1 Introduction

3-Lie algebras are special types of n-Lie algebras and have close relationships with many important fields in mathematics and mathematical physics [4, 5]. The structure of 3-Lie algebras is closely linked to the supersymmetry and gauge symmetry transformations of the world-volume theory of multiple coincident M2-branes and is applied to the study of the Bagger-Lambert theory. Moreover, the n-Jacobi identity can be regarded as a generalized Plucker relation in the physics literature. In particular, the metric 3-Lie algebras, or more generally, the 3-Lie algebras with invariant symmetric bilinear forms attract even more attention in physics. Recently, many more properties and structures of 3-Lie algebras have been developed, see [6, 8, 16, 22, 27, 31, 32] and references cited therein.

Derivations on algebraic structures were first started by Ritt [25] for commutative algebras and field. The structure is called a differential (commutative) algebra. For the notion of differential n-Lie algebras and related structures, see [7, 10, 21]. Derivations of types of algebra provide many important aspects of the algebraic structure. For example, Coll, Gertstenhaber, and Giaquinto [11] described explicitly a deformation formula for algebras whose Lie algebra of derivations contains the unique non-abelian Lie algebra of dimension two. Amitsur [1, 2] studied derivations of central simple algebras. Derivations are also used to construct homotopy Lie algebras [30] and play an important role in the study of differential Galois theory [24]. One may also look at some interesting roles played by derivations in control theory and gauge theory in quantum field theory [3]. In [15, 23], the authors studied algebras with derivations from an operadic point of view. Recently, Lie algebras with derivations (called LieDer pairs) are studied from a cohomological point of view [29] and extensions, deformations of LieDer pairs are considered. The results of [29] have been extended to associative algebras and Leibniz algebras with derivations in [12] and [13].

The deformation is a tool to study a mathematical object by deforming it into a family of the same kind of objects depending on a certain parameter. The deformation theory was introduced by Gerstenhaber for rings and algebras [18, 19], and by Zhang for 3-Lie color algebras [32]. They studied 1-parameter formal deformations and established the connection between the cohomology groups and infinitesimal deformations. Motivated by Tang’s [29] terminology of LieDer pairs. Due to the importance of 3-Lie algebras, cohomology, and deformation theories, Our main objective of this paper is to study the cohomology and deformation theory of 3-Lie algebra with a derivation.

The paper is organized as follows. In Sect. 2, we define a cohomology theory for 3-LieDer pair with coefficients in a representation. In Sect. 3, we study central extensions of a 3-LieDer pair and show that isomorphic classes of central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation. In Sect. 4, we study formal one-parameter deformations of 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivations.

Throughout this paper, we work over the field \({\mathbb {F}}\) of characteristics 0.

2 Cohomology of 3-LieDer pairs

In this section, we define a cohomology theory for 3-LieDer pair with coefficients in a representation.

Definition 2.1

[17] A 3-Lie algebra is a tuple \((L, [\cdot , \cdot , \cdot ])\) consisting of a vector space L, a 3-ary skew-symmetric operation \([\cdot ,\cdot ,\cdot ]: \wedge ^{3}L\rightarrow L\) satisfying the following Jacobi identity

$$\begin{aligned} {[}x, y, [u,v,w]]= & {} [[x,y,u], v, w] +[u, [x,y,v], w]+[u, v,[x,y,w]], \end{aligned}$$
(2.1)

for any \(x, y, u, v,w \in L\).

Definition 2.2

[20] A representation of a 3-Lie algebra \((L, [\cdot , \cdot , \cdot ])\) on the vector space M is a linear map \(\rho : L \wedge L \rightarrow {{\mathfrak {g}}}{{\mathfrak {l}}}(M)\), such that for any \(x, y, z, u\in L\), the following equalities are satisfied

$$\begin{aligned} \rho ([x,y,z], u)= & {} \rho (y, z)\rho (x,u)+\rho (z, x)\rho (y,u)+\rho (x, y)\rho (z,u),\\ \rho (x, y)\rho (z,u)= & {} \rho (z, u)\rho (x,y)+\rho ([x,y,z], u)+\rho (z, [x,y,u]). \end{aligned}$$

Then \((M, \rho )\) is called a representation of L, or M is an L-module.

Definition 2.3

[17] Let \((L, [\cdot , \cdot , \cdot ])\) be a 3-Lie algebra. A derivation on L is given by a linear map \(\phi _L: L \rightarrow L\) satisfying

$$\begin{aligned} \phi _L([x, y, z])=[\phi _L(x), y, z]+[x, \phi _L(y), z]+[x, y, \phi _L(z)],~~~~~\forall x, y, z \in L. \end{aligned}$$

We call the pair \((L, \phi _L)\) of a 3-Lie algebra and a derivation by a 3-LieDer pair.

Remark 2.4

Let \((L, [\cdot , \cdot , \cdot ])\) be a 3-Lie algebra. For all \(x_1, x_2 \in L\), the map defined by

$$\begin{aligned} ad_{x_1, x_2} x := [x_1, x_2, x],~\text {for all}~x\in L, \end{aligned}$$

is called the adjoint map. From the Eq. 2.1, it is clear that \(ad_{x_1, x_2}\) is a derivation. The linear map \(ad : L\wedge L \rightarrow {{\mathfrak {g}}}{{\mathfrak {l}}}(L)\) defines a representation of \((L, [\cdot , \cdot , \cdot ])\) on itself. This representation is called the adjoint representation.

Definition 2.5

Let \((L, \phi _L)\) be a 3-LieDer pair. A representation of \((L, \phi _L)\) is given by \((M, \phi _M)\) in which M is a representation of L and \(\phi _M: M\rightarrow M\) is a linear map satisfying

$$\begin{aligned}&\phi _M(\rho (x,y)(m))=\rho (\phi _L(x),y)(m)+\rho (x,\phi _L(y))(m)+\rho (x,y)(\phi _M(m)), \end{aligned}$$

for all \(x, y\in L\) and \(m\in M\).

Proposition 2.6

Let \((L, \phi _L)\) be a 3-LieDer pair and \((M, \phi _M)\) be a representation of it. Then \((L\oplus M, \phi _L\oplus \phi _{M})\) is a 3-LieDer pair where the 3-Lie algebra bracket on \(L\oplus M\) is given by the semi-direct product

$$\begin{aligned}{}[(x, m), (y, n), (z, p)]=([x, y, z], \rho (y,z)(m)+\rho (z,x)(n) + \rho (x,y)(p)), \end{aligned}$$

for any \(x, y, z\in L\) and \(m, n, p\in M\).

