# On the Deformations and Derivations ofn-Ary Multiplicative Hom-Nambu-Lie Superalgebras

Authors: Baoling Guan; Liangyun Chen; Yao Ma
Journal: Advances in Mathematical Physics
 (2014)

---

## Abstract

We introduce the relevant concepts ofn-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.


---

## Body

## 1. Introduction

The notion ofn-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 ofn-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 Section1, 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]).
Ann-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
(1)|[x1,…,xn]|=|x1|+⋯+|xn|,[x1,…,xi,xi+1,…,xn]=-(-1)|xi||xi+1|×[x1,…,xi+1,xi,…,xn],[x1,…,xn-1,[y1,…,yn]]=∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)·[y1,…,yi-1,[x1,…,xn-1,yi],yi+1,…,yn],
where |x|∈Z2 denotes the degree of a homogeneous element x∈g.Definition 2.
Ann-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=(αi)1≤i≤n-1 of even linear maps αi:g→g, satisfying
(2)|[x1,…,xn]|=|x1|+⋯+|xn|,(3)[x1,…,xi,xi+1,…,xn]=-(-1)|xi||xi+1|×[x1,…,xi+1,xi,…,xn],(4)[α1(x1),…,αn-1(xn-1),[y1,…,yn]]=∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)·[α1(y1),…,αi-1(yi-1),[x1,…,xn-1,yi],·αi(yi+1),…,αn-1(yn)],
where |x|∈Z2 denotes the degree of a homogeneous element x∈g.Ann-ary Hom-Nambu-Lie superalgebra (g,[·,…,·],α) is multiplicative, if α=(αi)1≤i≤n-1 with α1=⋯=αn-1=α and satisfying
(5)α[x1,…,xn]=[α(x1),…,α(xn)],∀x1,x2,…,xn∈g.If then-ary Hom-Nambu-Lie superalgebra (g,[·,…,·],α) is multiplicative, then (4) can be read as
(4′)[α(x1),…,α(xn-1),[y1,…,yn]]=∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)·[α(y1),…,α(yi-1),[x1,…,xn-1,yi],α(yi+1),…,α(yn)].It is clear thatn-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=(αi)1≤i≤n-1 and α′=(αi′)1≤i≤n-1. A linear map f:g→g is an n-ary Hom-Nambu-Lie superalgebra morphism if satisfies
(6)f[x1,…,xn]=[f(x1),…,f(xn)]′,f∘αi=αi′∘f,∀i=1,…,n-1.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
(7)(β∘α)[x1,…,xn]β=(β∘α)(β[x1,…,xn])=β∘(α∘β)[x1,…,xn]=β∘(β∘α)[x1,…,xn]=β[β∘α(x1),…,β∘α(xn)]=[β∘α(x1),…,β∘α(xn)]β;
that is, β∘α is a morphism of g. Moreover, we have
(8)[β∘α(x1),…,β∘α(xn-1),[y1,…,yn]β]β=β[β∘α(x1),…,β∘α(xn-1),β[y1,…,yn]]=β2[α(x1),…,α(xn-1),[y1,…,yn]]=β2(∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)×[α(y1),…,[x1,…,xn-1,yi],…,α(yn)]∑i=1n‍)=∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)×β[β∘α(y1),…,β[x1,…,xn-1,yi],…,β∘α(yn)]=∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)×[β∘α(y1),…,[x1,…,xn-1,yi]β,…,β∘α(yn)]β.
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,…,un]g∈H, ∀u1,…,un∈H.
A graded subspaceH⊆g is a Hom-ideal of (g,[·,…,·]g,α) if α(H)⊆H and [u1,u2,…,un]g∈H, ∀u1∈H, u2,…,un∈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. Gϕ={(x,ϕ(x))∣x∈g1}⊆g1⊕g2 is called as the graph of a linear map ϕ:g1→g2.Proposition 8.
