Abelian extensions of Lie triple systems with derivations

1.   Introduction

Lie triple systems were introduced by E. Cartan in his studies on Riemannian geometry. Since then, it has been studied by many scholars. For example, Jacobson studied Lie triple systems by an algebraic method in ^[1,2]. The cohomology theories have been given by Yamaguti in ^[3] and some problems about cohomology have been solved, see ^[4,5,6]. Note that Lie triple systems are closely related to Lie algebras, so it is natural to generalize some properties of Lie algebras to Lie triple systems.

Automorphisms are very important subjects in the research of algebras. In ^[7], the authors studied extension of a pair of automorphisms of Lie algebras, and they gave a necessary and sufficient condition for a pair of automorphisms to be extensible. Since derivations are infinitesimals of automorphisms, it is interesting to study extension of a pair of derivations. Recently, extension of a pair of derivations on Lie algebras, $3$-Lie algebras, Leibniz algebras and associative algebras have been studied, refer to ^[8,9,10,11]. We attempt to consider the same problems on Lie triple systems. Inspired by ^[11], we define a Lie algebra $G_{\theta_A},$ where $\theta_A$ is a representation of a Lie triple system $\mathfrak{L}$ on a Lie triple system $A,$ using compatible derivations of $\mathfrak{L}$ and $A.$ Then we show that a certain representation of $G_{\theta_A}$ can characterize the extensibility of above compatible derivations.

This paper is organized as follows. In Section 2, we recall some basic definitions and properties of Lie triple systems. Then we construct a Lie algebra $G_{\theta_A}$ and consider its representation on $H^3(\mathfrak{L}, A)$ in Section 3. In Section 4, we investigate the abelian extension and extensibility of a pair of derivations. We prove that a compatible pair $(\partial_a, \partial_l)$ is extensible if and only if $[{\rm Ob}^{\tilde{\mathfrak{L}}}_{(\partial_a, \partial_l)}]\in H^3(\mathfrak{L}, A)$ is trivial and $(\partial_a, \partial_l)$ is extensible in every reversible extension if and only if $\Phi(\partial_a, \partial_l) = 0.$

In this paper, all Lie triple systems $\mathfrak{L}$ are defined over a fixed but arbitrary field $\mathbb{F}$.

2.   Cohomology of Lie triple systems

In this section, we first recall some basic definitions and properties of Lie triple systems, then we show that $H^3(\mathfrak{L}, A) = 0$ for an abelian extension implies split property.

Definition 2.1. ^[2] A Lie triple system is a vector space $\mathfrak{L}$ endowed with a trilinear operation $[\cdot, \cdot, \cdot]: \mathfrak{L}\times \mathfrak{L}\times \mathfrak{L}\longrightarrow \mathfrak{L}$ satisfying

for all $a, b, c, d, e \in \mathfrak{L}$.

For a Lie triple system $\mathfrak{L},$ a linear map $\partial:\mathfrak{L}\longrightarrow \mathfrak{L}$ is called a derivation of $\mathfrak{L},$ if for all $a, b, c\in \mathfrak{L},$

Denote by ${\rm Der}(\mathfrak{L})$ the space of derivations of $\mathfrak{L}.$

Definition 2.2. ^[3] An $\mathfrak{L}$-module is a vector space $V$ with a bilinear map

such that the following conditions hold:

where $D_V(a, b) = \theta_V(b, a)-\theta_V(a, b),$ for all $a, b, c, d\in \mathfrak{L}.$ Also $\theta_V$ is called a representation of $\mathfrak{L}$ on $V.$

Definition 2.3. ^[3] Let $\theta_V$ be a representation of a Lie triple system $\mathfrak{L}.$ An $n$-linear map $f:\underbrace{\mathfrak{L}\times\cdots\times \mathfrak{L}}_{n\; {\rm times}}\rightarrow V$ satisfying

and

is called an $n$-cochain of $\mathfrak{L}$ on $V.$ Denote by $C^{n}(\mathfrak{L}, V)$ the set of all $n$-cochains, for $n\geq 0.$

