On the Deformations and Derivations of 𝑛-Ary Multiplicative Hom-Nambu-Lie Superalgebras

Abstract

We introduce the relevant concepts of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras and construct three classes of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras. As a generalization of the notion of derivations for 𝑛-ary multiplicative Hom-Nambu-Lie algebras, we discuss the derivations of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the theory of one parameter formal deformation of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras is developed by choosing a suitable cohomology.

1. Introduction

The notion of 𝑛-Lie algebras was introduced by Filippov in 1985 in [1]. The 𝑛-Lie algebra is a vector space endowed with an 𝑛-ary linear skew-symmetric product which satisfies the generalized Jacobi identity (also named Filippov identity). For 𝑛=3 this product is a special case of the Nambu bracket, introduced by Nambu in 1973 in [2], and was well known in physics, as a generalization of the Poisson bracket in Hamiltonian mechanics. 𝑛-Lie algebras are also useful in the research for M2-branes in the string theory and are closely linked to the Plücker relation in the literature in physics in [3–6].

In 1996, the concept of 𝑛-Lie superalgebras was firstly introduced by Daletskiĭ and Kushnirevich in [7]. Moreover, Cantarini and Kac gave a more general concept of 𝑛-Lie superalgebras again in 2010 in [8]. 𝑛-Lie superalgebras are more general structures including 𝑛-Lie algebras, 𝑛-ary Nambu-Lie superalgebras, and Lie superalgebras.

The general Hom-algebra structures arose first in connection with quasi-deformation and discretizations of Lie algebras of vector fields. These quasi-deformations lead to quasi-Lie algebras, a generalized Lie algebra structure in which the skew-symmetry and Jacobi conditions are twisted. Hom-Lie algebras, Hom-associative algebras, Hom-Lie superalgebras, Hom-bialgebras, 𝑛-ary Hom-Nambu-Lie algebras, and quasi-Hom-Lie algebras are discussed in [9–20]. Generalizations of 𝑛-ary algebras of Lie type and associative type by twisting the identities using linear maps have been introduced in [21].

The mathematical theory of deformations has proved to be a powerful tool in modeling physical reality. For example, (algebras associated with) classical quantum mechanics (and field theory) on a Poisson phase space can be deformed to (algebras associated with) quantum mechanics (and quantum field theory). The deformation of algebraic systems has been one of the problems that many mathematical researchers are interested in; Gerstenhaber studied the deformation theory of algebras in a series of papers [22–26]. For example, it has been extended to covariant functors from a small category to algebras. In [27, 28], it is, respectively, extended to algebra systems, bialgebras, Hopf algebras, Leibniz pairs, Poisson algebras, and so forth. In [23], Gerstenhaber developed a theory of deformation of associative and Lie algebras. His theory links cohomologies of these algebras and the Gerstenhaber bracket giving obstructions to deformations. Nijenhuis and Richardson noticed strong similarities between Gerstenhaber theory and the deformations of complex analytic structures on compact manifolds [29]. They axiomatized the theory of deformations via the introduction of graded Lie algebras [30]. One such example was given by the theory of deformations of homomorphisms [31]. Inspired by these works, we study the deformation theory of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras in this paper. In addition, the paper also discusses derivations of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notions of derivations for 𝑛-ary multiplicative Hom-Nambu-Lie algebras.

This paper is organized as follows. In Section 1, we introduce the relevant concepts of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras and construct three classes of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras. In Section 2, the notion of derivation introduced for 𝑛-ary multiplicative Hom-Nambu-Lie algebras in [10] is extended to 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras. In Section 3, the theory of deformations of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras is developed by choosing a suitable cohomology.

Definition 1 (see [32]). An 𝑛-ary Nambu-Lie superalgebra is a pair (𝔤,[⋅,…,⋅]) consisting of a ℤ₂-graded vector space 𝔤=𝔤₀⊕𝔤₁ and a multilinear mapping [⋅,…,⋅]∶𝔤×⋯×𝔤_𝑛→𝔤, satisfying 𝑥₁,…,𝑥_𝑛=𝑥₁+⋯+𝑥_𝑛,𝑥₁,…,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛=−(−1)^{|𝑥𝑖||𝑥𝑖+1|}×𝑥₁,…,𝑥_𝑖+1,𝑥_𝑖,…,𝑥_𝑛,𝑥₁,…,𝑥_𝑛−1,𝑦₁,…,𝑦_𝑛=^𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}⋅𝑦₁,…,𝑦_𝑖−1,𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,𝑦_𝑖+1,…,𝑦_𝑛, (1) where |𝑥|∈ℤ₂ denotes the degree of a homogeneous element 𝑥∈𝔤.

Definition 2. An 𝑛-ary Hom-Nambu-Lie superalgebra is a triple (𝔤,[⋅,…,⋅],𝛼) consisting of a ℤ₂-graded vector space 𝔤=𝔤₀⊕𝔤₁, a multilinear mapping [⋅,…,⋅]∶𝔤×⋯×𝔤_𝑛→𝔤, and a family 𝛼=(𝛼_𝑖)_{1≤𝑖≤𝑛−1} of even linear maps 𝛼_𝑖∶𝔤→𝔤, satisfying 𝑥₁,…,𝑥_𝑛=𝑥₁+⋯+𝑥_𝑛,𝑥₁,…,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛=−(−1)^{|𝑥𝑖||𝑥𝑖+1|}×𝑥₁,…,𝑥_𝑖+1,𝑥_𝑖,…,𝑥_𝑛,𝛼₁𝑥₁,…,𝛼_𝑛−1𝑥_𝑛−1,𝑦₁,…,𝑦_𝑛=^𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}⋅𝛼₁𝑦₁,…,𝛼_𝑖−1𝑦_𝑖−1,𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,⋅𝛼_𝑖𝑦_𝑖+1,…,𝛼_𝑛−1𝑦_𝑛, (2)(3)(4) where |𝑥|∈ℤ₂ denotes the degree of a homogeneous element 𝑥∈𝔤.