Proof

It is known that \(L\oplus M\) equipped with the above product is a 3-Lie algebra. Moreover, we have

$$\begin{aligned}&(\phi _L\oplus \phi _{M})([(x, m), (y, n), (z, p)])\\&\quad = (\phi _L([x, y, z]), \phi _{M}(\rho (y,z)(m))+\phi _{M}(\rho (z,x)(n))+\phi _M(\rho (x,y)(p)))\\&\quad = ([\phi _L(x), y, z], \rho (y,z)(\phi _M(m))+\rho (\phi _L(x),z)(n) + \rho (\phi _L(x),y)(p))\\&\qquad + ([x, \phi _L(y), z], \rho (\phi _L(y),z)(m)+\rho (z,x)(\phi _M(n)) + \rho (x,\phi _L(y))(p))\\&\qquad +([x, y, \phi _T(z)], \rho (y,\phi _L(z))(m)+\rho (z,\phi _L(x))(n) + \rho (x,y)(\phi _{M}(p)))\\&\quad = [(\phi _L\oplus \phi _{M})(x, m), (y, n), (z, p)]+[(x, m), (\phi _L\oplus \phi _{M})(y, n), (z, p)]\\&\qquad +[(x, m), (y, n), (\phi _L\oplus \phi _{M})(z, p)]. \end{aligned}$$

Hence the proof is finished. \(\square \)

Recall from [28] that let \(\rho \) be a representation of \((L,[\cdot , \cdot ,\cdot ])\) on M. Denote by \(C^n(L,M)\) the set of all n-cochains and defined as

$$\begin{aligned} C^n(L,M) = \text {Hom}((\wedge ^2 L)^{\otimes n-1}, M),~n\ge 1. \end{aligned}$$

Let \(d^n: C^{n}(L, M) \rightarrow C^{n+1}(L, M)\) be defined by

$$\begin{aligned}&d^n f(X_1,\ldots , X_n, x_{n+1})\\&\quad =(-1)^{n+1}\rho (y_{n}, x_{n+1})f(X_1,\ldots , X_{n-1}, x_{n})\\&\qquad +(-1)^{n+1}\rho ( x_{n+1}, x_{n})f(X_1,\ldots ,X_{n-1},y_{n})\\&\qquad +\sum _{j=1}^n(-1)^{j+1}\rho (x_{j}, y_{j}) f(X_1,\ldots ,\hat{X}_{j},\ldots , X_{n}, x_{n+1})\\&\qquad +\sum _{j=1}^n(-1)^{j} f(X_1,\ldots ,\hat{X}_{j},\ldots , X_n, [x_{j}, y_{j}, x_{n+1}]),\\&\qquad + \sum _{1\le j<k\le n}(-1)^{j}f(X_1,\ldots ,\hat{X}_{j},\ldots , X_{k-1}, [x_{j}, y_{j}, x_{k}]\wedge y_k\\&\qquad +x_k\wedge [x_{j}, y_{j}, x_{k}],X_{k+1},\ldots ,X_{n}, x_{n+1}), \end{aligned}$$

for all \(X_i=x_i\wedge y_i\in \otimes ^2L,i=1, 2, \ldots , n\) and \(x_{n+1}\in L\), it was proved that \(d^{n+1}\circ d^n=0\). Therefore, \((C^*(L, M), d^*)\) is a cochain complex.

Observe that for trivial representation coboundary maps \(d^1\) and \(d^2\) are explicitly given as follows:

$$\begin{aligned} d^1(f)(a,b,c)= & {} [f(a),b,c] + [a, f(b), c] + [a, b, f(c)]- f([a,b,c]),~f\in C^1(L, M). \\ d^2(f)(a,b,c,d,e)= & {} [a,b, f(c,d,e)]- f([a,b,c],d,e) + f(a,b,[c,d,e])\\&-[f(a,b,c),d,e],~f\in C^2(L, M). \end{aligned}$$

In [26], the graded space \(C^{*} (L, L) = \bigoplus _{n\ge 0}C^{n+1} (L, L)\) of cochain groups carries a degree -1 graded Lie bracket given by \([f, g] = f \circ g- (-1)^{mn} g \circ f\), for \(f \in C^{m+1} (L, L), g \in C^{n+1} (L, L)\), where \(f \circ g \in C^{m+n+1} (L, L)\), and defined as follows:

$$\begin{aligned}&f\circ g(X_1,\ldots , X_{m+n}, x)\\&\quad = \sum _{k=1}^{m}(-1)^{(k-1)n} \sum _{\sigma \in {\mathbb {S}}(k-1, n)}f(X_{\sigma (1)}, \ldots , X_{\sigma (k-1)}, g(X_{\sigma (k)}, \cdots , X_{\sigma (k+n-1)}, x_{k+n})\\&\qquad \wedge y_{k+n},X_{\sigma (k+n+1)}, \ldots , X_{\sigma (m+n)}, x) \\&\qquad +\sum _{k=1}^{m}(-1)^{(k-1)n} \sum _{\sigma \in {\mathbb {S}}(k-1, n)}(-1)^{\sigma } f(X_{\sigma (1)}, \ldots , X_{\sigma (k-1)},x_{k+n}\\&\qquad \wedge g(X_{\sigma (k)}, \ldots , X_{\sigma (k+n-1)},y_{k+n}), X_{k+n+1}, \ldots , X_{m+n}, x)\\&\qquad \sum _{\sigma \in {\mathbb {S}}(m, n)}(-1)^{mn}(-1)^{\sigma }f(X_{\sigma (1)}, \ldots , X_{\sigma (m)}, g(X_{\sigma (m+1)}, \ldots , X_{\sigma (m+n-1)},X_{\sigma (m+n)}, x)), \end{aligned}$$

for all \(X_i=x_i\wedge y_i\in \otimes ^2L,i=1, 2, \ldots , m+n\) and \(x\in L\). Here \({\mathbb {S}}(k-1, n)\) denotes the set of all \((k-1, n)\)-shuffles. Moreover, \(\mu : \otimes ^3L\rightarrow L\) is a 3-Lie bracket if and only if \([\mu , \mu ] = 0\), i.e. \(\mu \) is a Maurer–Cartan element of the graded Lie algebra \((C^{*} (L, L), [\cdot , \cdot ]\). where \(\mu \) is considered as an element in \(C^2 (L, L)\). With this notation, the differential (with coefficients in L) is given by

$$\begin{aligned} df=(-1)^{n}[\mu , f], ~~~~\text {for all}~ f\in C^{n} (L, L). \end{aligned}$$

In the next, we introduce cohomology for a 3-LieDer pair with coefficients in a representation.