Definition 2.4. ^[3] Let $\mathfrak{L}$ be a Lie triple system and $\theta_V$ a representation of $\mathfrak{L}$ on $V.$ The coboundary operator $\delta:C^{2n-1}(\mathfrak{L}, V)\rightarrow C^{2n+1}(\mathfrak{L}, V)$ is given by

for any $x_1, x_2, ..., x_{2n+1}\in \mathfrak{L},$ $f\in C^{2n-1}(\mathfrak{L}, V).$

Let $\mathfrak{L}$ be a Lie triple system and $V$ an $\mathfrak{L}$-module. The set

is called the space of $(2n+1)$-cocycles of $\mathfrak{L}$ on $V$.

The set

is called the space of $(2n+1)$-coboundaries of $\mathfrak{L}$ on $V$.

The $n$-th cohomology group is

Note that, by Eq 2.1, for $f\in C^3(\mathfrak{L}, V)$ we have

Definition 2.5. ^[6] Suppose that $\mathfrak{L}$ and $A$ are Lie triple systems. If

is an exact sequence of Lie triple systems, and $[\tilde{\mathfrak{L}}, A, A] = 0$ (which implies that $[A, \tilde{\mathfrak{L}}, A] = [A, A, \tilde{\mathfrak{L}}] = 0),$ then we call $\tilde{\mathfrak{L}}$ an abelian extension of $\mathfrak{L}$ by $A.$ A linear map $s:\mathfrak{L} \rightarrow \tilde{\mathfrak{L}}$ is called a section of $\tilde{\mathfrak{L}}$ if it satisfies $\pi\circ s = {\rm id}_{\mathfrak{L}}.$ If there exists a section which is also a homomorphism between Lie triple systems, we say that the abelian extension is split.

The following properties have been proved in ^[6], which will be used in Sections 3 and 4.

Let $\tilde{\mathfrak{L}}$ be an abelian extension of $\mathfrak{L}$ by $A.$ We construct a representation of $\mathfrak{L}$ on $A.$ Fix any section $s:\mathfrak{L}\longrightarrow \tilde{\mathfrak{L}}$ of $\pi$ and define $\theta_A:\mathfrak{L}\times\mathfrak{L}\longrightarrow {\rm End}(A)$ by

for all $x, y\in \mathfrak{L},$ $v \in A.$ In particular, $D_A(x, y)(v) = (\theta_A(y, x)-\theta_A(x, y))(v) = [s(x), s(y), v]_{\tilde{\mathfrak{L}}}.$ Note that $\theta_A$ is independent on the choice of $s.$ Moreover, since

for any $x, y, z\in\mathfrak{L},$ we have a map $\omega: \mathfrak{L}\times\mathfrak{L}\times\mathfrak{L}\longrightarrow A$ given by

for all $x, y, z\in\mathfrak{L}.$

Lemma 2.6. Let $\tilde{\mathfrak{L}}$ be an abelian extension of a Lie triple system $\mathfrak{L}$ by $A.$ Then

$(1)$ $\theta_A$ given by Eq $2.3$ is a representation of $\mathfrak{L}$ on $A;$

$(2)$ $\omega$ given by Eq $2.4$ is a $3$-cocycle associated to $\theta_A.$

Corollary 2.7. Let $\tilde{\mathfrak{L}}$ be an abelian extension of a Lie triple system $\mathfrak{L}$ by $A.$ Keep the same notations as in Lemma $2.6$. Then the cohomology class $[\omega]$ does not depend on the choice of $s.$