An 𝑛-ary Hom-Nambu-Lie superalgebra (𝔤,[⋅,…,⋅],𝛼) is multiplicative, if 𝛼=(𝛼_𝑖)_{1≤𝑖≤𝑛−1} with 𝛼₁=⋯=𝛼_𝑛−1=𝛼 and satisfying 𝛼𝑥₁,…,𝑥_𝑛=𝛼𝑥₁,…,𝛼𝑥_𝑛,∀𝑥₁,𝑥₂,…,𝑥_𝑛∈𝔤. (5)

If the 𝑛-ary Hom-Nambu-Lie superalgebra (𝔤,[⋅,…,⋅],𝛼) is multiplicative, then (4) can be read as 𝛼𝑥₁,…,𝛼𝑥_𝑛−1,𝑦₁,…,𝑦_𝑛=^𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}⋅𝛼𝑦₁,…,𝛼𝑦_𝑖−1,𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,𝛼𝑦_𝑖+1,…,𝛼𝑦_𝑛. 4^′

It is clear that 𝑛-ary Hom-Nambu-Lie algebras and Hom-Lie superalgebras are particular cases of 𝑛-ary Hom-Nambu-Lie superalgebras. In the sequel, when the notation “|𝑥|” appears, it means that 𝑥 is a homogeneous element of degree |𝑥|.

Definition 3. Let (𝔤,[⋅,…,⋅],𝛼) and (𝔤^′,[⋅,…,⋅]^′,𝛼^′) be two 𝑛-ary Hom-Nambu-Lie superalgebras, where 𝛼=(𝛼_𝑖)_{1≤𝑖≤𝑛−1} and 𝛼^′=(𝛼^′_𝑖)_{1≤𝑖≤𝑛−1}. A linear map 𝑓∶𝔤→𝔤 is an 𝑛-ary Hom-Nambu-Lie superalgebra morphism if satisfies 𝑓𝑥₁,…,𝑥_𝑛=𝑓𝑥₁,…,𝑓𝑥_𝑛^′,𝑓∘𝛼_𝑖=𝛼^′_𝑖∘𝑓,∀𝑖=1,…,𝑛−1. (6)

Theorem 4. Let (𝔤,[⋅,…,⋅],𝛼) be an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra and let 𝛽∶𝔤→𝔤 be a morphism of 𝑔 such that 𝛽∘𝛼=𝛼∘𝛽. Then (𝔤,𝛽∘[⋅,…,⋅],𝛽∘𝛼) is an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra.

Proof. Put [⋅,…,⋅]_𝛽∶=𝛽∘[⋅,…,⋅]. Then 𝛽∘𝛼𝑥₁,…,𝑥_𝑛𝛽=𝛽∘𝛼𝛽𝑥₁,…,𝑥_𝑛=𝛽∘𝛼∘𝛽𝑥₁,…,𝑥_𝑛=𝛽∘𝛽∘𝛼𝑥₁,…,𝑥_𝑛=𝛽𝛽∘𝛼𝑥₁,…,𝛽∘𝛼𝑥_𝑛=𝛽∘𝛼𝑥₁,…,𝛽∘𝛼𝑥_𝑛𝛽; (7) that is, 𝛽∘𝛼 is a morphism of 𝑔. Moreover, we have 𝛽∘𝛼𝑥₁,…,𝛽∘𝛼𝑥_𝑛−1,𝑦₁,…,𝑦_𝑛𝛽𝛽=𝛽𝛽∘𝛼𝑥₁,…,𝛽∘𝛼𝑥_𝑛−1,𝛽𝑦₁,…,𝑦_𝑛=𝛽²𝛼𝑥₁,…,𝛼𝑥_𝑛−1,𝑦₁,…,𝑦_𝑛=𝛽^2𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}×𝛼𝑦₁,…,𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,…,𝛼𝑦_𝑛=^𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}×𝛽𝛽∘𝛼𝑦₁,…,𝛽𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,…,𝛽∘𝛼𝑦_𝑛=^𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}×𝛽∘𝛼𝑦₁,…,𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖𝛽,…,𝛽∘𝛼𝑦_𝑛𝛽. (8) Therefore, (𝔤,𝛽∘[⋅,…,⋅],𝛽∘𝛼) is an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra.

In particular, we have the following example.

Example 5. Let (𝔤,[⋅,…,⋅]) be an 𝑛-ary Nambu-Lie superalgebra and let 𝜌∶𝔤→𝔤 be an 𝑛-ary Nambu-Lie superalgebra endomorphism. Then (𝔤,𝜌∘[⋅,…,⋅],𝜌) is an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra.

Definition 6. Let (𝔤,[⋅,…,⋅]_𝔤,𝛼) be an 𝑛-ary Hom-Nambu-Lie superalgebra. A graded subspace 𝐻⊆𝔤 is a Hom-subalgebra of (𝔤,[⋅,…,⋅]_𝔤,𝛼) if 𝛼(𝐻)⊆𝐻 and 𝐻 is closed under the bracket operation [⋅,…,⋅]_𝔤,; that is, [𝑢₁,…,𝑢_𝑛]_𝔤∈𝐻, ∀𝑢₁,…,𝑢_𝑛∈𝐻.
A graded subspace 𝐻⊆𝔤 is a Hom-ideal of (𝔤,[⋅,…,⋅]_𝔤,𝛼) if 𝛼(𝐻)⊆𝐻 and [𝑢₁,𝑢₂,…,𝑢_𝑛]_𝔤∈𝐻, ∀𝑢₁∈𝐻, 𝑢₂,…,𝑢_𝑛∈𝔤.

Definition 7. Let (𝔤₁,[⋅,…,⋅]₁,𝛼) and (𝔤₂,[⋅,…,⋅]₂,𝛽) be two 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras. Suppose that 𝜙∶𝔤₁→𝔤₂ is a linear map. 𝔊_𝜙={(𝑥,𝜙(𝑥))∣𝑥∈𝔤₁}⊆𝔤₁⊕𝔤₂ is called as the graph of a linear map 𝜙∶𝔤₁→𝔤₂.

Proposition 8. Given two 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras (𝔤₁,[⋅,…,⋅]_𝔤1,𝛼) and (𝔤₂,[⋅,…,⋅]_𝔤2,𝛽), there is an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra (𝔤₁⊕𝔤₂,[⋅,…,⋅]_{𝔤1⊕𝔤2},𝛼+𝛽), where the bilinear map [⋅,…,⋅]_{𝔤1⊕𝔤2}∶∧²(𝔤₁⊕𝔤₂)→𝔤₁⊕𝔤₂ is given by 𝑢₁+𝑣₁,…,𝑢_𝑛+𝑣_{𝑛𝔤1⊕𝔤2}=𝑢₁,…,𝑢_𝑛𝔤1+𝑣₁,…,𝑣_𝑛𝔤2,∀𝑢_𝑖∈𝔤₁,𝑣_𝑖∈𝔤₂(𝑖=1,2,…,𝑛), (9) and the linear map (𝛼+𝛽)∶𝔤₁⊕𝔤₂→𝔤₁⊕𝔤₂ is given by 𝛼+𝛽(𝑢+𝑣)=𝛼(𝑢)+𝛽(𝑣),∀𝑢∈𝔤₁,𝑣∈𝔤₂. (10)