Let \((L, \phi _L)\) be a 3-LieDer pair and \((M, \phi _{M})\) be a representation of it. For any \(n\ge 2\), we define cochain groups for 3-LieDer pair as follows:

$$\begin{aligned} C^n_{\text {3-LieDer}}(L, M) := C^n(L, M)\oplus C^{n-1}(L, M). \end{aligned}$$

Define the space \(C^0_{\text {3-LieDer}} (L,M)\) of 0-cochains to be 0 and the space \(C^1_{\text {3-LieDer}} (L,M)\) of 1-cochains to be Hom(L, M). Note that \(\mu = [\cdot ,\cdot ,\cdot ] \in C^2(L, L)\) and derivation \(\phi _L \in C^1(L, L) \). Thus, the pair \((\mu , \phi _L) \in C^2_{\text {3-LieDer}}(L, L)\). To define the coboundary map for 3-LieDer pair, we need following map \(\delta : C^n(L, M)\rightarrow C^n(L, M)\) by

$$\begin{aligned} \delta f=\sum _{i=1}^n f\circ (Id_L\otimes \cdot \cdot \cdot \otimes \phi _L\otimes \cdot \cdot \cdot \otimes Id_L)-\phi _{M}\circ f. \end{aligned}$$

The following lemma shows maps \(\partial \) and \(\delta \) commute, and is useful to define the coboundary operator of the cohomology of 3-LieDer pair.

Lemma 2.7

The map \(\delta \) commute with d, i.e, \(d\circ \delta =\delta \circ d\).

Proof

Note that in case of self representation, that is, when \((M, \phi _M) = (L, \phi _L)\), we have

$$\begin{aligned} \delta (f) = - [\phi _L, f],~\text {for all}~f\in C^n(L, L). \end{aligned}$$

Therefore, we have

$$\begin{aligned} {}(d\circ \delta )(f)&= -d[\phi _L, f{]}\\&= (-1)^n[\mu , [\phi _L, f]{]}\\&= (-1)^n [[\mu ,\phi _L], f] + (-1)^n[\phi _L,[\mu ,f]{]}\\&= (-1)^n[\phi _L,[\mu ,f]{]}\\&= (\delta \circ \delta )(f) \end{aligned}$$

\(\square \)

We are now in a position to define the cohomology of the 3-LieDer pair. We define a map \(\partial : C^n_{\text {3-LieDer}} (L,M)\rightarrow C^{n+1}_{\text {3-LieDer}} (L,M)\) by

$$\begin{aligned} \partial f= & {} (d f, -\delta f), ~~~\text{ for } \text{ all }~~~ f\in C^1_{\text {3-LieDer}} (L,M),\\ \partial (f_n, \overline{f_n})= & {} (df_n, d{\overline{f}}_n+(-1)^{n}\delta f_n),~~~\text{ for } \text{ all }~~~(f_n, {\overline{f}}_n)\in C^{n}_{\text {3-LieDer}} (L,M). \end{aligned}$$

Proposition 2.8

The map \(\partial \) satisfies \(\partial \circ \partial =0\).

Proof

For any \(f\in C^1_{\text {3-LieDer}} (L,M)\), we have

$$\begin{aligned} (\partial \circ \partial ) f= \partial (d f, -\delta f)=((d\circ d)f, -(d\circ \delta ) f+ (\delta \circ d) f )=0. \end{aligned}$$

Similarly, for any \((f_n, {\overline{f}}_n)\in C^{n}_{\text {3-LieDer}} (L,M)\), we have

$$\begin{aligned} \qquad \qquad \qquad (\partial \circ \partial ) (f_n, \overline{f_n})= & {} \partial (df_n, df_n+(-1)^{n}f_n)\\= & {} (d^2f_n, d^2\overline{f_n}+(-1)^{n}d\delta f_n+(-1)^{n+1}\delta df_n)\\= & {} 0.\square \end{aligned}$$

Therefore, \((C^{*}_{\text {3-LieDer}} (L,M), \partial )\) forms a cochain complex. We denote the corresponding cohomology groups by \(H^{*}_{\text {3-LieDer}} (L,M)\).

3 Central extensions of 3-LieDer pairs

In this section, we study central extensions of a 3-LieDer pair. Similar to the classical cases, we show that isomorphic classes of central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation.

Let \((L, \phi _L)\) be a 3-LieDer pair and \((M, \phi _{M})\) be an abelian 3-LieDer pair i.e, the 3-Lie algebra bracket of M is trivial.

Definition 3.1

A central extension of \((L, \phi _L)\) by \((M, \phi _{M})\) is an exact sequence of 3-LieDer pairs

(3.1)

such that \([i(m), {\hat{x}}, {\hat{y}}]=0\), for all \(m\in M\) and \({\hat{x}}, {\hat{y}}\in {\hat{L}}\).

In a central extension, using the map i we can identify M with the corresponding subalgebra of \({\hat{L}}\) and with this \(\phi _{M}=\phi _{{\hat{L}}}|_{M}\).

Definition 3.2

Two central extensions \(({\hat{L}}, \phi _{{\hat{T}}})\) and \((\hat{L'}, \phi _{\hat{L'}})\) are said to be isomorphic if there is an isomorphism \(\eta : ({\hat{L}}, \phi _{{\hat{L}}})\rightarrow (\hat{L'}, \phi _{\hat{L'}})\) of 3-LieDer pairs that makes the following diagram commutative

Let Eq. (3.1) be a central extension of \((L, \phi _L)\). A section of the map p is given by a linear map \(s : L\rightarrow {\hat{L}}\) such that \(p\circ s=Id_L\).