Proposition 2.8. If $\theta_A$ is a representation of $\mathfrak{L}$ on $A$ and $\omega$ is a $3$-cocycle i.e., $\delta \omega = 0$ in Eq $2.2$ then $\mathfrak{L}\oplus A$ is a Lie triple system with the bracket given by

where $x, y, z\in \mathfrak{L},$ $u, v, w\in A.$

Corollary 2.9. Retain all the notations and assumptions in Proposition $2.8.$ Let $\pi:\mathfrak{L}\oplus A\longrightarrow \mathfrak{L}$ be the canonical projection. Then there is an abelian extension $\tilde{\mathfrak{L}} = \mathfrak{L}\oplus A$ of Lie triple systems $\mathfrak{L}$ by $A.$

Based on the previous notations, we have the following

Proposition 2.10. Let $\theta_A$ be a representation of a Lie triple system $\mathfrak{L}$ on a Lie triple system $A.$ If $H^3(\mathfrak{L}, A) = 0$ then any abelian extension of $\mathfrak{L}$ by $A$ is split.

Proof. It suffices to show that there is a section $s$ of $\pi$ which is a homomorphism. Recall that $\theta_A$ given by Eq 2.3 is independent on the choice of $s.$ Consider the $3$-cocycle $\omega$ given by Lemma 2.6, since $H^3(\mathfrak{L}, A) = 0,$ there exists an $\alpha\in C^1(\mathfrak{L}, A)$ such that $\omega = \delta\alpha.$ For any $x, y, z\in \mathfrak{L},$ it holds that

Define a linear map $s^\prime:\mathfrak{L}\longrightarrow \tilde{\mathfrak{L}}$ by $s^\prime(x) = s(x)-\alpha(x).$ Note that $s^\prime$ is also a section of $\pi.$ Then for any $x, y, z\in \mathfrak{L},$ we have

Therefore, $s^\prime$ is a homomorphism.

3.   Cohomology classes and a Lie algebra

In this section, $\mathfrak{L}$ and $A$ denote Lie triple systems. We choose derivations of $\mathfrak{L}$ and $A,$ and use these derivations to construct third-order cohomology classes. Based on these preparations, we construct a Lie algebra $G_{\theta_A}$ and its representation on $H^3(\mathfrak{L}, A).$

Given a representation $\theta_A$ of $\mathfrak{L}$ on $A.$ Suppose $\omega\in C^3(\mathfrak{L}, A).$ For any pair $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L}),$ define a $3$-cochain ${\rm Ob}_{(\partial_a, \partial_l)}^\omega\in C^3(\mathfrak{L}, A)$ as

or equivalently,

for all $x, y, z\in \mathfrak{L}.$

Lemma 3.1. Let $\theta_A$ be a representation of $\mathfrak{L}$ on $A$ and $\omega\in C^3(\mathfrak{L}, A)$ with respect to the representation $\theta_A.$ Assume that a pair $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L})$ satisfies that

for all $x, y\in \mathfrak{L}.$ If $\omega$ is a $3$-cocycle then ${\rm Ob}_{(\partial_a, \partial_l)}^\omega$ given by Eq $3.1$ is also a $3$-cocycle.

Proof. We only need to prove that $\delta{\rm Ob}_{(\partial_a, \partial_l)}^\omega = 0.$ Since $\omega$ is a $3$-cocycle, $\delta\omega = 0,$ by Eq 2.2 it follows that, for any $x_1, x_2, x_3, x_4, x_5\in\mathfrak{L},$

Then for any $x_1, x_2, x_3, x_4, x_5\in\mathfrak{L},$ we have

Since $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L})$ and ${\rm Ob}_{(\partial_a, \partial_l)}^\omega$ satisfies Eq 3.1, we have

For Eqs 3.5–3.12, by suitable combination and with the aid of Eq 3.3, we get the following

Next, let us substituting Eqs 3.13–3.17 into Eq 3.4 having the following

By Eq 3.3, we have

Here, inserting Eq 3.19 into Eq 3.18 gives that

Since $(\partial_a, \partial_l)$ satisfies Eq 3.2, we only need to show that

by$D_A(x, y) = \theta_A(y, x)-\theta_A(x, y),$ we have

Then we have $\delta{\rm Ob}^\omega_{(\partial_a, \partial_l)} = 0$ as required.