Proof. For any 𝑢_𝑖∈𝔤₁,𝑣_𝑖∈𝔤₂, we have 𝑢₁+𝑣₁,…,𝑢_𝑖+𝑣_𝑖,𝑢_𝑖+1+𝑣_𝑖+1,…,𝑢_𝑛+𝑣_{𝑛𝔤1⊕𝔤2}=𝑢₁,…,𝑢_𝑖,𝑢_𝑖+1,…,𝑢_𝑛𝔤1+𝑣₁,…,𝑣_𝑖,𝑣_𝑖+1,…,𝑣_𝑛𝔤2=−(−1)^{|𝑢𝑖||𝑢𝑖+1|}𝑢₁,…,𝑢_𝑖+1,𝑢_𝑖,…,𝑢_𝑛𝔤1−(−1)^{|𝑣𝑖||𝑣𝑖+1|}𝑣₁,…,𝑣_𝑖+1,𝑣_𝑖,…,𝑣_𝑛𝔤2=−(−1)^{|𝑢𝑖||𝑢𝑖+1|}×𝑢₁,…,𝑢_𝑖,𝑢_𝑖+1,…,𝑢_𝑛𝔤1+𝑣₁,…,𝑣_𝑖,𝑣_𝑖+1,…,𝑣_𝑛𝔤2=−(−1)^{|𝑢𝑖||𝑢𝑖+1|}×𝑢₁+𝑣₁,…,𝑢_𝑖+1+𝑣_𝑖+1,𝑢_𝑖+𝑣_𝑖,…,𝑢_𝑛+𝑣_{𝑛𝔤1⊕𝔤2}. (11) The bracket is obviously supersymmetric. By a direct computation we have 𝛼+𝛽𝑢₁+𝑣₁,…,𝛼+𝛽𝑢_𝑛−1+𝑣_𝑛−1,𝑥₁+𝑦₁,…,𝑥_𝑛+𝑦_{𝑛𝔤1⊕𝔤2𝔤1⊕𝔤2}=𝛼𝑢₁+𝛽𝑣₁,…,𝛼𝑢_𝑛−1+𝛽𝑣_𝑛−1,𝑥₁+𝑦₁,…,𝑥_𝑛+𝑦_{𝑛𝔤1⊕𝔤2𝔤1⊕𝔤2}=𝛼𝑢₁,…,𝛼𝑢_𝑛−1,𝑥₁,…,𝑥_{𝑛𝔤1𝔤1}+𝛽𝑣₁,…,𝛽𝑣_𝑛−1,𝑦₁,…,𝑦_{𝑛𝔤2𝔤2}=^𝑛_𝑖=1(−1)^{(|𝑢1|+⋯+|𝑢𝑛−1|)(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝛼𝑥₁,…,𝛼𝑥_𝑖−1,𝑢₁,…,𝑢_𝑛−1,𝑥_𝑖𝔤1,𝛼𝑥_𝑖+1,…,𝛼𝑥_𝑛𝔤1+𝛽𝑦₁,…,𝛽𝑦_𝑖−1,𝑣₁,…,𝑣_𝑛−1,𝑦_𝑖𝔤2,𝛽𝑦_𝑖+1,…,𝛽𝑦_𝑛𝔤2=^𝑛_𝑖=1(−1)^{(|𝑢1|+⋯+|𝑢𝑛−1|)(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝛼𝑥₁+𝛽𝑦₁,…,𝛼𝑥_𝑖−1+𝛽𝑦_𝑖−1,𝑢₁,…,𝑢_𝑛−1,𝑥_𝑖𝔤1+𝑣₁,…,𝑣_𝑛−1,𝑦_𝑖𝔤2,𝛼(𝑥_𝑖+1)+𝛽(𝑦_𝑖+1),…,𝛼(𝑥_𝑛)+𝛽𝑦_{𝑛𝔤1⊕𝔤2}=^𝑛_𝑖=1(−1)^{(|𝑢1|+⋯+|𝑢𝑛−1|)(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝛼+𝛽𝑥₁+𝑦₁,…,𝛼+𝛽𝑥_𝑖−1+𝑦_𝑖−1,𝑢₁+𝑣₁,…,𝑢_𝑛−1+𝑣_𝑛−1,𝑥_𝑖+𝑦_{𝑖𝔤1⊕𝔤2},(𝛼+𝛽)(𝑥_𝑖+1+𝑦_𝑖+1),…,𝛼+𝛽𝑥_𝑛+𝑦_{𝑛𝔤1⊕𝔤2}=^𝑛_𝑖=1(−1)^{(|𝑢1|+⋯+|𝑢𝑛−1|)(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝛼+𝛽𝑥₁+𝑦₁,…,𝛼+𝛽𝑥_𝑖−1+𝑦_𝑖−1,𝑢₁+𝑣₁,…,𝑢_𝑛−1+𝑣_𝑛−1,𝑥_𝑖+𝑦_{𝑖𝔤1⊕𝔤2},(𝛼+𝛽)(𝑥_𝑖+1+𝑦_𝑖+1),…,𝛼+𝛽𝑥_𝑛+𝑦_{𝑛𝔤1⊕𝔤2}. (12) The result follows.

Proposition 9. A linear map 𝜙∶(𝔤₁,[⋅,…,⋅]_𝔤1,𝛼)→(𝔤₂,[⋅,…,⋅]_𝔤2,𝛽) is a morphism of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras if and only if the graph 𝔊_𝜙⊆𝔤₁⊕𝔤₂ is a Hom-subalgebra of (𝔤₁⊕𝔤₂,[⋅,…,⋅]_{𝔤1⊕𝔤2},𝛼+𝛽).