Given twon-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
(9)[u1+v1,…,un+vn]g1⊕g2=[u1,…,un]g1+[v1,…,vn]g2,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀ui∈g1,vi∈g2(i=1,2,…,n),
and the linear map (α+β):g1⊕g2→g1⊕g2 is given by
(10)(α+β)(u+v)=α(u)+β(v),∀u∈g1,v∈g2.Proof.
For anyui∈g1,vi∈g2, we have
(11)[u1+v1,…,ui+vi,ui+1+vi+1,…,un+vn]g1⊕g2=[u1,…,ui,ui+1,…,un]g1+[v1,…,vi,vi+1,…,vn]g2=-(-1)|ui||ui+1|[u1,…,ui+1,ui,…,un]g1-(-1)|vi||vi+1|[v1,…,vi+1,vi,…,vn]g2=-(-1)|ui||ui+1|×([u1,…,ui,ui+1,…,un]g1+[v1,…,vi,vi+1,…,vn]g2)=-(-1)|ui||ui+1|×[u1+v1,…,ui+1+vi+1,ui+vi,…,un+vn]g1⊕g2.
The bracket is obviously supersymmetric. By a direct computation we have
(12)[[x1+y1,…,xn+yn]g1⊕g2(α+β)(u1+v1),…,(α+β)(un-1+vn-1),[x1+y1,…,xn+yn]g1⊕g2]g1⊕g2=[[x1+y1,…,xn+yn]g1⊕g2α(u1)+β(v1),…,α(un-1)+β(vn-1),[x1+y1,…,xn+yn]g1⊕g2]g1⊕g2=[α(u1),…,α(un-1),[x1,…,xn]g1]g1+[β(v1),…,β(vn-1),[y1,…,yn]g2]g2=∑i=1n‍(-1)(|u1|+⋯+|un-1|)(|x1|+⋯+|xi-1|)×(β(yi+1),…,β(yn)[u1+v1,…,un-1+vn-1,xi+yi]g1⊕g2]g2[[u1+v1,…,un-1+vn-1,xi+yi]g1⊕g2α(x1),…,α(xi-1),[u1,…,un-1,xi]g1,α(xi+1),…,α(xn)[u1+v1,…,un-1+vn-1,xi+yi]g1⊕g2]g1+[β(y1),…,β(yi-1),[v1,…,vn-1,yi]g2,β(yi+1),…,β(yn)[u1+v1,…,un-1+vn-1,xi+yi]g1⊕g2]g2)=∑i=1n‍(-1)(|u1|+⋯+|un-1|)(|x1|+⋯+|xi-1|)×[α(x1)+β(y1),…,α(xi-1)+β(yi-1),[u1,…,un-1,xi]g1+[v1,…,vn-1,yi]g2,α(xi+1)+β(yi+1),…,α(xn)+β(yn)]g1⊕g2=∑i=1n‍(-1)(|u1|+⋯+|un-1|)(|x1|+⋯+|xi-1|)×[(α+β)(x1+y1),…,(α+β)(xi-1+yi-1),[u1+v1,…,un-1+vn-1,xi+yi]g1⊕g2,(α+β)(xi+1+yi+1),…,(α+β)(xn+yn)]g1⊕g2=∑i=1n‍(-1)(|u1|+⋯+|un-1|)(|x1|+⋯+|xi-1|)×[[u1+v1,…,un-1+vn-1,xi+yi]g1⊕g2(α+β)(x1+y1),…,(α+β)(xi-1+yi-1),[u1+v1,…,un-1+vn-1,xi+yi]g1⊕g2,(α+β)(xi+1+yi+1),…,(α+β)(xn+yn)[u1+v1,…,un-1+vn-1,xi+yi]g1⊕g2]g1⊕g2.
The result follows.Proposition 9.
A linear mapϕ:(g1,[·,…,·]g1,α)→(g2,[·,…,·]g2,β) is a morphism of n-ary multiplicative Hom-Nambu-Lie superalgebras if and only if the graph Gϕ⊆g1⊕g2 is a Hom-subalgebra of (g1⊕g2,[·,…,·]g1⊕g2,α+β).Proof.
Letϕ:(g1,[·,…,·]g1,α)→(g2,[·,…,·]g2,β) be a morphism of n-ary multiplicative Hom-Nambu-Lie superalgebras. Then
(13)[u1+ϕ(u1),…,un+ϕ(un)]g1⊕g2=[u1,…,un]g1+[ϕ(u1),…,ϕ(un)]g2=[u1,…,un]g1+ϕ[u1,…,un]g2.
Then the graphGϕ is closed under the bracket operation [·,…,·]g1⊕g2. Furthermore, we obtain
(14)(α+β)(u+ϕ(u))=α(u)+β∘ϕ(u)=α(u)+ϕ∘α(u),
which implies that (α+β)(Gϕ)⊆Gϕ. Thus, Gϕ is a Hom-subalgebra of (g1⊕g2,[·,…,·]g1⊕g2,α+β).
Conversely, if the graphGϕ⊆g1⊕g2 is a Hom-subalgebra of (g1⊕g2,[·,…,·]g1⊕g2,α+β), then we have
(15)[u1+ϕ(u1),…,un+ϕ(un)]g1⊕g2=[u1,…,un]g1+ϕ[u1,…,un]g2,
which implies that
(16)ϕ[u1,…,un]g2=ϕ[u1,…,un]g1.
Furthermore, (α+β)(Gϕ)⊂Gϕ yields that
(17)(α+β)(u+ϕ(u))=α(u)+β∘ϕ(u)∈Gϕ,
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 ofn-Ary Multiplicative Hom-Nambu-Lie Superalgebras

