1. Introduction
The notion of
n-Lie algebras was introduced by Filippov [
1] in 1985. It is the algebraic structure corresponding to Nambu mechanics [
2,
3,
4]. If
, then we get Lie algebra structure. Nanmbu’s used 3-Lie algebras to describe simultaneous classical dynamics of three particles in [
3]. Takhtajan [
5] systematically developed a foundemental theory of
n-Poisson or Nambu–Poisson manifolds, and established a connection between Nambu mechanics and Filippov algebras. Numerous works have been devoted to various aspects of
n-Lie algebras in both mathematics and physics, see [
6,
7,
8,
9,
10,
11,
12,
13,
14] and their references.
The method of deformation is ubiquitous in mathematics and physics. Gerstenhaber developed a deformation theory of associative algebras in [
15,
16]. Subsequently, Nijenhuis and Richardson generalized it to Lie algebras in [
17,
18]. Both associative algebras and Lie algebras are algebras over some quadratic operads. Balavoine [
19] investigated a deformation theory of quadratic operads. Cohomology and deformation of
n-Lie algebras have been studied from several directions. In [
7], central trivial extensions and infinitesimal deformations are considered. The two-order deformations of 3-Lie algebras are discussed in [
1]. In general,
-order deformations of
n-Lie algebras were studied in [
12], meanwhile, Nijenhuis operators were obtained from a trivial deformation.
A Nijenhuis algebra is a nonunitary associative algebra
A with a linear endomorphism
N satisfying the Nijenhuis equation:
, where
N is called a Nijenhuis operator. The concept of a Nijenhuis operator on a Lie algebra originated from the important concept of a Nijenhuis tensor that was introduced by Nijenhuis [
20] in the study of pseudo-complex manifolds in the 1950s and was related to the well-known concepts of the Schouten–Nijenhuis bracket, the Frölicher–Nijenhuis bracket [
21], and the Nijenhuis–Richardson bracket. Nijenhuis operators on a Lie algebra appeared in a more general study of Poisson–Nijenhuis manifolds [
22] and in the context of the classical Yang–Baxter equation [
23,
24]. A Nijenhuis operator on a Lie algebra is related to its deformation. Nijenhuis operators on
n-Lie algebras have been studied in [
12]. They can be used to construct a deformation of
n-Lie algebras. As a generalization of the classical Yang–Baxter equation (CYBE) on Lie algebras [
25], the
-operator provides a solution of the CYBE on a Lie algebra [
26]. The
-operator on Lie algebras was generalized to
n-Lie algebras in [
12].
Deformations of algebras are described by cohomology groups. Derivations of algebras are also controlled by a cohomological group. Derivations are a basic concept in homotopy Lie algebras [
27], deformation formulas [
28] and differential Galois theory [
29]. They are of vital importance to control theories and gauge theories in quantum field theory [
30,
31]. In [
32,
33], the authors study algebras with derivations, which are a kind of homotopy algebra, from the operadic point of view. Lie algebras with derivations are usually called LieDer pairs. Recently, LieDer pairs have been studied from the cohomological point of view. Extensions and deformations of LieDer pairs are considered in [
34]. These results have been extended to associative algebras with derivations [
35], Leibniz algebras with derivations [
36] and Pre-Lie algebras with derivations [
4].
Inspired by the previous works, we would like to investigate the cohomological theory of n-Lie algebras with derivations.
The paper is organized as follows. In
Section 2, we introduce the notion of an
n-LieDer pair
and its representation
. In
Section 3, we consider the cohomology theory of
n-LieDer pairs. The relation between the cohomogy of
n-LieDer pair and associated LeibDer pair is also characterized. In
Section 4, we study
-order deformations of
n-LieDer pairs. We also describe the notions of a Nijenhuis operator and an
-operator on
n-LieDer pair
. Moreover, we show that
becomes an
n-LieDer pair and
is a representation of
. Finally, we discuss central extensions of an
n-LieDer pair in terms of the first cohomology group with coefficients in its trivial representation.
Unless otherwise specified, all vector spaces, linear maps, and tensor products are studied over an algebraically closed field .