Proof. Let 𝜙∶(𝔤₁,[⋅,…,⋅]_𝔤1,𝛼)→(𝔤₂,[⋅,…,⋅]_𝔤2,𝛽) be a morphism of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras. Then 𝑢₁+𝜙𝑢₁,…,𝑢_𝑛+𝜙𝑢_{𝑛𝔤1⊕𝔤2}=𝑢₁,…,𝑢_𝑛𝔤1+𝜙𝑢₁,…,𝜙𝑢_𝑛𝔤2=𝑢₁,…,𝑢_𝑛𝔤1+𝜙𝑢₁,…,𝑢_𝑛𝔤2. (13)
Then the graph 𝔊_𝜙 is closed under the bracket operation [⋅,…,⋅]_{𝔤1⊕𝔤2}. Furthermore, we obtain 𝛼+𝛽𝑢+𝜙(𝑢)=𝛼(𝑢)+𝛽∘𝜙(𝑢)=𝛼(𝑢)+𝜙∘𝛼(𝑢), (14) which implies that (𝛼+𝛽)(𝔊_𝜙)⊆𝔊_𝜙. Thus, 𝔊_𝜙 is a Hom-subalgebra of (𝔤₁⊕𝔤₂,[⋅,…,⋅]_{𝔤1⊕𝔤2},𝛼+𝛽).
Conversely, if the graph 𝔊_𝜙⊆𝔤₁⊕𝔤₂ is a Hom-subalgebra of (𝔤₁⊕𝔤₂,[⋅,…,⋅]_{𝔤1⊕𝔤2},𝛼+𝛽), then we have 𝑢₁+𝜙𝑢₁,…,𝑢_𝑛+𝜙𝑢_{𝑛𝔤1⊕𝔤2}=𝑢₁,…,𝑢_𝑛𝔤1+𝜙𝑢₁,…,𝑢_𝑛𝔤2, (15) which implies that 𝜙𝑢₁,…,𝑢_𝑛𝔤2=𝜙𝑢₁,…,𝑢_𝑛𝔤1. (16) Furthermore, (𝛼+𝛽)(𝔊_𝜙)⊂𝔊_𝜙 yields that 𝛼+𝛽𝑢+𝜙(𝑢)=𝛼(𝑢)+𝛽∘𝜙(𝑢)∈𝔊_𝜙, (17) which is equivalent to the condition 𝛽∘𝜙(𝑢)=𝜙∘𝛼(𝑢); that is, 𝛽∘𝜙=𝜙∘𝛼. Therefore, 𝜙 is a morphism of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras.

2. Derivations of 𝑛-Ary Multiplicative Hom-Nambu-Lie Superalgebras

Let (𝔤,[⋅,…,⋅],𝛼) be an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra. We denote by 𝛼^𝑘 the 𝑘-times compositions of 𝛼. In particular, we set 𝛼⁰=id.

Definition 10. For 𝑘≥0, we call 𝐷∈End(𝔤) an 𝛼^𝑘-derivation of the 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra (𝔤,[⋅,…,⋅],𝛼) if 𝐷∘𝛼=𝛼∘𝐷 (18) and for 𝑥_𝑖∈𝔤(𝑖=1,…,𝑛), 𝐷𝑥₁,…,𝑥_𝑛=^𝑛_𝑖=1(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝛼^𝑘𝑥₁,…,𝛼^𝑘𝑥_𝑖−1,𝐷𝑥_𝑖,𝛼^𝑘𝑥_𝑖+1𝛼,…,_𝑘𝑥_𝑛. (19)

We denote by Der_𝛼𝑘(𝔤) the set of 𝛼^𝑘-derivations of the 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra (𝔤,[⋅,…,⋅],𝛼). Notice that we obtain classical derivations for 𝑘=0.

For 𝒳∈𝔤^∧𝑛−1 satisfying 𝛼(𝒳)=𝒳 and 𝑘≥0, we define the map ad_𝑘(𝒳)∈End(𝔤) by ad_𝑘(𝒳)𝑦=𝑥₁,…,𝑥_𝑛−1,𝛼^𝑘𝑦,∀𝑦∈𝔤. (20) Then one has the following.

Lemma 11. The map ad_𝑘(𝒳) is an 𝛼^𝑘+1-derivation and is called an inner 𝛼^𝑘+1-derivation.

We denote by Inn_𝛼𝑘(𝔤) the 𝕂-vector space generated by all inner 𝛼^𝑘+1-derivations. For any 𝐷∈Der_𝛼𝑘(𝔤) and 𝐷^′∈Der_𝛼𝑘′(𝔤), we define their commutator [𝐷,𝐷^′]=𝐷∘𝐷^′−(−1)^{|𝐷||𝐷′|}𝐷^′∘𝐷. Set Der(𝔤)=⊕_𝑘≥0Der_𝛼𝑘(𝔤) and Inn(𝔤)=⊕_𝑘≥0Inn_𝛼𝑘(𝔤).

Lemma 12. For any D∈Der_𝛼𝑘(𝔤) and 𝐷^′∈Der_𝛼𝑘′(𝔤), one has [𝐷,𝐷^′]∈Der_{𝛼′𝑘+𝑘}(𝔤).

