2})\cong\Sha(A_{1})\times\Sha(A_{2})$. The result follows from Theorem 3.5.
∎

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