For any section s, we define linear maps \(\psi : L\wedge L\wedge L \rightarrow M\) and \(\chi : L\rightarrow M\) by

$$\begin{aligned} \psi (x, y, z):= & {} [s(x), s(y), s(z)]-s([x, y, z]),~~~~~\chi (x)=\phi _{{\hat{L}}}(s(x))-s(\phi _L(x)),\\&\text {for all}~ x, y, z\in L. \end{aligned}$$

Note that the vector space \({\hat{L}}\) is isomorphic to the direct sum \(L\oplus M\) via the section s. Therefore, we may transfer the structures of \({\hat{L}}\) to \(L\oplus M\). The product and linear maps on \(L\oplus M\) are given by

$$\begin{aligned}{}[(x, m), (y, n), (z, p)]_{\psi }= & {} ([x, y, z], \psi (x, y, z)), \\ \phi _{L\oplus M}(x, m)= & {} (\phi _L(x), \phi _{M}(m)+\chi (x)). \end{aligned}$$

Proposition 3.3

The vector space \(L\oplus M\) equipped with the above product and linear maps \(\phi _{L\oplus M}\) forms a 3-LieDer pair if and only if \((\psi , \chi )\) is a 2-cocycle in the cohomology of the 3-LieDer pair \((L, \phi _L)\) with coefficients in the trivial representation M. Moreover, the cohomology class of \((\psi , \chi )\) does not depend on the choice of the section s.

Proof

The tuple \((L \oplus M, \phi _{L\oplus M})\) is a 3-LieDer pair if and only if the following equations holds:

$$\begin{aligned}&[(x, m), (y, n), [(z, p), (v, k), (w, l)]_{\psi }]_\psi \nonumber \\&\quad = [[(x, m),(y,n),(z,p)]_\psi , (v,k), (w,l)]_\psi + [ (z,p),[(x,m),(y,n),(v,k)]_\psi ,(w,l)]_\psi \nonumber \\&\qquad +[(z,p), (v,k),[(x,m),(y,n),(w,l)]_\psi ]_\psi , \end{aligned}$$
(3.2)
$$\begin{aligned} and, \end{aligned}$$
(3.3)
$$\begin{aligned}&\phi _{L\oplus M}[(x, m), (y, n), (z, p)]_{\psi }\nonumber \\&\quad = [\phi _{L\oplus M}(x, m), (y, n), (z, p)]_{\psi }+[(x, m), \phi _{L\oplus M}(y, n), (z, p)]_{\psi }\nonumber \\&\qquad +[(x, m), (y, n), \phi _{L\oplus M}(z, p)]_{\psi }, \end{aligned}$$
(3.4)

for all \(x \oplus m, y\oplus n, z\oplus p, v\oplus k, w\oplus l\in L\oplus M\). The condition Eq. (3.2) is equivalent to

$$\begin{aligned} \psi (x, y, [z, v, w]) = \psi ([x, y, z], v, w)+ \psi (z, [x, y, v], w)+\psi (z, v, [x, y, w]), \end{aligned}$$

or, equivalently, \(d(\psi )=0\), as we are considering only trivial representation. The condition Eq. (3.3) is equivalent to

$$\begin{aligned}&\phi _{M}(\psi (x, y, z))+\chi ([x, y, z])= \psi (\phi _L(x), y, z)+\psi (x, \phi _L(y), z)+\psi (x, y, \phi _L(z)). \end{aligned}$$

This is same as \(d(\chi ) + \delta \psi = 0\). This implies \((\psi , \chi )\) is a 2-cocycle.

Let \(s_1, s_2\) be two sections of p. Define a map \(u: L \rightarrow M\) by \(u(x):= s_1(x)-s_2(x)\). Observe that

$$\begin{aligned} \psi (x, y, z)= & {} [s_1(x), s_1(y), s_1(z)]-s_1([x, y, z])\\= & {} [s_2(x)+u(x), s_2(y)+u(y), s_2(z)+u(z)]-s_2([x, y, z])-u([x, y, z])\\= & {} \psi '(x, y, z)-u([x, y, z]), \end{aligned}$$

as \(u(x), u(y), u(z)\in M\) and \((M,\phi _M)\) be an abelian 3-LieDer pair.

Also note that

$$\begin{aligned} \chi (x)= & {} \phi _{{\hat{L}}}(s_1(x))-s_1(\phi _L(x))\\= & {} \phi _{{\hat{L}}}(s_2(x)+u(x))-s_2(\phi _L(x))-u(\phi _L(x))\\= & {} \chi '(x)+\phi _{M}(u(x))-u(\phi _L(x)). \end{aligned}$$

This shows that \((\psi , \chi )-(\psi ', \chi ')=\partial u\). Hence they correspond to the same cohomology class. \(\square \)

Theorem 3.4

Let \((L, \phi _L)\) be a 3-LieDer pair and \((M, \phi _{M})\) be an abelian 3-LieDer pair. Then the isomorphism classes of central extensions of L by M are classified by the second cohomology group \(H^{2}_{\text {3-LieDer}} (L, M)\).

Proof

Let \(({\hat{L}}, \phi _{{\hat{L}}})\) and \((\hat{L'}, \phi _{\hat{L'}})\) be two isomorphic central extensions and the isomorphism is given by \(\eta : {\hat{L}}\rightarrow \hat{L'}\). Let \(s:L\rightarrow {\hat{L}}\) be a section of p. Then

$$\begin{aligned} p'\circ (\eta \circ s)=(p'\circ \eta )\circ s=p\circ s=Id_L. \end{aligned}$$

This shows that \(s':=\eta \circ s\) is a section of \(p'\). Since \(\eta \) is a morphism of 3-LieDer pairs, we have \(\eta |_M = Id_M\). Thus,

$$\begin{aligned} \psi '(x, y, z)= & {} [s'(x), s'(y), s'(z)]-s'([x, y, z])\\= & {} \eta ([s(x), s(y), s(z)]-[x, y, z])\\= & {} \psi (x, y, z), \end{aligned}$$

and

$$\begin{aligned} \chi '(x)= & {} \phi _{\hat{L'}}(s'(x))-s'(\phi _L(x))\\= & {} \phi _{\hat{L'}}(\eta \circ s(x))-\eta \circ s(\phi _L(x))\\= & {} \phi _{{\hat{L}}}(s(x))- s(\phi _L(x))\\= & {} \chi (x). \end{aligned}$$

Therefore, isomorphic central extensions give rise to the same 2-cocycle, hence, correspond to the same element in \(H^{2}_{3-LieDer} (L, M)\).

