1. Introduction
3-Lie algebra plays an important role in string theory, and it is also used to study supersymmetry and gauge symmetry transformation of the world-volume theory of multiple coincident M2-branes [
1,
2]. The concept of 3-Lie algebra, general
n-Lie algebra, was first introduced by Filippov [
3] and can be regarded as a generalization of Lie algebra to higher-order algebra. 3-Lie algebra has attracted the attention of scholars from mathematics and physics [
4,
5,
6]. Representation theory, cohomology theory, deformations, Nijenhuis operators and extension theory of
n-Lie algebras have been widely studied by scholars [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17].
Derivations play important roles in the study of homotopy Lie algebras [
18], differential Galois theory [
19], control theory and gauge theories of quantum field theory [
20]. Recently, authors have studied algebras with derivations in [
21,
22] from the operadic point of view. In [
23], Tang et al. investigated the deformation and extension of Lie algebras with derivations from the cohomological point of view. The results of [
23] have been extended to 3-Lie algebras with derivations [
24,
25]. More research on algebraic structures with derivations has been developed; see [
26,
27,
28,
29,
30,
31] and references cited therein.
In recent years, scholars have increasingly focused on structures with arbitrary weights, thanks to the important work of [
32,
33,
34,
35,
36,
37]. The papers [
38,
39,
40] established the cohomology, extensions and deformations of Rota–Baxter 3-Lie algebras with any weight
, as well as the differential 3-Lie algebras with any weight
. Additionally, the cohomology and deformation of modified Rota–Baxter algebras were studied by Das [
41]. The works [
42,
43] provided insights into the cohomology and deformation of modified Rota–Baxter Leibniz algebras with weight
. Furthermore, Peng et al. [
44] introduced the concept of modified
-differential Lie algebras.
Motivated by the mentioned work on the modified -differential operator on Lie algebras and considering the importance of 3-Lie algebra, representation and cohomology, in this paper, our main objective is to study modified -differential 3-Lie algebras. We develop a cohomology theory of modified -differential 3-Lie algebras that controls the deformation and extensions of modified -differential 3-Lie algebras.
The paper is organized as follows. In
Section 2, we introduce the representation and cohomology of modified
-differential 3-Lie algebras (MD
3-LieAs). In
Section 3, we consider linear deformations of MD
3-LieAs. In
Section 4, we study abelian extensions of MD
3-LieAs. In
Section 5, we study
-extensions of MD
3-LieAs.
Throughout this paper, denotes a field of characteristic zero. All the vector spaces and linear maps are taken over .
2. Representations and Cohomologies of MD3-LieAs
In this section, we introduce the concept of a MD3-LieA and present some examples. Then, we give representations and cohomologies of MD3-LieAs.
Definition 1 ([
3])
. (i) A 3-Lie algebra is a tuple in which is a vector space together with a skew-symmetric ternary operation such thatfor all .- (ii)
A homomorphism between two 3-Lie algebras and is a linear map satisfying
Definition 2. (i) Let and be a 3-Lie algebra. A modified λ-differential operator on 3-Lie algebra is a linear map , such that - (ii)
A modified λ-differential 3-Lie algebra (MD3-LieA) is a triple consisting of a 3-Lie algebra and a modified λ-differential operator .
- (iii)
A homomorphism between two MD3-LieAs and is a 3-Lie algebra homomorphism such that . Furthermore, if η is nondegenerate, then η is called an isomorphism from to .
Let
be a 3-Lie algebra; then, the elements in
are called fundamental objects of the 3-Lie algebra
. There is a bilinear operation
on
, which is given by
It is shown in [
45] that
is a Leibniz algebra. Furthermore, by direct calculation, we have the following result.
Proposition 1. Let be a MD3-LieA. Then, is a Leibniz algebra with a derivation, where , for all . See [26] for more details about Leibniz algebras with derivations.
Remark 1. Let be a modified λ-differential operator on a 3-Lie algebra . If then is a derivation on the 3-Lie algebra . See [46] for various derivations of 3-Lie algebras. Example 1. Let be a 3-Lie algebra. Then, a linear map is a modified λ-differential operator if and only if is a derivation on the 3-Lie algebra .
Example 2. Let be a MD3-LieA. Then, for , is a MD3-LieA.
Definition 3. (i) (see [8]) A representation of a 3-Lie algebra on a vector space is a skew-symmetric bilinear map , such thatfor all . We also denote a representation of on by . (ii) A representation of a MD3-LieA is a triple , where is a representation of the 3-Lie algebra and is a linear operator on , satisfying the following equationfor any and Remark 2. Let be a representation of a MD3-LieA . If then is a representation of the 3-Lie algebra with a derivation . See [24,25,29] for more details about 3-Lie algebras with derivations. Example 3. Let be a representation of a 3-Lie algebra . Then, is a representation of the MD3-LieA if and only if is a representation of the 3-Lie algebra with a derivation .