Definition 3.2. Let $\theta_A$ be a representation of $\mathfrak{L}$ on $A.$ A pair $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L})$ is called compatible with respect to $\theta_A$ if Eq 3.2 holds.

Based on the previous works, we are ready to construct a Lie algebra and its representation on the third cohomology group. Set

Lemma 3.3. There is a linear map $\Phi:G_{\theta_A}\longrightarrow {\rm End}(H^3(\mathfrak{L}, A))$ given by

where ${\rm Ob}^\omega_{(\partial_a, \partial_l)}$ is given by Eq $3.1.$

Proof. By Lemma 3.1 and $(\partial_a, \partial_l)$ is compatible with respect to $\theta_A$ it follows that ${\rm Ob}^\omega_{(\partial_a, \partial_l)}$ is a $3$-cocycle whenever $\omega$ is a $3$-cocycle. Therefore, we only need to show that if $\delta\lambda$ is a $3$-coboundary, then $\Phi(\partial_a, \partial_l)(\delta\lambda) = 0,$ which means that $\Phi$ is well-defined.

Since $\partial_l$ is a derivation, we have

Then

it is obvious that $\Phi$ is a linear map.

We will end this section with the following conclusion.

Theorem 3.4. Let $\theta_A$ be a representation of $\mathfrak{L}$ on $A.$ Then $G_{\theta_A}$ is a Lie subalgebra of ${\rm Der}(A)\times {\rm Der}(\mathfrak{L})$ and the map $\Phi$ given by Eq $3.20$ is a Lie algebra homomorphism.

Proof. Note that $G_{\theta_A}$ is a subalgebra of ${\rm Der}(A)\times {\rm Der}(\mathfrak{L}).$ First, we prove that if $(\partial_{a_1}, \partial_{l_1}),$ $(\partial_{a_2}, \partial_{l_2})\in G_{\theta_A},$ then the commutator $[(\partial_{a_1}, \partial_{l_1}), (\partial_{a_2}, \partial_{l_2})]\in G_{\theta_A}.$ For all $x, y\in \mathfrak{L},$ we have

By Eq 3.2 it follows that

Then taking Eqs 3.22 and 3.23 into Eq 3.21 gives that

which implies that $[(\partial_{a_1}, \partial_{l_1}), (\partial_{a_2}, \partial_{l_2})]$ is compatible.

Next, for any $(\partial_{a_1}, \partial_{l_1}),$ $(\partial_{a_2}, \partial_{l_2})\in G_{\theta_A},$ $[\omega]\in H^3(\mathfrak{L}, A),$ we have

By Eqs 3.1 and 3.20, we have

Similarly, one obtains

Inserting the above two identities into Eq 3.24, we deduce

as desired.

4.   Abelian extensions and extensibility of derivations

In this section, we construct obstruction classes for extensibility of derivations by Lemma 3.1. Also, for $G_{\theta_A}$ we give a representation in terms of extensibility of derivations.

Lemma 4.1. Keep notation as above. The cohomology class $[{\rm Ob}^\omega_{(\partial_a, \partial_l)}]\in H^3(\mathfrak{L}, A)$ does not depend on the choice of sections of $\pi.$ $($Hence we will denote $[{\rm Ob}^\omega_{(\partial_a, \partial_l)}]$ by $[{\rm Ob}^{\tilde{\mathfrak{L}}}_{(\partial_a, \partial_l)}]$ $).$

Proof. Suppose that $s_1$ and $s_2$ are sections of $\pi$ and $\omega_1,$ $\omega_2$ are defined by Eq 2.4, while ${\rm Ob}^{\omega_1}_{(\partial_a, \partial_l)},$ ${\rm Ob}^{\omega_2}_{(\partial_a, \partial_l)}$ are defined by Eq 3.1 with respect to $\omega_1,$ $\omega_2.$ Then