Proof. Let 𝑥_𝑖∈𝔤, 1≤𝑖≤𝑛. 𝐷∈Der_𝛼𝑘(𝔤) and 𝐷^′∈Der_𝛼𝑘′(𝔤), and then 𝐷∘𝐷^′𝑥₁,…,𝑥_𝑛=𝐷^𝑛_𝑖=1(−1)^{|𝐷′|(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝛼^𝑘′𝑥₁,…,𝛼^𝑘′𝑥_𝑖−1,𝐷^′𝑥_𝑖,𝛼^𝑘′𝑥_𝑖+1,…,𝛼^𝑘′𝑥_𝑛=^𝑛_𝑖=1(−1)^{|𝐷′|(|𝑥1|+⋯+|𝑥𝑖−1|)}𝐷×𝛼^𝑘′𝑥₁,…,𝛼^𝑘′𝑥_𝑖−1,𝐷^′𝑥_𝑖,𝛼^𝑘′𝑥_𝑖+1,…,𝛼^𝑘′𝑥_𝑛=^𝑛_𝑖=1(−1)^{|𝐷′|(|𝑥1|+⋯+|𝑥𝑖−1|)}(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}⋅𝛼^𝑘+𝑘′𝑥₁,…,𝛼^𝑘+𝑘′𝑥_𝑖−1,𝐷∘𝐷^′𝑥_𝑖,𝛼^𝑘+𝑘′𝑥_𝑖+1,…,𝛼^𝑘+𝑘′𝑥_𝑛+_𝑖<𝑗(−1)^{|𝐷′|(|𝑥1|+⋯+|𝑥𝑖−1|)}(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑗−1|+|𝐷′|)}⋅𝛼^𝑘+𝑘′𝑥₁,…,𝛼^𝑘𝐷^′𝑥_𝑖𝛼,…,_𝑘′𝐷𝑥_𝑗,…,𝛼^𝑘+𝑘′𝑥_𝑛+_𝑖>𝑗(−1)^{|𝐷′|(|𝑥1|+⋯+|𝑥𝑖−1|)}(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑗−1|)}⋅𝛼^𝑘+𝑘′𝑥₁,…,𝛼^𝑘′𝐷𝑥_𝑗𝛼,…,_𝑘𝐷^′𝑥_𝑖,…,𝛼^𝑘+𝑘′𝑥_𝑛,−(−1)^{|𝐷||𝐷′|}𝐷^′∘𝐷𝑥₁,…,𝑥_𝑛=−(−1)^{|𝐷||𝐷′|}𝐷^′×^𝑛_𝑖=1(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝛼^𝑘𝑥₁,…,𝐷𝑥_𝑖,…,𝛼^𝑘𝑥_𝑛=−(−1)^{|𝐷||𝐷′|𝑛}_𝑖=1(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝐷^′𝛼^𝑘𝑥₁,…,𝐷𝑥_𝑖,…,𝛼^𝑘𝑥_𝑛=−(−1)^{|𝐷||𝐷′|𝑛}_𝑖=1(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}(−1)^{|𝐷′|(|𝑥1|+⋯+|𝑥𝑖−1|)}⋅𝛼^𝑘′+𝑘𝑥₁,…,𝛼^𝑘′+𝑘𝑥_𝑖−1,𝐷^′∘𝐷𝑥_𝑖,𝛼^𝑘′+𝑘𝑥_𝑖+1,…,𝛼^𝑘′+𝑘𝑥_𝑛−(−1)^{|𝐷||𝐷′|}_𝑖<𝑗(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}(−1)^{|𝐷′|(|𝑥1|+⋯+|𝑥𝑗−1|+|𝐷|)}⋅𝛼^𝑘′+𝑘𝑥₁,…,𝛼^𝑘′𝐷𝑥_𝑖𝛼,…,_𝑘𝐷^′𝑥_𝑗,…,𝛼^𝑘′+𝑘𝑥_𝑛−(−1)^{|𝐷||𝐷′|}_𝑖>𝑗(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}(−1)^{|𝐷′|(|𝑥1|+⋯+|𝑥𝑗−1|)}⋅𝛼^𝑘′+𝑘𝑥₁,…,𝛼^𝑘𝐷^′𝑥_𝑗𝛼,…,_𝑘′𝐷𝑥_𝑖,…,𝛼^𝑘′+𝑘𝑥_𝑛. (21) Then we obtain 𝐷,𝐷^′𝑥₁,…,𝑥_𝑛=𝐷∘𝐷^′−(−1)^{|𝐷||𝐷′|}𝐷^′∘𝐷𝑥₁,…,𝑥_𝑛=^𝑛_𝑖=1(−1)^{|𝐷′|(|𝑥1|+⋯+|𝑥𝑖−1|)}(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝛼^𝑘+𝑘′𝑥₁,…,𝛼^𝑘+𝑘′𝑥_𝑖−1,𝐷∘𝐷^′−(−1)^{|𝐷||𝐷′|}𝐷^′∘𝐷𝑥_𝑖,𝛼^𝑘+𝑘′𝑥_𝑖+1,…,𝛼^𝑘+𝑘′𝑥_𝑛=^𝑛_𝑖=1(−1)^{(|𝐷′|+|𝐷|)(|𝑥1|+⋯+|𝑥𝑖−1|)}⋅𝛼^𝑘+𝑘′𝑥₁,…,𝛼^𝑘+𝑘′𝑥_𝑖−1,𝐷,𝐷^′𝑥_𝑖,𝛼^𝑘+𝑘′𝑥_𝑖+1,…,𝛼^𝑘+𝑘′𝑥_𝑛, (22) which yields that [𝐷,𝐷^′]∈Der_{𝛼′𝑘+𝑘}(𝔤).

Proposition 13. The pair (Der(𝔤),[⋅,⋅]), where the bracket is the usual commutator, defines a Lie superalgebra and Inn(𝔤) constitutes an ideal of it.

Proof. (Der(𝔤),[⋅,⋅]) is a Lie superalgebra by using Lemma 12. We show that Inn(𝔤) is an ideal. Let ad_𝑘−1(𝒳)(𝑦)=[𝑥₁,…,𝑥_𝑛−1,𝛼^𝑘−1(𝑦)] be an inner 𝛼^𝑘-derivation on 𝔤 and 𝐷∈Der_𝛼𝑘′(𝔤) for 𝑘≥1 and 𝑘^′≥0. Then [𝐷,ad_𝑘−1(𝒳)]∈Der_{𝛼′𝑘+𝑘}(𝔤) and for any 𝑦∈𝔤𝐷,ad_𝑘−1(𝒳𝑦𝑥)=𝐷₁,…,𝑥_𝑛−1,𝛼^𝑘−1𝑦−(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑛−1|)}𝑥₁,…,𝑥_𝑛−1,𝛼^𝑘−1𝐷𝑦=𝐷𝛼^𝑘𝑥₁,…,𝛼^𝑘𝑥_𝑛−1,𝛼^𝑘−1𝑦−(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑛−1|)}⋅𝛼^𝑘+𝑘′𝑥₁,…,𝛼^𝑘+𝑘′𝑥_𝑛−1,𝛼^𝑘−1𝐷𝑦=_{𝑖≤𝑛−1}(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝛼^𝑘+𝑘′𝑥₁,…,𝐷𝛼^𝑘𝑥_𝑖𝛼,…,_𝑘+𝑘′𝑥_𝑛−1,𝛼^{𝑘+𝑘′−1}𝑦=_{𝑖≤𝑛−1}(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}×𝑥₁,…,𝐷𝑥_𝑖,…,𝑥_𝑛−1,𝛼^{𝑘+𝑘′−1}𝑦=_{𝑖≤𝑛−1}(−1)^{|𝐷|(|𝑥1|+⋯+|𝑥𝑖−1|)}ad_{𝑘+𝑘′−1}×𝑥₁∧⋯∧𝐷𝑥_𝑖∧⋯∧𝑥_𝑛−1𝑦. (23) Therefore, [𝐷,ad_𝑘−1(𝒳)]∈Inn_{𝛼′𝑘+𝑘−1}(𝔤).