Conversely, let \((\psi , \chi )\) and \((\psi ', \chi ')\) be two cohomologous 2-cocycles. Therefore, there exists a map \(v: L \rightarrow M\) such that

$$\begin{aligned} (\psi , \chi )-(\psi ', \chi ')=\partial v. \end{aligned}$$

The 3-LieDer pair structures on \(L \oplus M\) corresponding to the above 2-cocycles are isomorphic via the map \(\eta : L \oplus M\rightarrow L \oplus M\) given by \(\eta (x, m) = (x, m+v(x))\). This proves our theorem. \(\square \)

4 Extensions of a pair of derivations

It is well-known that derivations are infinitesimals of automorphisms, and a study [9] has been done on extensions of a pair of automorphisms of Lie-algebras. In this section, we study extensions of a pair of derivations and see how it is related to the cohomology of the 3-LieDer pair.

Let

(4.1)

be a fixed central extensions of 3-Lie algebras. Given a pair of derivations \((\phi _L, \phi _M)\in Der(L)\times Der(M)\), here we study extensions of them to a derivation \(\phi _{{\hat{L}}}\in Der({\hat{L}})\) which makes

(4.2)

into an exact sequence of 3-LieDer pairs. In such a case, the pair \((\phi _L, \phi _M)\in Der(L)\times Der(M)\) is said to be extensible.

Let \(s: L\rightarrow {\hat{L}}\) be a section of Eq. (4.1), we define a map \(\psi : L\otimes L\otimes L \rightarrow M\) by

$$\begin{aligned} \psi (x, y, z):=[s(x), s(y), s(z)]-s([x, y, z]),~~~~~\chi (x)=\phi _{{\hat{L}}}(s(x))-s(\phi _L(x)),~~\forall x, y, z\in L. \end{aligned}$$

Given a pair of derivations \((\phi _L, \phi _M)\in Der(L)\times Der(M)\), we define another map \(Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}: L\otimes L\otimes L \rightarrow M\) by

$$\begin{aligned} Ob_{(\phi _L, \phi _M)}^{M}( x, y, z):=\phi _{M}(\psi (x, y, z))-\psi (\phi _L(x), y, z)-\psi (x, \phi _L(y), z)-\psi (x, y, \phi _L(z)). \end{aligned}$$

Proposition 4.1

The map \(Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}: L\otimes L\otimes L \rightarrow M\) is a 2-cocycle in the cohomology of the 3-Lie algebra L with coefficients in the trivial representation a. Moreover, the cohomology class \([Ob_{(\phi _L, \phi _M)}^{{\hat{L}}} ] \in H^2(L, M)\) does not depend on the choice of sections.

Proof

First observe that \(\psi \) is a 1-cocycle in the cohomology of the 3-Lie algebra L with coefficients in the trivial representation M. Thus, we have

$$\begin{aligned}&(dOb_{(\phi _L, \phi _M)}^{M})( x, y, u, v, w)\\&\quad = -Ob_{(\phi _L, \phi _M)}^{M}(x, y, [u,v,w])+Ob_{(\phi _L, \phi _M)}^{M}([x,y,u], v, w)\\&\qquad +Ob_{(\phi _L, \phi _M)}^{M}(u, [x,y,v], w)+Ob_{(\phi _L, \phi _M)}^{M}(u, v,[x,y,w])\\&\quad = -\phi _{M}(\psi (x, y, [u,v,w]))+\psi (\phi _L(x), y, [u,v,w])+\psi (x, \phi _L(y), [u,v,w])\\&\qquad +\psi (x, y, \phi _L([u,v,w]))+\phi _{M}(\psi ([x,y,u], v, w))-\psi (\phi _L([x,y,u]), v, w) \\&\qquad -\psi ([x,y,u], \phi _L(v), w)-\psi ([x,y,u], v, \phi _L(w)) +\phi _{M}(\psi (u, [x,y,v], w))\\&\qquad -\psi (\phi _L(u), [x,y,v], w)-\psi (u, \phi _L([x,y,v]), w)-\psi (u, [x,y,v], \phi _L(w)) \\&\qquad +\phi _{M}(\psi (u, v,[x,y,w]))-\psi (\phi _L(u), v, [x,y,w])-\psi (u, \phi _L(v), [x,y,w])\\&\qquad -\psi (u, v, \phi _L([x,y,w])) \\&\quad = \psi (\phi _L(x), y, [u,v,w])+\psi (x, \phi _L(y), [u,v,w])+\psi (x, y, \phi _L([u,v,w]))\\&\qquad -\psi (\phi _L([x,y,u]), v, w) -\psi ([x,y,u], \phi _L(v), w)-\psi ([x,y,u], v, \phi _L(w)) \\&\qquad -\psi (\phi _L(u), [x,y,v], w)-\psi (u, \phi _L([x,y,v]), w)-\psi (u, [x,y,v], \phi _L(w)) \\&\qquad -\psi (\phi _L(u), v, [x,y,w])-\psi (u, \phi _L(v), [x,y,w])-\psi (u, v, \phi _L([x,y,w])) \\&\quad = 0. \end{aligned}$$

Therefore, \(Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}\) is a 2-cocycle. To prove the second part, let \(s_1\) and \(s_2\) be two sections of Eq. (4.1). Consider the map \(u:L\rightarrow M\) given by \(u(x):= s_1 (x)-s_2 (x)\). Then

$$\begin{aligned} \psi _1(x, y, z)=\psi _2(x, y, z)-u[x, y, z]. \end{aligned}$$

If \(^1Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}\) and \(^2Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}\) denote the one cocycles corresponding to the sections \(s_1\) and \(s_2\), then

$$\begin{aligned}&^1Ob_{(\phi _L, \phi _M)}^{M}(x, y, z)\\&\quad = \phi _{M}(\psi _1(x, y, z))-\psi _1(\phi _L(x), y, z)-\psi _1(x, \phi _L(y), z)-\psi _1(x, y, \phi _L(z))\\&\quad = \phi _{M}(\psi _2(x, y, z))- \phi _{M}(u(x, y, z))-\psi _2(\phi _L(x), y, z)+u(\phi _L(x), y, z)\\&\qquad -\psi _2(x, \phi _L(y), z)+u(x, \phi _L(y), z)-\psi _2(x, y, \phi _L(z))+u(x, y, \phi _L(z))\\&\quad = ^2Ob_{(\phi _L, \phi _M)}^{M}(x, y, z)+d(\phi _M\circ u-u\circ \phi _L)(x, y, z). \end{aligned}$$

This shows that the 2-cocycles \(^1Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}\) and \(^2Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}\) are cohomologous. Hence they correspond to the same cohomology class in \(\in H^2(L, M)\). \(\square \)

The cohomology class \([Ob_{(\phi _L, \phi _M)}^{{\hat{L}}} ] \in H^2(L, M)\) is called the obstruction class to extend the pair of derivations \((\phi _L, \phi _M)\).