A map $\lambda:\mathfrak{L}\longrightarrow A$ is defined by $\lambda(x) = s_1(x)-s_2(x),$ for all $x\in \mathfrak{L}.$ Recall the proof of Corollary 2.7, we have

for any $x, y, z\in \mathfrak{L}.$ So

Similarly, we have

and

By $I_1,$ $I_2,$ $I_3,$ $I_4$ and Eq $4.1$ have

Since $(\partial_a, \partial_l)$ satisfies Eq 3.2 it follows that

i.e., $[{\rm Ob}^{\omega_1}_{(\partial_a, \partial_l)}] = [{\rm Ob}^{\omega_2}_{(\partial_a, \partial_l)}]\in H^3(\mathfrak{L}, A),$ so we have done the proof.

Next, we will define the extensibility of derivations, and we will give a necessary and sufficient condition that $(\partial_a, \partial_l)$ is extensible.

Definition 4.2. Let $\tilde{\mathfrak{L}}$ be an abelian extension of a Lie triple system $\mathfrak{L}$ by $A.$ A pair $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L})$ is called extensible if there is a derivation $\partial_{\tilde{l}}\in {\rm Der}(\tilde{\mathfrak{L}})$ such that the diagram

is commutative, where $\iota:A\longrightarrow \tilde{\mathfrak{L}}$ is the inclusion map.

Proposition 4.3. Assume that $\tilde{\mathfrak{L}}$ is an abelian extension of a Lie triple system $\mathfrak{L}$ by $A.$ If $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L})$ is extensible, then $(\partial_a, \partial_l)$ is compatible with respect to $\theta_A.$

Proof. Since $(\partial_a, \partial_l)$ is extensible, there is a derivation $\partial_{\tilde{l}}\in {\rm Der}(\tilde{\mathfrak{L}})$ such that the diagram $(4.2)$ is commutative, i.e., $\iota\circ \partial_a = \partial_{\tilde{l}}\circ \iota,$ hence there is a map $\mu:\mathfrak{L}\longrightarrow A$ given by

Since $\iota\circ \partial_a = \partial_{\tilde{l}}\circ \iota,$ (equivalent, $\partial_{\tilde{l}}|_{A} = \partial_a$) and $\partial_{\tilde{l}}\in {\rm Der}(\tilde{\mathfrak{L}}),$ we obtain

hence $(\partial_a, \partial_l)$ is compatible.

Theorem 4.4. Let $\tilde{\mathfrak{L}}$ be an abelian extension of a Lie triple system $\mathfrak{L}$ by $A$ and $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L})$ compatible with respect to $\theta_A.$ Then $(\partial_a, \partial_l)$ is extensible if and only if $[{\rm Ob}^{\tilde{\mathfrak{L}}}_{(\partial_a, \partial_l)}]\in H^3(\mathfrak{L}, A)$ is trivial.

Proof. $(\Rightarrow)$ Fix any linear section $s$ of $\pi.$ Since $(\partial_a, \partial_l)$ is extensible, there exists a derivation $\partial_{\tilde{l}}\in {\rm Der}(\tilde{\mathfrak{L}})$ such that the associated diagram (4.2) is commutative. By $\pi\circ \partial_{\tilde{l}} = \partial_l\circ \pi,$ we have $\partial_{\tilde{l}}(s(x))-s(\partial_l(x))\in A,$ for $x\in \mathfrak{L},$ then there exists a map $\mu:\mathfrak{L}\longrightarrow A$ given byEq 4.3. It sufficient to show that

for all $x, y, z\in\mathfrak{L}.$ Next, we compute both hand sides of the following identity

for any $x_1, x_2, x_3\in \mathfrak{L},$ $v_1, v_2, v_3\in A.$

At first, since $\tilde{\mathfrak{L}}$ is an abelian extension of $\mathfrak{L}$ by $A,$ we have $[\tilde{\mathfrak{L}}, A, A] = 0$ and

by Eq 2.4 the left-hand side of Eq 4.5 is

Since the diagram $(4.2)$ is commutative, we have

Now we compute the right-hand side of Eq 4.5. Note that, since $\partial_{\tilde{l}}|_A = \partial_a,$ it holds that