3. Deformations of 𝑛-Ary Multiplicative Hom-Nambu-Lie Superalgebras

Definition 14 (see [33]). For 𝑚≥1, we call 𝑚-coboundary operator of the 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra (𝔤,[⋅,…,⋅],𝛼) the even linear map 𝛿^𝑚∶𝐶^𝑚(𝔤,𝑉)→𝐶^𝑚+1(𝔤,𝑉) by 𝛿^𝑚𝑓𝒳₁,…,𝒳_𝑚,𝒳_𝑚+1,𝑧=_𝑖<𝑗(−1)^𝑖(−1)^{|𝒳𝑖|(|𝒳𝑖+1|+⋯+|𝒳𝑗−1|)}×𝑓𝛼𝒳₁,…,𝛼𝒳_𝑖𝒳,…,_𝑖,𝒳_𝑗𝛼,…,𝛼𝒳_𝑚+1(𝑧+,𝛼)_𝑚+1𝑖=1(−1)^𝑖(−1)^{|𝒳𝑖|(|𝒳𝑖+1|+⋯+|𝒳𝑚+1|)}×𝑓𝛼𝒳₁,…,𝛼𝒳_𝑖,…,𝛼𝒳_𝑚+1,𝒳_𝑖+⋅𝑧_𝑚+1𝑖=1(−1)^𝑖+1(−1)^{|𝒳𝑖|(|𝑓|+|𝒳1|+⋯+|𝒳𝑖−1|)}𝛼^𝑚𝒳_𝑖⋅𝑓𝒳₁,…,𝒳_𝑖,…,𝒳_𝑚+1+(),𝑧−1^𝑚𝑓𝒳₁,…,𝒳_𝑚,⋅𝒳_𝑚+1•_𝛼𝛼^𝑚(𝑧), (24) where 𝒳_𝑖=𝒳¹_𝑖∧⋯∧𝒳_𝑖^𝑛−1∈𝔤^∧𝑛−1, 𝑖=1,…,𝑚+1,𝑧∈𝔤, and the last term is defined by 𝑓𝒳₁,…,𝒳_𝑚,⋅𝒳_𝑚+1•_𝛼𝛼^𝑚(𝑧)=^𝑛−1_𝑖=1(−1)^{(|𝑓|+|𝒳1|+⋯+|𝒳𝑚|)(|𝒳1𝑚+1|+⋯+|𝒳𝑖−1𝑚+1|)}⋅𝛼^𝑚𝒳¹_𝑚+1,…,𝑓𝒳₁,…,𝒳_𝑚,𝒳^𝑖_𝑚+1𝛼,…,_𝑚𝒳^𝑛−1_𝑚+1,𝛼^𝑚(𝑧). (25)

Theorem 15 (see [33]). Let 𝑓∈𝐶^𝑚(𝔤,𝑉) be an 𝑚-cochain. Then 𝛿^𝑚+1∘𝛿^𝑚(𝑓)=0.

In [33], it also points out that the map 𝑓∈𝐶^𝑚(𝔤,𝑉) is called an 𝑚-supercocycle if 𝛿^𝑚𝑓=0. We denote by 𝑍^𝑚(𝔤,𝑉) the graded subspace spanned by 𝑚-supercocycles. Since 𝛿^𝑚+1∘𝛿^𝑚(𝑓)=0 for all 𝑓∈𝐶^𝑚(𝔤,𝑉), 𝛿^𝑚−1𝐶^𝑚−1(𝔤,𝑉) is a graded subspace of 𝑍^𝑚(𝔤,𝑉). Therefore, we can define a graded cohomology space 𝐻^𝑚(𝔤,𝑉) of 𝔤 as the graded factor space 𝑍^𝑚(𝔤,𝑉)/𝛿^𝑚−1𝐶^𝑚−1(𝔤,𝑉).

We next will discuss the deformation of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras. Let 𝕂[[𝑡]] denote the power series ring in one variable 𝑡 with coefficients in 𝕂 and let 𝔤[[𝑡]] be the set of formal series whose coefficients are elements of the vector space 𝔤.

Definition 16. Let (𝔤,[⋅,…,⋅],𝛼) be an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra over 𝕂. A deformation of (𝔤,[⋅,…,⋅],𝛼) is given by 𝕂[[𝑡]]-𝑛-linear map 𝑓_𝑡=_𝑝≥0𝑓_𝑝𝑡^𝑝∶𝔤[[𝑡]]×⋯×𝔤[[𝑡]]⟶𝔤[[𝑡]] (26) such that (𝔤[[𝑡]],𝑓_𝑡,𝛼) is also an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra. We call 𝑓₁ the infinitesimal deformation of (𝔤,[⋅,…,⋅],𝛼).

Since (𝔤[[𝑡]],𝑓_𝑡,𝛼) is an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra, 𝑓_𝑡 satisfies 𝛼∘𝑓_𝑡𝑥₁,…,𝑥_𝑛=𝑓_𝑡𝛼𝑥₁,…,𝛼𝑥_𝑛,𝑓_𝑡𝑥₁,…,𝑥_𝑛=𝑥₁+⋯+𝑥_𝑛,𝑓_𝑡𝛼𝑥₁,…,𝛼𝑥_𝑛−1,𝑓_𝑡𝑦₁,…,𝑦_𝑛=^𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}⋅𝑓_𝑡𝛼𝑦₁,…,𝛼𝑦_𝑖−1,𝑓_𝑡𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,𝛼𝑦_𝑖+1,…,𝛼𝑦_𝑛. (27)(28)(29) Equations (27)–(29) are, respectively, equivalent to 𝛼∘𝑓_𝑝𝑥₁,…,𝑥_𝑛=𝑓_𝑝𝛼𝑥₁,…,𝛼𝑥_𝑛,𝑓_𝑝𝑥₁,…,𝑥_𝑛=𝑥₁+⋯+𝑥_𝑛,_{𝑝+𝑞=𝑙}𝑓_𝑝𝛼𝑥₁,…,𝛼𝑥_𝑛−1,𝑓_𝑞𝑦₁,…,𝑦_𝑛=^𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}⋅_{𝑝+𝑞=𝑙}𝑓_𝑝𝛼𝑦₁,…,𝛼𝑦_𝑖−1,𝑓_𝑞𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,𝛼𝑦_𝑖+1,…,𝛼𝑦_𝑛. 27^′28^′29^′ We call these the deformation equations for an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra.

Equations 27^′ and 28^′ show that 𝑓_𝑝∈𝐶¹(𝔤,𝔤)₀. In 29^′, set 𝑙=1, and then 𝛼𝑥₁,…,𝛼𝑥_𝑛−1,𝑓₁𝑦₁,…,𝑦_𝑛+𝑓₁𝛼𝑥₁,…,𝛼𝑥_𝑛−1,𝑦₁,…,𝑦_𝑛−^𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}×𝛼𝑦₁,…,𝛼𝑦_𝑖−1,𝑓₁𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,𝛼𝑦_𝑖+1,…,𝛼𝑦_𝑛+𝑓₁𝛼𝑦₁,…,𝛼𝑦_𝑖−1,𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,𝛼𝑦_𝑖+1,…,𝛼𝑦_𝑛=0; (30) that is, 𝛿¹𝑓₁(𝑥₁,…,𝑥_𝑛−1,[𝑦₁,…,𝑦_𝑛−1,𝑦_𝑛])=0. Hence the infinitesimal deformation 𝑓₁∈𝑍¹(𝔤,𝔤)₀.