Theorem 4.2

Let Eq. (4.1) be a central extension of 3-Lie algebras. A pair of derivations \((\phi _L, \phi _M)\in Der(L)\times Der(M)\) is extensible if and only if the obstruction class \([Ob_{(\phi _L, \phi _M)}^{{\hat{L}}} ] \in H^2(L, M)\) is trivial.

Proof

Suppose there exists a derivations \(\phi _{{\hat{L}}}\in Der({\hat{L}})\) such that Eq. (4.2) is an exact sequence of 3-LieDer pairs. For any \(x \in L\), we observe that \(p(\phi _{{\hat{L}}} (s(x)) - s(\phi _{L} (x))) = 0\). Hence \(\phi _{{\hat{L}}} (s(x))- s(\phi _{L} (x))\in ker(p) = im(i)\). We define \(\lambda : L\rightarrow M\) by

$$\begin{aligned} \lambda (x)=\phi _{{\hat{L}}} (s(x))- s(\phi _{L} (x)). \end{aligned}$$

For any \(s(x) + a \in {\hat{L}}\), we have

$$\begin{aligned} \phi _{{\hat{L}}} (s(x) + a)=s(\phi _{L} (x))+\lambda (x)+\phi _{{\hat{L}}} (a). \end{aligned}$$

Since \(\phi _{{\hat{L}}}\) is a derivation, for any \(s(x) + a, s(y) + b \in {\hat{L}}\), we have

$$\begin{aligned} \phi _M (\psi (x, y, z)) - \psi (\phi _L (x), y, z) - \psi (x,\phi _L (y), z)- \psi (x, y, \phi _L (z))= -\lambda ([x, y, z]), \end{aligned}$$

or, equivalently, \(Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}=\partial \lambda \) is a coboundary. Hence the obstruction class \([Ob_{(\phi _L, \phi _M)}^{{\hat{L}}} ] \in H^2(L, M)\) is trivial. \(\square \)

To prove the converse part, suppose \(Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}\) is given by a coboundary, say \(Ob_{(\phi _L, \phi _M)}^{{\hat{L}}}=\partial \lambda \). We define a map \(\phi _{{\hat{L}}}: {\hat{L}}\rightarrow {\hat{L}}\) by

$$\begin{aligned} \phi _{{\hat{L}}} (s(x) + a)=s(\phi _{L} (x))+\lambda (x)+\phi _{{\hat{L}}} (a). \end{aligned}$$

Then \(\phi _{{\hat{L}}}\) is a derivation on \({\hat{L}}\) and Eq. (4.2) is an exact sequence of 3-LieDer pairs. Hence the pair \((\phi _L, \phi _M)\) is extensible. Thus, we obtain the following.

Theorem 4.3

If \(H^2(L, M)=0\), then any pair of derivations \((\phi _L, \phi _M)\in Der(L)\times Der(M)\) is extensible.

5 Formal deformations of 3-LieDer pairs

In this section, we study one-parameter formal deformations of 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivations.

Let \((L, \phi _L)\) be a 3-LieDer pair. We denote the 3-Lie bracket on L by \(\mu \), i.e, \(\mu (x,y, z) = [x, y, z]\), for all \(x, y, z \in L\). Consider the space L[[t]] of formal power series in t with coefficients from L. Then L[[t]] is a \({\mathbb {F}}[[t]]\)-module.

A formal one-parameter deformation of the 3-LieDer pair \((L, \phi _L)\) consist of formal power series

$$\begin{aligned} \mu _t= & {} \sum _{i=0}^{\infty }t^i\mu _i\in \text{ Hom }(L^{\otimes 3}, L)[[t]] ~\text{ with }~\mu _0=\mu ,\\ \phi _{t}= & {} \sum _{i=0}^{\infty }t^i\phi _{i}\in \text{ Hom }(L, L)[[t]] ~\text{ with }~\phi _{0}=\phi _L, \end{aligned}$$

such that L[[t]] together with the bracket \(\mu _t\) forms a 3-Lie algebra over \({\mathbb {F}}[[t]]\) and \(\phi _t\) is a derivation on L[[t]].

Therefore, in a formal one-parameter deformation of 3-LieDer pair, the following relations hold:

$$\begin{aligned} \mu _t(x, y, \mu _t(z, v, w))= & {} \mu _t(\mu _t(x, y, z), v, w)+ \mu _t(z, \mu _t(x, y, v), w)\nonumber \\&+\mu _t(z, v, \mu _t(x, y, w)), \end{aligned}$$
(5.1)
$$\begin{aligned} \phi _{t}(\mu _t(x, y, z))= & {} \mu _t(\phi _{t}(x),y, z)+\mu _t(x,\phi _{t}(y), z)+\mu _t(x, y, \phi _{t}(z)). \end{aligned}$$
(5.2)

Conditions Eqs.(5.1)–(5.2) are equivalent to the following equations:

$$\begin{aligned}&\sum _{i+j=n}\mu _i(x, y, \mu _j(z, v, w))\nonumber \\&\quad = \sum _{i+j=n}\mu _i(\mu _j(x, y, z), v, w)+ \mu _i(z, \mu _j(x, y, v), w)+\mu _i(z, v, \mu _j(x, y, w)), \end{aligned}$$
(5.3)

and,

$$\begin{aligned}&\sum _{i+j=n}\phi _{i}(\mu _j(x, y, z))\nonumber \\&\quad =\sum _{i+j=n}\mu _i(\phi _{j}(x),y, z)+\mu _i(x,\phi _{j}(y), z)+\mu _i(x, y, \phi _{j}(z)). \end{aligned}$$
(5.4)