By this the right-hand side of Eq 4.5 is

By Eqs 4.6 and 4.7, we have

Then

Since $(\partial_a, \partial_l)$ is compatible and by the proof of Lemma 3.1,

Thus we have

which is exactly Eq 4.4 due to Eq 3.1 and the definition of $3$-cohomology group. So $[{\rm Ob}^{\tilde{\mathfrak{L}}}_{(\partial_a, \partial_l)}] = 0$ as required.

$(\Leftarrow)$ Assume that $[{\rm Ob}^{\tilde{\mathfrak{L}}}_{(\partial_a, \partial_l)}]$ is trivial. Then there is a map $\mu:\mathfrak{L}\longrightarrow A$ such that ${\rm Ob}^{\tilde{\mathfrak{L}}}_{(\partial_a, \partial_l)} = \delta\mu.$ For any $s(x)+v\in \tilde{\mathfrak{L}},$ define $\partial_{\tilde{l}}:\tilde{\mathfrak{L}}\longrightarrow \tilde{\mathfrak{L}}$ by

then the associated diagram in (4.2) is commutative: for any $x\in \mathfrak{L},$ $v\in A,$

Moreover, since $(\partial_a, \partial_l)$ satisfies Eq 3.2, by Eqs 4.8 and 4.9, it follows that Eq 4.5 holds by Eqs 4.6 and 4.7, that is, $\partial_{\tilde{l}}\in {\rm Der}(\tilde{\mathfrak{L}}).$

Hence, for a compatible pair $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L}),$ the cohomology class $[{\rm Ob}^{\tilde{\mathfrak{L}}}_{(\partial_a, \partial_l)}]$ can be regarded as an obstruction class to extensibility.

By Proposition 4.3 and Theorem 4.4 we have the following result.

Corollary 4.5. Let $\tilde{\mathfrak{L}}$ be an abelian extension of a Lie triple system $\mathfrak{L}$ by $A.$ If $H^3(\mathfrak{L}, A) = 0,$ then any pair $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L})$ is compatible if and only if it is extensible.

We know that the condition $H^3(\mathfrak{L}, A) = 0$ is in general not equivalent to split property of extensions. However, we still have the following result.

Corollary 4.6. Let $\tilde{\mathfrak{L}}$ be a split abelian extension of a Lie triple system $\mathfrak{L}$ by $A.$ Then any pair $(\partial_a, \partial_l)\in {\rm Der}(A)\times {\rm Der}(\mathfrak{L})$ is compatible if and only if it is extensible.

Proof. $(\Leftarrow)$ It holds due to Proposition 4.3.

$(\Rightarrow)$ Since the extension is split there exists a section $s^\prime,$ which is a homomorphism. Let $\theta_{s^\prime},$ (resp. $\omega_{s^\prime}$) be defined by Eq 2.3 (resp. Eq 2.4) with respect to $s^\prime.$ Then we get $\omega_{s^\prime} = 0.$ By Eq 3.1, we have ${\rm Ob}^{\omega_{s^\prime}}_{(\partial_a, \partial_l)} = 0.$ In view of Lemmas 3.1 and 4.1, one has $[{\rm Ob}^{\tilde{\mathfrak{L}}}_{(\partial_a, \partial_l)}] = [{\rm Ob}^{\omega_{s^\prime}}_{(\partial_a, \partial_l)}] = 0.$ Then by Theorem 4.4, we deduce that $(\partial_a, \partial_l)$ is extensible as required.