Definition 17. Two deformations 𝑓_𝑡 and 𝑓_𝑡′ of the 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra (𝔤,[⋅,⋯,⋅],𝛼) are said to be equivalent, if there exists an isomorphism of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras Φ_𝑡∶(𝔤,𝑓_𝑡,𝛼)→(𝔤,𝑓^′_𝑡,𝛼), where Φ_𝑡=∑_𝑖≥0𝜑_𝑖𝑡^𝑖, 𝜑_𝑖∶𝔤→𝔤 is a linear map such that 𝜑₀=id_𝔤;𝜑_𝑖∘𝛼=𝛼∘𝜑_𝑖;Φ_𝑡∘𝑓_𝑡𝑥₁,…,𝑥_𝑛=𝑓^′_𝑡Φ_𝑡𝑥₁,…,Φ_𝑡𝑥_𝑛 (31) and is denoted by 𝑓_𝑡∼𝑓_𝑡′. When 𝑓₁=𝑓₂=⋯=0,𝑓_𝑡=𝑓₀ is called the null deformation; if 𝑓_𝑡∼𝑓₀, then 𝑓_𝑡 is called the trivial deformation.

Theorem 18. Let 𝑓_𝑡 and 𝑓_𝑡′ be two equivalent deformations of the 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra (𝔤,[⋅,…,⋅],𝛼). Then the infinitesimal deformations 𝑓₁ and 𝑓^′₁ belong to the same cohomology class in the cohomology group 𝐻²(𝔤,𝔤).

Proof. Put 𝐵²(𝔤,𝔤)∶=𝛿¹𝐶¹(𝔤,𝔤). It is enough to prove that 𝑓₁−𝑓^′₁∈𝐵²(𝔤,𝔤). Let Φ_𝑡∶(𝔤,𝑓_𝑡,𝛼)→(𝔤,𝑓^′_𝑡,𝛼) be an isomorphism of 𝑛-ary multiplicative Hom-Nambu-Lie superalgebras. Then 𝜑₁∈𝐶¹(𝔤,𝔤)₀ and _𝑖≥0𝜑_𝑖𝑗≥0𝑓_𝑗𝑥₁,…,𝑥_𝑛𝑡^𝑖+𝑗=_𝑖≥0𝑓^′_{𝑖𝑗1≥0}𝜑_𝑗1𝑥₁,…,_𝑗𝑛≥0𝜑_𝑗𝑛𝑥_𝑛𝑡^{𝑖+𝑗1+⋯+𝑗𝑛}, (32) and comparing with the coefficients of 𝑡¹ for two sides of the above equation, we obtain 𝑓₁𝑥₁,…,𝑥_𝑛+𝜑₁𝑥₁,𝑥₂,…,𝑥_𝑛=𝜑₁𝑥₁,𝑥₂,…,𝑥_𝑛+𝑥₁,𝜑₁𝑥₂,𝑥₃,…,𝑥_𝑛+⋯+𝑥₁,…,𝑥_𝑛−1,𝜑₁𝑥_𝑛+𝑓^′₁𝑥₁,…,𝑥_𝑛. (33) Furthermore, one gets 𝑓₁𝑥₁,…,𝑥_𝑛−𝑓^′₁𝑥₁,…,𝑥_𝑛=−𝜑₁𝑥₁,𝑥₂,…,𝑥_𝑛+𝜑₁𝑥₁,𝑥₂,…,𝑥_𝑛+𝑥₁,𝜑₁𝑥₂,𝑥₃,…,𝑥_𝑛+⋯+𝑥₁,…,𝑥_𝑛−1,𝜑₁𝑥_𝑛=−𝜑₁𝑥₁,…,𝑥_𝑛+^𝑛_𝑖=1(−1)^𝑛−𝑖(−1)^{|𝑥𝑖|(|𝑥𝑖+1|+⋯+|𝑥𝑛|)}×𝑥₁,…,𝑥_𝑖,…,𝑥_𝑛,𝜑₁𝑥_𝑖=𝛿¹𝜑₁𝑥₁,…,𝑥_𝑛. (34) Therefore, 𝑓₁−𝑓^′₁=𝛿¹𝜑₁∈𝛿¹𝐶¹(𝔤,𝔤)₀; that is, 𝑓₁−𝑓^′₁∈𝐵²(𝔤,𝔤).

An 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra (𝔤,[⋅,⋯,⋅],𝛼) is analytically rigid if every deformation 𝑓_𝑡 is equivalent to the null deformation 𝑓₀. We have a fundamental theorem.

Theorem 19. If (𝔤,[⋅,…,⋅],𝛼) is an 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra with 𝐻²(𝔤,𝔤)=0, then (𝔤,[⋅,…,⋅],𝛼) is analytically rigid.