Example 4. Let be a representation of a 3-Lie algebra . Then, for , is a representation of the MD3-LieA .
Example 5. Let be a representation of a MD3-LieA . Then, for , is a representation of the MD3-LieA .
Proposition 2. Let be a 3-Lie algebra, and be a representation of it. Then, is a representation of MD3-LieA if and only if is a MD3-LieA under the following maps:for all and . Proof. Assume that
is a MD
3-LieA, for any
and
; we have
which implies that
Therefore, is a representation of .
The converse can be proved similarly. □
Let
be a representation of a MD
3-LieA
and
be a dual space of
. We define a bilinear map
and a linear map
respectively by
for any
and
Proposition 3. With the above notations, is a representation of the MD3-LieA .
Proof. First, It has been proved that [
47]
is a representation of the 3-Lie algebra
. Furthermore, for any
and
by Equations (5) and (6), we have
which implies that
. So, we obtain the result. □
Example 6. Let be a MD3-LieA and define by . Then, is a representation of the MD3-LieA , which is called the adjoint representation of . Furthermore, is a dual adjoint representation of , which is called the coadjoint representation of .
Next, we will study the cohomology of a MD3-LieA with coefficients in its representation.
Recall from [
14] that let
be a representation of a 3-Lie algebra
. Denote the
n—cochains of
with coefficients in representation
by
The coboundary operator
, for
and
, as
it was proved that
Lemma 1. Let be a representation of a MD3-LieA . For any , we define a linear map byfor any and . Then, Φ is a cochain map; i.e., Proof. It follows by a straightforward tedious calculations. □
Let
be a representation of a MD
3-LieA
; we define
n-cochains for MD
3-LieA as follows:
We define a linear map
by
In view of Lemma 1, we have the following theorem.
Theorem 1. The linear map ∂ is a coboundary operator; that is,
Therefore, we obtain a cochain complex , for , and we denote the set of n-cocycles by , the set of n-coboundaries by and the n-th cohomology group of the MD3-LieA with coefficients in the representation by .
Lastly, we calculate the 1-cocycle and 2-cocycle.
For
,
f is a 1-cocycle if
i.e.,
and
For
,
is a 2-cocycle if
i.e.,
and
3. Linear Deformations of MD3-LieAs
In this section, we study linear deformations of MD3-LieAs.
Let
be a MD
3-LieA. Denote
and
. Consider a family of linear maps:
Definition 4. A linear deformation of the MD3-LieA is a pair which endows with the MD3-LieA.
Proposition 4. The pair generates a linear deformation of the MD3-LieA if and only if the following equations hold:for any and . Proof. is a MD
3-LieA if and only if
Comparing the coefficients of on both sides of the above equations, Equations (9) and (10) are equivalent to Equations (7) and (8), respectively. □
Corollary 1. Let be a linear deformation of a MD3-LieA . Then, is a 2-cocycle of with the coefficient in the adjoint representation .
Proof. For
, Equations (7) and (8) are equivalent to
which imply that
, respectively. Hence,
is a 2-cocycle of
with the coefficient in the adjoint representation
. □
Definition 5. The 2-cocycle is called the infinitesimal of the linear deformation of .
Definition 6. (i) Two linear deformations and of the MD3-LieA are said to be equivalent if there exists a linear map , such that satisfyingfor any . (ii) A linear deformation of the MD3-LieA is said to be trivial if is equivalent to .
Comparing the coefficients of
t on both sides of the above Equations (11) and (12), we have
Thus, we have the following theorem.
Theorem 2. The infinitesimals of two equivalent linear deformations of are in the same cohomological class in .
Let
be a trivial deformation of
. Then, there exists a linear map
, such that
satisfying
Compare the coefficients of
on both sides of Equations (13) and (14), and we can obtain
Thus, from a trivial deformation, we can obtain the following definition of Nijenhuis operator.
Definition 7. Let be a MD3-LieA. A linear map is called a Nijenhuis operator if the following equations hold:for any Proposition 5. Let be a MD3-LieA, and a Nijenhuis operator. Then, is a MD3-LieA, where Proof. In the light of [
9],
is a 3-Lie algebra. Next, we prove that
is a modified
-differential operator of
, for any
by Equations (2) and (19), and we have
So, we obtain the conclusion. □
Definition 8. A linear map is called an -operator on the MD3-LieA with respect to the representation if the following equations hold:for any Remark 3. Obviously, an invertible linear map is an -operator if and only if is a 1-cocycle of the MD3-LieA with coefficients in the representation .