For \(n = 0\) we simply get \((L, \phi _L)\) is a 3-LieDer pair. For \(n = 1\), we have

$$\begin{aligned}&\mu _1(x, y, [z, v, w])+[x, y, \mu _1(z, v, w)]\nonumber \\&\quad =\mu _1([x, y, z], v, w)+[\mu _1(x, y, z), v, w]+[z, \mu _1(x, y, v), w]\nonumber \\&\qquad +\mu _1(z, [x, y, v], w)+[z, v, \mu _1(x, y, w)]+\mu _1(z, v, [x, y, w]), \end{aligned}$$
(5.5)
$$\begin{aligned} and,\nonumber \\&\phi _1([x, y, z])+\phi _{L}(\mu _1(x, y, z))\nonumber \\&\quad =\mu _1(\phi _{L}(x),y, z)+[\phi _{1}(x),y, z]+\mu _1(x,\phi _{L}(y), z)+[x,\phi _{1}(y), z]\nonumber \\&\qquad +\mu _1(x, y, \phi _{L}(z))+[x, y, \phi _{1}(z)]. \end{aligned}$$
(5.6)

The condition Eq. (5.5) is equivalent to \(d(\mu _1)=0\) whereas the condition Eq. (5.6) is equivalent to \( d(\phi _{1}) + \delta (\mu _1)= 0\). Therefore, we have

$$\begin{aligned} \partial (\mu _1, \phi _{1})=0. \end{aligned}$$

Definition 5.1

Let \((\mu _t, \phi _t)\) be a one-parameter formal deformation of 3-LieDer pair \((L, \phi _L)\). Suppose \((\mu _n, \phi _n)\) is the first non-zero term of \((\mu _t, \phi _t)\) after \((\mu _0, \phi _0)\), then such \((\mu _n, \phi _n)\) is called the infinitesimal of the deformation of \((L, \phi _L)\).

Hence, from the above observations, we have the following proposition.

Proposition 5.2

Let \((\mu _t, \phi _{t})\) be a formal one-parameter deformation of a 3-LieDer pair \((L, \phi _L)\). Then the linear term \((\mu _1, \phi _{1})\) is a 1-cocycle in the cohomology of the 3-LieDer pair L with coefficients in itself.

Proof

We have showed that

$$\begin{aligned} \partial (\mu _1, \phi _{1})=0. \end{aligned}$$

If \((\mu _1, \phi _{1})\) be the first non-zero term, then we are done. If \((\mu _n, \phi _{n})\) be the first non-zero term after \((\mu _0, \phi _0)\), then exactly the same way, one can show that

$$\begin{aligned} \partial (\mu _n, \phi _{n})=0. \end{aligned}$$

\(\square \)

Next, we define a notion of equivalence between formal deformations of 3-LieDer pairs.

Definition 5.3

Two deformations \((\mu _t, \phi _{t})\) and \((\mu '_t, \phi '_{t})\) of a 3-LieDer pair \((L, \phi _L)\) are said to be equivalent if there exists a formal isomorphism \(\Phi _t=\sum _{i=0}^{\infty }t^{i}\phi _i: L[[t]]\rightarrow L[[t]]\) with \(\Phi _0=Id_L\) such that

$$\begin{aligned} \Phi _t \circ \mu _t=\mu '_t\circ (\Phi _t\otimes \Phi _t\otimes \Phi _t), ~~~~~\Phi _t \circ \phi _t=\phi '_t \circ \Phi _t. \end{aligned}$$

By comparing coefficients of \(t^n\) from both the sides, we have

$$\begin{aligned} \sum _{i+j=n} \phi _i \circ \mu _j= & {} \sum _{p+q+r+l=n}\mu '_p\circ (\phi _q\otimes \phi _r\otimes \phi _l),\\ \sum _{i+j=n} \phi '_{i}\circ \phi _j= & {} \sum _{p+q=n}\phi _p\circ \phi _{q}. \end{aligned}$$

Easy to see that the above identities hold for \(n = 0\). For \(n = 1\), we get

$$\begin{aligned} \mu _1+\phi _1\circ \mu= & {} \mu '_1+\mu \circ (\phi _1\otimes Id\otimes Id)+\mu \circ (Id\otimes Id \otimes \phi _1), \end{aligned}$$
(5.7)
$$\begin{aligned} \phi _L \circ \Phi _1+\phi '_{1}= & {} \phi _{1}+\phi _1 \circ \phi _{L}. \end{aligned}$$
(5.8)

These two identities together imply that

$$\begin{aligned} (\mu _1, \phi _{1})-(\mu '_1, \phi '_{1})=\partial \phi _1. \end{aligned}$$

Thus, we have the following.

Proposition 5.4

The infinitesimals corresponding to equivalent deformations of the 3-LieDer pair \((L, \phi _L)\) are cohomologous.

Definition 5.5

A deformation \((\mu _t, \phi _{t})\) of a 3-LieDer pair is said to be trivial if it is equivalent to the undeformed deformation \((\mu '_t=\mu , \phi '_{t}=\phi _L)\).

Definition 5.6

A 3-LieDer pair \((L,\phi _L)\) is called rigid, if every 1-parameter formal deformation \(\mu _t\) is equivalent to the trivial deformation.

Theorem 5.7

Every formal deformation of the 3-LieDer pair \((L, \phi _L)\) is rigid if the second cohomology group of the 3-LieDer pair vanishes, that is, \(H^{2}_{\text {3-LieDer}} (L, L) = 0\).

Proof

Let \((\mu _t, \phi _{t})\) be a deformation of the 3-LieDer pair \((L, \phi _L)\). From the Proposition 5.2, the linear term \((\mu _1, \phi _{1})\) is a 2-cocycle. Therefore, \((\mu _1, \phi _{1})=\partial \Phi _1\) for some \(\phi _1 \in C^1_{\text {3-LieDer}}(L, L) = \text {Hom}(L, L)\).

We set \(\Phi _t = Id_L + t\Phi _1: L[[t]]\rightarrow L[[t]]\) and define

$$\begin{aligned} \mu '_t=\Phi _t^{-1}\circ \mu _t\circ (\Phi _t\otimes \Phi _t\otimes \Phi _t),~~~~\phi '_{t}=\Phi _t^{-1}\circ \phi _{t}\circ \Phi _t. \end{aligned}$$
(5.9)

By definition, \((\mu '_t, \phi '_{t})\) is equivalent to \((\mu _t, \phi _{t})\). Moreover, it follows from Eq. (5.7) that

$$\begin{aligned}&\mu '_t=\mu +t^2\mu '_2+\cdot \cdot \cdot ~~~~~~\text{ and }~~~ \phi '_{t}=\phi _L+t^{2}\phi '_{2}+\cdot \cdot \cdot . \end{aligned}$$

In other words, the linear terms are vanish. By repeating this argument, we get \((\mu _t, \phi _t)\) is equivalent to \((\mu , \phi _L)\). \(\square \)

Next, we consider finite order deformations of a 3-LieDer pair \((L, \phi _L)\), and show that how obstructions of extending deformation of order N to deformation of order \((N+1)\) depends on the third cohomology class of the 3-LieDer pair \((L, \phi _L)\) .