Let us end this section with the relation between the representation $\Phi$ of $G_{\theta_A}$ and extensibility of derivations.

Definition 4.7. Let $\theta_A$ be a representation of $\mathfrak{L}$ on $A$ and $\tilde{\mathfrak{L}}$ an abelian extension of Lie triple systems $\mathfrak{L}$ by $A.$ So there exists a section $s$ of $\pi$ such that the representation $\theta_s$ defined by Eq 2.3 is the same as $\theta_A,$ then the extension is called reversible with respect to $\theta_A.$

Example 4.8. Let $\theta_A$ be a representation of $\mathfrak{L}$ on $A.$ By Proposition 2.8 we have a Lie triple system $\tilde{\mathfrak{L}}: = \mathfrak{L}\oplus A$ with the bracket given by

where $x, y, z\in \mathfrak{L},$ $u, v, w\in A.$ By Corollary 2.9, we have an abelian extension

and $p$ is the canonical projection. Choose a section $s$ of $p$ given by $s(x) = x,$ $x\in \mathfrak{L}.$ Then $s:\mathfrak{L}\longrightarrow \tilde{\mathfrak{L}}$ is a homomorphism, and hence the representation $\theta_s$ given by Eq 2.3 is the same as $\theta_A$ which means that the extension

is reversible with respect to $\theta_A.$

Theorem 4.9. Let $\theta_A$ be a representation of $\mathfrak{L}$ on $A.$ The pair $(\partial_a, \partial_l)\in G_{\theta_A}$ is extensible in every reversible extension if and only if $\Phi(\partial_a, \partial_l) = 0.$

Proof. $(\Rightarrow)$ For any $[\varphi]\in H^3(\mathfrak{L}, A),$ by Corollary 2.9, there is an abelian extension

where $p$ is the canonical projection and the bracket on $\tilde{\mathfrak{L}}: = \mathfrak{L}\oplus A$ is given by

for any $x, y, z\in\mathfrak{L},$ $u, v, w\in A.$ Choose a section $s$ of $p$ defined by $s(x) = x,$ for any $x\in \mathfrak{L}.$ Then the representation $\theta_s$ is given by Eq 2.3. Let $H^n_{\theta_s}(\mathfrak{L}, A)$ denote the cohomology group with respect to $\theta_s.$ We first show that $\theta_s$ is the same as $\theta_A$ for $x, y\in \mathfrak{L},$ $v\in A,$

Hence we obtain that the cohomology group $H^n_{\theta_s}(\mathfrak{L}, A)$ associated to $\theta_s$ is the same as $H^n(\mathfrak{L}, A).$ Since we define $\omega_s$ by Eq 2.4 (resp. $\varphi$) is a $3$-cocycle in $H^n_{\theta_s}(\mathfrak{L}, A)$ (resp. in $H^n(\mathfrak{L}, A)$), we have $[\omega_s] = [\varphi].$ Then

$(\Leftarrow)$ Suppose $\Phi(\partial_a, \partial_l) = 0.$ For any reversible abelian extension

there exists a section $s$ of $\pi$ such that the representation $\theta_s$ is the same as $\theta_A.$ Therefore, $\omega_s$ defined by Eq 2.4 is a $3$-cocycle in $H^3(\mathfrak{L}, A).$ Then we have

By Theorem 4.4, $(\partial_a, \partial_l)$ is extensible. This completes the proof.

The following corollary is straightforward.

Corollary 4.10. Let $\theta_A$ be a representation of $\mathfrak{L}$ on $A.$ Then any pair $(\partial_a, \partial_l)\in G_{\theta_A}$ is extensible in every reversible extension if and only if $\Phi\equiv0.$

Acknowledgments

The second author is supported by NNSF of China (Nos. 11801066, 11771410). The third author is supported by NNSF of China (Nos.11771069, 12071405).

Conflict of interest

The authors declare there is no conflicts of interest.