Proposition 6. Let be a representation of a MD3-LieA . Then, is an -operator if and only if is a Nijenhuis operator on semidirect product MD3-LieA .
Proof. For any
and
, by
we have
which implies that
R is an
-operator if and only if
is a Nijenhuis operator. □
4. Abelian Extensions of MD3-LieAs
In this section, we study abelian extensions of MD3-LieAs and show that they are classified by the second cohomology groups.
Notice that a vector space together with a linear map is naturally a MD3-LieA where the bracket on is defined to be
Definition 9. An abelian extension of by is a short exact sequence of homomorphisms of MD3-LieAsi.e., there exists a commutative diagram:where the MD3-LieA satisfies , for all . We will call an abelian extension of by .
A section of an abelian extension of by is a linear map such that .
Let
be a representation of a MD
3-LieA
. Assume that
. Define
and
, respectively, by
Proposition 7. The triple is a MD3-LieA if and only if is a 2-cocycle in the cohomology of the MD3-LieA with the coefficient in . In this case,is an abelian extension. Proof. is a MD3-LieA if and only if
for any
. Furthermore, Equations (23) and (24) are equivalent to
using Equations (25) and (26), we obtain
and
, respectively. Therefore,
which implies that
is a 2-cocycle.
Conversely, if satisfying Equations (25) and (26), then is a MD3-LieA. □
Let
be an abelian extension of
by
and
a section. Define
,
and
, respectively, by
Note that is independent on the choice of s.
Proposition 8. With the above notations, is a representation of the MD3-LieA and is a 2-cocycle in the cohomology of the MD3-LieA with the coefficient in . Furthermore, the cohomological class of the 2-cocycle is independent of the choice of sections of p.
Proof. First, for any
, by Equation (1), we obtain
In addition, by Equation (2), we have
Hence, is a representation over .
Since
is an abelian extension of
by
, by Proposition 7,
is a 2-cocycle. Moreover, let
be two distinct sections providing 2-cocycles
and
, respectively. Define linear map
by
. Then,
and
which implies that
. So
. □
Definition 10. Let and be two abelian extensions of by . They are said to be equivalent if there is an isomorphism of MD3-LieA such that the following diagram is commutative: Now, we are ready to classify abelian extensions of a MD3-LieA.
Theorem 3. There is a one-to-one correspondence between equivalence classes of abelian extensions of a MD3-LieA by and the second cohomology group of with coefficients in the representation .
Proof. Assume that
and
are two equivalent abelian extensions of
by
with the associated isomorphism
. Let
be a section of
. As
, we have
That is,
is a section of
. Denote
. Since
is an isomorphism of MD
3-LieAs such that
, we obtain
and
Hence, all equivalent abelian extensions give rise to the same element in .
Conversely, suppose that
, and we can construct two abelian extensions
and
via Equations (21) and (22). Then, there exists a linear map
such that
Define linear map by Then, is an isomorphism of these two abelian extensions and . □
Remark 4. In particular, any vector space with linear transformation can serve as a trivial representation of . In this situation, central extensions of by are classified by the second cohomology group of with the coefficient in the trivial representation
5. -Extensions of MD3-LieAs
The
-extension of a 3-Lie algebra was studied in [
11]. In this section, we consider
-extensions of MD
3-LieAs by the second cohomology groups with the coefficient in a coadjoint representation.
Let
be a MD
3-LieA and
be the dual space of
. By Example 6,
is a coadjoint representation of
. Suppose that
. Define a trilinear map
and a linear map
, respectively, by
Similar to Proposition 7, we have the following result.
Proposition 9. With the above notations, is a MD3-LieA if and only if is a 2-cocycle in the cohomology of the MD3-LieA with the coefficient in the representation .
Definition 11. The MD3-LieA is called the -extension of the MD3-LieA . Denote the -extension by .
Definition 12. Let be a MD3-LieA. is said to be metrised if it has a non-degenerate symmetric bilinear form which satisfies We may also say that is a metric MD3-LieA.
Define a bilinear map
by
Proposition 10. With the above notations, is a metric MD3-LieA if and only if Proof. For any
, using Equations (6), (27)–(31) we have
Thus, we obtain the result. □
Let
be a metric MD
3-LieA, then
induces an isomorphism
defined by
Proposition 11. With the above notations, is an isomorphism from the adjoint representation to the coadjoint representation
Proof. For any
by Equations (6) and (29), we have
which implies that
In addition, for any
by Equations (6) and (30), we have
which implies that
Therefore,
is an isomorphism from
to
. □