Definition 5.8

A deformation of order N of a 3-LieDer pair \((L, \phi _L)\) consist of finite sums \(\mu _t = \sum _{i=0}^N t^i\mu _i\) and \(\phi _t =\sum _{i=0}^N t^i\phi _i\) such that \(\mu _t\) defines 3-Lie bracket on \(L[[t]]/(t^{N+1})\) and \(\phi _t\) is a derivation on it.

Therefore, we have

$$\begin{aligned}&\sum _{i+j=n}\mu _i(x, y, \mu _j(z, v, w)) \\&\quad = \sum _{i+j=n}\mu _i(\mu _j(x, y, z), v, w)+ \mu _i(z, \mu _j(x, y, v), w)+\mu _i(z, v, \mu _j(x, y, w)),~~~~~~~~\\&and,\\&\sum _{i+j=n}\phi _{i}(\mu _j(x, y, z))=\sum _{i+j=n}\mu _i(\phi _{j}(x),y, z)+\mu _i(x,\phi _{j}(y), z)+\mu _i(x, y, \phi _{j}(z)), \end{aligned}$$

for \(n = 0, 1,\ldots , N\). These identities are equivalent to

$$\begin{aligned}{}[\mu , \mu _n]= & {} -\frac{1}{2}\sum _{i+j=n, i, j> 0}[\mu _i, \mu _j], \end{aligned}$$
(5.10)
$$\begin{aligned}&-[\phi _L, \mu _n]+[\mu , \phi _n]=\sum _{i+j=n, i, j> 0}[\phi _i,\mu _j]. \end{aligned}$$
(5.11)

Definition 5.9

A deformation (\(\mu _t = \sum _{i=0}^N t^i\mu _i, \phi _t =\sum _{i=0}^N t^i\phi _i\)) of order N is said to be extendable if there is an element \((\mu _{N+1}, \phi _{N+1} )\in C^2_{\text {3-LieDer}} (L, L)\) such that \((\mu '_t=\mu _t+t^{N+1}\mu _{N+1}, \phi '_t=\phi _t+t^{N+1}\phi _{N+1})\) is a deformation of order \(N + 1\).

Thus, the following two equations need to be satisfied-

$$\begin{aligned}&\sum _{i+j=N+1}\mu _i(x, y, \mu _j(z, v, w)) \nonumber \\&\quad = \sum _{i+j=N+1}\mu _i(\mu _j(x, y, z), v, w)+ \mu _i(z, \mu _j(x, y, v), w)+\mu _i(z, v, \mu _j(x, y, w)),\nonumber \\ \end{aligned}$$
(5.12)

and,

$$\begin{aligned}&\sum _{i+j=N+1}\phi _{i}(\mu _j(x, y, z)) \nonumber \\&\quad =\sum _{i+j=N+1}\mu _i(\phi _{j}(x),y, z)+\mu _i(x,\phi _{j}(y), z)+\mu _i(x, y, \phi _{j}(z)). \end{aligned}$$
(5.13)

The above two equations can be equivalently written as

$$\begin{aligned}&d(\mu _{N+1})=-\frac{1}{2}\sum _{i+j=N+1, i, j> 0}[\mu _i, \mu _j]=Ob^3 \end{aligned}$$
(5.14)
$$\begin{aligned}&d(\phi _{N+1})+\delta (\mu _{N+1})=-\sum _{i+j=N+1, i, j> 0}[\phi _i, \mu _j]=Ob^2. \end{aligned}$$
(5.15)

Using the Eqs. 5.14 and 5.15, it is routine but lengthy work to prove the following proposition. Thus, we choose to omit the proof.

Proposition 5.10

The pair \((Ob^3, Ob^2 ) \in C^3_{\text {3-LieDer}} (L, L)\) is a 3-cocycle in the cohomology of the 3-LieDer pair \((L, \phi _L )\) with coefficients in itself.

Definition 5.11

Let \((\mu _t, \phi _t)\) be a deformation of order N of a 3-LieDer pair \((L, \phi _L)\). The cohomology class \([(Ob^3, Ob^2 )]\in H^3_{\text {3-LieDer}} (L, L)\) is called the obstruction class of \((\mu _t, \phi _t)\).

Theorem 5.12

A deformation \((\mu _t, \phi _t )\) of order N is extendable if and only if the obstruction class \([(Ob^3, Ob^2 )]\in H^3_{\text {3-LieDer}} (L, L)\) is trivial.

Proof

Suppose that a deformation \((\mu _t, \phi _t )\) of order N of the 3-LieDer pair \((L, \phi _L)\) extends to a deformation of order \(N + 1\). Then we have

$$\begin{aligned} \partial (\mu _{N+1}, \phi _{N+1})=(Ob^3, Ob^2 ). \end{aligned}$$

Thus, the obstruction class \([(Ob^3, Ob^2 )]\in H^3_{\text {3-LieDer}} (L, L)\) is trivial.

Conversely, if the obstruction class \([(Ob^3, Ob^2)]\in H^3_{\text {3-LieDer}} (L, L)\) is trivial, suppose that

$$\begin{aligned} (Ob^3, Ob^2 )=\partial (\mu _{N+1}, \phi _{N+1}), \end{aligned}$$

for some \((\mu _{N+1}, \phi _{N+1})\in C^{2}_{\text {3-LieDer}}(L, L)\). Then it follows from the above observation that \((\mu '_t=\mu _t+t^{N+1}\mu _{N+1}, \phi '_t=\phi _t+t^{N+1}\phi _{N+1})\) is a deformation of order \(N + 1\), which implies that \((\mu _t, \phi _t )\) is extendable. \(\square \)

Theorem 5.13

If \(H^3_{\text {3-LieDer}} (L, L)\), then every finite order deformation of \((L, \phi _L)\) is extendable.

Corollary 5.14

If \(H^3_{\text {3-LieDer}} (L, L)=0\), then every 2-cocycle in the cohomology of the 3-LieDer pair \((L, \phi _L)\) with coefficients in itself is the infinitesimal of a formal deformation of \((L, \phi _L)\).