Proof. Let 𝑓_𝑡 be a deformation of the 𝑛-ary multiplicative Hom-Nambu-Lie superalgebra (𝔤,[⋅,…,⋅],𝛼) with 𝑓_𝑡=𝑓₀+𝑓_𝑟𝑡^𝑟+𝑓_𝑟+1𝑡^𝑟+1+⋯; that is; 𝑓₁=𝑓₂=⋯=𝑓_𝑟−1=0. Then setting 𝑙=𝑟 in 29^′, we have 𝑓_𝑟𝛼𝑥₁,…,𝛼𝑥_𝑛−1,𝑦₁,…,𝑦_𝑛+𝛼𝑥₁,…,𝛼𝑥_𝑛−1,𝑓_𝑟𝑦₁,…,𝑦_𝑛−^𝑛_𝑖=1(−1)^{(|𝑥1|+⋯+|𝑥𝑛−1|)(|𝑦1|+⋯+|𝑦𝑖−1|)}⋅𝛼𝑦₁,…,𝛼𝑦_𝑖−1,𝑓_𝑟𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,𝛼𝑦_𝑖+1,…,𝛼𝑦_𝑛+𝑓_𝑟𝛼𝑦₁,…,𝛼𝑦_𝑖−1,𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,𝛼𝑦_𝑖+1,…,𝛼𝑦_𝑛=0, (35) that is, 𝛿²𝑓_𝑟(𝑥₁,…,𝑥_𝑛−1,[𝑦₁,…,𝑦_𝑛])=0, 𝛿²(𝑓_𝑟)=0; that is, 𝑓_𝑟∈𝑍²(𝔤,𝔤)₀. By our assumption 𝐻²(𝔤,𝔤)=0, one gets 𝑓_𝑟∈𝐵²(𝔤,𝔤)₀, and thus we can find ℎ_𝑟∈𝐶¹(𝔤,𝔤)₀ such that 𝑓_𝑟=𝛿¹ℎ_𝑟. Putting Φ_𝑡=id_𝔤−ℎ_𝑟𝑡^𝑟, then Φ_𝑡∘id_𝔤+ℎ_𝑟𝑡^𝑟+ℎ²_𝑟𝑡^2𝑟+ℎ³_𝑟𝑡^3𝑟=+⋯id_𝔤−ℎ_𝑟𝑡^𝑟∘id_𝔤+ℎ_𝑟𝑡^𝑟+ℎ²_𝑟𝑡^2𝑟+ℎ³_𝑟𝑡^3𝑟=+⋯id_𝔤+ℎ_𝑟𝑡^𝑟+ℎ²_𝑟𝑡^2𝑟+ℎ³_𝑟𝑡^3𝑟−ℎ+⋯_𝑟𝑡^𝑟+ℎ²_𝑟𝑡^2𝑟+ℎ³_𝑟𝑡^3𝑟=+⋯id_𝔤, (36) and moreover, (id_𝔤+ℎ_𝑟𝑡^𝑟+ℎ²_𝑟𝑡^2𝑟+ℎ³_𝑟𝑡^3𝑟+⋯)∘Φ_𝑡=id_𝔤. Hence Φ_𝑡∶𝔤→𝔤 is a linear isomorphism and Φ_𝑡∘𝛼=𝛼∘Φ_𝑡. Set 𝑓^′_𝑡(𝑥₁,…,𝑥_𝑛)=Φ_𝑡⁻¹𝑓_𝑡(Φ_𝑡(𝑥₁),…,Φ_𝑡(𝑥_𝑛)), and then 𝑓^′_𝑡 is also a deformation of (𝔤,[⋅,…,⋅],𝛼) and 𝑓_𝑡∼𝑓^′_𝑡. Note that Φ_𝑡𝑓^′_𝑡(𝑥₁,…,𝑥_𝑛)=𝑓_𝑡(Φ_𝑡(𝑥₁),…,Φ_𝑡(𝑥_𝑛)). Let 𝑓^′_𝑡=∑_𝑖≥0𝑓^′_𝑖𝑡^𝑖. Then id_𝔤−ℎ_𝑟𝑡^𝑟_𝑖≥0𝑓^′_𝑖𝑥₁,…,𝑥_𝑛𝑡^𝑖=𝑓₀+_𝑖≥𝑟𝑓_𝑖𝑡^𝑖𝑥₁−ℎ_𝑟𝑥₁𝑡^𝑟,…,𝑥_𝑛−ℎ_𝑟𝑥_𝑛𝑡^𝑟. (37) So _𝑖≥0𝑓^′_𝑖𝑥₁,…,𝑥_𝑛𝑡^𝑖−_𝑖≥0ℎ_𝑟∘𝑓^′_𝑖𝑥₁,…,𝑥_𝑛𝑡^𝑖+𝑟=𝑓₀𝑥₁,…,𝑥_𝑛−^𝑛_𝑖=1𝑓₀𝑥₁,…,ℎ_𝑟𝑥_𝑖,…,𝑥_𝑛𝑡^𝑟+_{1≤𝑖<𝑗≤𝑛}𝑓₀𝑥₁,…,ℎ_𝑟𝑥_𝑖,…,ℎ_𝑟𝑥_𝑗,…,𝑥_𝑛𝑡^2𝑟−_{1≤𝑖<𝑗<𝑘≤𝑛}𝑓₀𝑥₁,…,ℎ_𝑟𝑥_𝑖,…,ℎ_𝑟𝑥_𝑗ℎ,…,_𝑟𝑥_𝑘,…,𝑥_𝑛𝑡^3𝑟+()+⋯−1^𝑛𝑓₀ℎ_𝑟𝑥₁,ℎ_𝑟𝑥₂,…,ℎ_𝑟𝑥_𝑛𝑡^𝑛𝑟+_𝑖≥𝑟𝑓_𝑖𝑥₁,…,𝑥_𝑛𝑡^𝑖−^𝑛_{𝑖≥𝑟𝑗=1}𝑓_𝑖𝑥₁,…,ℎ_𝑟𝑥_𝑗,…,𝑥_𝑛𝑡^𝑖+𝑟+_{𝑖≥𝑟1≤𝑗≤𝑘≤𝑛}𝑓_𝑖𝑥₁,…,ℎ_𝑟𝑥_𝑗,…,ℎ_𝑟𝑥_𝑘𝑥,…,_𝑛𝑡^𝑖+2𝑟+⋯. (38) By the above equation, one gets 𝑓^′₀𝑥₁,…,𝑥_𝑛=𝑓₀𝑥₁,…,𝑥_𝑛=𝑥₁,…,𝑥_𝑛;𝑓^′₁𝑥₁,…,𝑥_𝑛=⋯=𝑓^′_𝑟−1𝑥₁,…,𝑥_𝑛𝑓=0;_′𝑟𝑥₁,…,𝑥_𝑛−ℎ_𝑟𝑥₁,…,𝑥_𝑛=−^𝑛_𝑖=1𝑥₁,…,ℎ_𝑟𝑥_𝑖,…,𝑥_𝑛+𝑓_𝑟𝑥₁,…,𝑥_𝑛. (39) Furthermore, we have 𝑓^′_𝑟𝑥₁,…,𝑥_𝑛=−𝛿¹ℎ_𝑟𝑥₁,…,𝑥_𝑛+𝑓_𝑟𝑥₁,…,𝑥_𝑛=0, (40) and hence, 𝑓^′_𝑡=𝑓₀+∑_{𝑖≥𝑟+1}𝑓^′_𝑖𝑡^𝑖. By induction, one can prove 𝑓_𝑡∼𝑓₀; that is, (𝔤,[⋅,…,⋅],𝛼) is analytically rigid.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by NNSF of China (no. 11171055), NSF of Jilin (no. 201115006), Scientific Research Fund of Heilongjiang Provincial Education Department (no. 12541900), Scientific Research Foundation for Returned Scholars Ministry of Education of China, and the Fundamental Research Funds for the Central Universities (no. 12SSXT139).