2. -LieDer Pair and Representation
In this section, we introduce the concept of
n-LieDer pairs and representations of an
n-LieDer pair. An
n-LieDer pair is an
n-Lie algebra with a derivation, namely, we have the following
Definition 1. A derivation of an n-Lie algebra is a linear map satisfyingfor any . Suppose that is a derivation of an n-Lie algebra . Then we call the datum an n-LieDer pair. For any n-Lie algebra , is a Leibniz algebra with Leibniz bracket given byfor all and . Furthermore, if is an n-LieDer pair, then is a LeibDer pair (see [36]), whereand I is the identity endomorphism of . A representation of an n-Lie algebra consists of a vector space V together with a linear map such thatandfor all and . Next, we give the definition of representations of an n-LieDer pair.
Definition 2. Let be an n-LieDer pair and be a representation of the n-Lie algebra . Suppose is an endomorphism of V. Then is called a representation of iffor any , where Suppose is a representation of an n-LieDer pair and is the dual space of V. Define and viafor any and respectively. Then can be endowed with a representation of the n-LieDer pair , as follows. Proposition 1. Let be a representation of an n-LieDer pair . Then is also a representation of .
Proof. We only need to check that (1) holds for
and
. In fact, for any
and
, in view of (1) and (2), we have
Hence is also a representation of . □
Example 1. Let be an n-LieDer pair and define linear map by for any . Then is a representation and is a dual representation.
Similar to the trivial extension of a Lie algebra by its representation, we can check the following proposition.
Proposition 2. Let be a representation of n-LieDer pair . Given operations byand by Then is an n-LieDer pair.
Suppose that are 3-Lie algebras, and and are two linear mappings. Recall that is said to be a matched pair of 3-Lie algebras [37] if is a representation of A and is a representation of and satisfying the following:for any and . Then, there is a 3-Lie algebra structure on given by Then we have the following result.
Proposition 3. Let and be two 3-LieDer pairs such that is a matched pair of 3-Lie algebras with and . Furthermore, if is a representation of 3-LieDer pair and is a representation of 3-LieDer pair . Definefor any . Then is a 3-LieDer pair with We call the matched pair of 3-LieDer pairs. Proof. It is straightforward. □
3. Cohomology of -LieDer Pair
In this section, we will define the cohomology of an
n-LieDer pair with coefficients in its representation. For this purpose, let us recall the cohomology of Leibniz algebras and LeibDer pairs in [
19,
36,
38]. Let
be a representation of a Leibniz algebra
, and
for any
, the
p-cochains group. Suppose
is given by
for any
. Then
d is a coboundary operator and cohomological groups of
with coefficients in
V is determined by the coboundary operator
d.
Next recall the definition of the cohomology of LeibDer pairs. Let
be a representation of a LeibDer pair
and
Define a linear map
by
for
. Then
,
is a coboundary operator.
Finally, let us recall the cohomology of
n-Lie algebras. Let
be an
n-Lie algebra and
be the associated Leibniz algebra. Suppose that
is a representation of
, the space
of
p-cochains (
) is the set of multilinear maps of the form
and the coboundary operator
is as follows:
for all
and
.
Based on the previous cohomologies, we introduce a cohomology of an
n-LieDer pair
. Let
be a representation of
n-LieDer pair
and
which is called the
p-cochain group. Define a map
by
Then we have the following.
Theorem 1. The map given byis a coboundary operator, that is, . Proof. By direct calculation, we get the required result. □
By Theorem 1, we can obtain cohomological groups of
with the coefficients in
. It is well-know that any
n-LieDer pair
associates a LeibDer pair
. Then it has the cohomology of the LeibDer pair
Are there some relations between these two cohomologies? Suppose that
(resp.
) is the set of
p-cochains of
n-LieDer pair (resp. LeibDer pair). Define
as follows. For
,
and
,
If
,
and
,
With these notations, we have an important result. However, let us prove the following result first.
Theorem 2. The linear map given byis a cochain map, that is, Proof. For any
, we get
and
See ref. [
13] (Theorem 3), we only need to check that
. In fact, for the case of
, it is clear. When
, for any
,
,
Hence, and thus □
The following corollary gives another proof of Theorem 1.
Corollary 1. If , then
Proof. In view of , we have □
Let
,
and
. In the light of [
13], we know that both
and
have graded Lie algebra structures, we denote them by
and
respectively. In view of [
36],
is a graded Lie algebra with the Lie algebra structure
given by
Then we have the following.
Proposition 4. Proof. It can be induced directly from Corollary 1 and [
13] (Lemma 1). □
From the previous proposition, we obtain the following theorem immediately.
Theorem 3. is a graded Lie algebra. Its Maurer–Cartan elements are n-LieDer pair .
Proof. According to Proposition 4, is a graded Lie algebra.
For any
,
So
is a Maurer–Cartan element if and only if
and
In the light of [
13],
if and only if
is an
n-Lie algebra.
At the same time, for any
, suppose
, we get
It follows that is a Maurer–Cartan element if and only if is an n-LieDer pair. □
Let
be an
n-LieDer pair. Define the linear map
by
then
. Hence, we get the following.
Proposition 5. is a differential graded Lie algebra.
4. Deformation of -LieDer Pairs
In the next two sections, we give some applications of the cohomology of an n-LieDer pair. In this section we use it to study the deformation of n-Lie Der pairs. First of all, let us introduce the deformations of an n-LieDer pair.
Let
be an
n-LieDer pair. Denote
Let
be skew-symmetric multilinear maps and linear maps
for any
. Consider the space
of formal power series in
t with coefficients in
and a family of linear maps:
and
where
.
If all
are
n-LieDer pairs, we say that
generate an
-order deformation of the
n-LieDer pair
. We also denote by
For any
, in the next proposition, we use
to denote the element
Proposition 6. generate an -order deformation of the n-LieDer pair if and only if the following holds for any and andwhere is given by Proof. is an
n-LieDer pair if and only if
and
In the light of [
12] (Proposition 1]), we only need to check that (7) is equivalent to (5). Combining (4) and (7), we achieve that (5) holds. Hence, we get the results. □
Corollary 2. If generate an -order deformation of the n-LieDer pair , then is a one-cocycle of the n-LieDer pair with the coefficients in the adjoint representation .
Proof. For any
,
For , (6) is equivalent to , and is equivalent to (5). So the conclusion holds. □
Definition 3. An -order deformation of the n-LieDer pair is said to be trivial if there is a linear map such that satisfiesand Similarly to the case of
n-Lie algebra [
12], we can define the Nijenhuis operator of an
n-LieDer pair.
Definition 4. Let be an n-LieDer pair. A linear map is called a Nijenhuis operator if the following holds:andfor any , where Definition 5. Let be an n-LieDer pair and be its representation. A linear map is called an -operator if it satisfies:andfor any . Using the above concepts, we can define some new n-LieDer pairs.
Proposition 7. Let be a representation of n-LieDer pair and be an -operator. Then is an n-LieDer pair, where Proof. We only need to check that
is a derivation of
V. In fact, according to (1) and (8), we have
and
Therefore,
that is,
is a derivation of
V. □
Similarly to the case of an
n-Lie algebra [
12], we have the following.
Proposition 8. Let a representation of an n-LieDer pair . Then, a linear map is an -operator operator if and only if with is a Nijenhuis operator on semidirect product .
Let be a linear map given by Then is a representation of by Proposition 8. Furthermore, we have the following result.
Proposition 9. Let be an -operator of n-LieDer pair with representation . Then is a representation of n-LieDer pair .
Proof. In view of (1) and (9), for any
and
, we have
and
□
Finally, in this section, let us study the cohomology of the new n-LieDer pair with coefficients in the representation . Suppose that is the associated Leibniz algebra. Then the space of p-cochains () is the set of multilinear maps of the form and the coboundary operator is given by:
for all
and
.
Define a map
by
Now we are ready to define cohomology of the
n-LieDer pair
with coefficients in the representation
. Denote the set of
p-cochains (
) by
, and the coboundary operator
is given by
Denote by the cohomology group of this cochain complex, which is called the cohomology of the n-LieDer pair with coefficients in the representation .
We calculate the 0-cocycle.
For any
,
By direct computation,
and
for any
.
It follows that
is a 0-cocycle if and only
and
Clearly, -operator on n-LieDer pair associated to the representation satisfying the above conditions. Hence, we have the following conclusion.
Proposition 10. The -operator on n-LieDer pair associated with the representation is a 0-cocycle of the cochain complex .