Let(g,[·,…,·],α) be an n-ary multiplicative Hom-Nambu-Lie superalgebra. We denote by αk the k-times compositions of α. In particular, we set α0=id.Definition 10.
Fork≥0, we call D∈End(g) an αk-derivation of the n-ary multiplicative Hom-Nambu-Lie superalgebra (g,[·,…,·],α) if
(18)D∘α=α∘D
and for xi∈g(i=1,…,n),
(19)D[x1,…,xn]=∑i=1n‍(-1)|D|(|x1|+⋯+|xi-1|)×[αk(x1),…,αk(xi-1),D(xi),αk(xi+1),…,αk(xn)].We denote byDerαk(g) the set of αk-derivations of the n-ary multiplicative Hom-Nambu-Lie superalgebra (g,[·,…,·],α). Notice that we obtain classical derivations for k=0.ForX∈g∧n-1 satisfying α(X)=X and k≥0, we define the map adk(X)∈End(g) by
(20)adk(X)(y)=[x1,…,xn-1,αk(y)],∀y∈g.
Then one has the following.Lemma 11.
The mapadk(X) is an αk+1-derivation and is called an inner αk+1-derivation.We denote byInnαk(g) the K-vector space generated by all inner αk+1-derivations. For any D∈Derαk(g) and D′∈Derαk′(g), we define their commutator [D,D′]=D∘D′-(-1)|D||D′|D′∘D. Set Der(g)=⊕k≥0Derαk(g) and Inn(g)=⊕k≥0Innαk(g).Lemma 12.
For anyD∈Derαk(g) and D′∈Derαk′(g), one has [D,D′]∈Derαk+k′(g).Proof.
Letxi∈g, 1≤i≤n. D∈Derαk(g) and D′∈Derαk′(g), and then
(21)D∘D′([x1,…,xn])=D(∑i=1n‍(-1)|D′|(|x1|+⋯+|xi-1|)×[αk′(x1),…,αk′(xi-1),D′(xi),αk′(xi+1),…,αk′(xn)]∑i=1n‍)=∑i=1n‍(-1)|D′|(|x1|+⋯+|xi-1|)D×[αk′(x1),…,αk′(xi-1),D′(xi),αk′(xi+1),…,αk′(xn)]=∑i=1n‍(-1)|D′|(|x1|+⋯+|xi-1|)(-1)|D|(|x1|+⋯+|xi-1|)·[αk+k′(x1),…,αk+k′(xi-1),D∘D′(xi),αk+k′(xi+1),…,αk+k′(xn)]+∑i<j‍(-1)|D′|(|x1|+⋯+|xi-1|)(-1)|D|(|x1|+⋯+|xj-1|+|D′|)·[αk+k′(x1),…,αk(D′(xi)),…,αk′(D(xj)),…,αk+k′(xn)]+∑i>j‍(-1)|D′|(|x1|+⋯+|xi-1|)(-1)|D|(|x1|+⋯+|xj-1|)·[αk+k′(x1),…,αk′(D(xj)),…,αk(D′(xi)),…,αk+k′(xn)],-(-1)|D||D′|D′∘D([x1,…,xn])=-(-1)|D||D′|D′×(∑i=1n‍(-1)|D|(|x1|+⋯+|xi-1|)×[αk(x1),…,D(xi),…,αk(xn)]∑i=1n‍)=-(-1)|D||D′|∑i=1n‍(-1)|D|(|x1|+⋯+|xi-1|)×D′[αk(x1),…,D(xi),…,αk(xn)]=-(-1)|D||D′|∑i=1n‍(-1)|D|(|x1|+⋯+|xi-1|)(-1)|D′|(|x1|+⋯+|xi-1|)·[αk′+k(x1),…,αk′+k(xi-1),D′∘D(xi),αk′+k(xi+1),…,αk′+k(xn)]-(-1)|D||D′|∑i<j‍(-1)|D|(|x1|+⋯+|xi-1|)(-1)|D′|(|x1|+⋯+|xj-1|+|D|)·[αk′+k(x1),…,αk′(D(xi)),…,αk(D′(xj)),…,αk′+k(xn)]-(-1)|D||D′|∑i>j‍(-1)|D|(|x1|+⋯+|xi-1|)(-1)|D′|(|x1|+⋯+|xj-1|)·[αk′+k(x1),…,αk(D′(xj)),…,αk′(D(xi)),…,αk′+k(xn)].
Then we obtain
(22)[D,D′]([x1,…,xn])=(D∘D′-(-1)|D||D′|D′∘D)([x1,…,xn])=∑i=1n‍(-1)|D′|(|x1|+⋯+|xi-1|)(-1)|D|(|x1|+⋯+|xi-1|)×[αk+k′(x1),…,αk+k′(xi-1),(D∘D′-(-1)|D||D′|D′∘D)(xi),αk+k′(xi+1),…,αk+k′(xn)]=∑i=1n‍(-1)(|D′|+|D|)(|x1|+⋯+|xi-1|)·[αk+k′(x1),…,αk+k′(xi-1),[D,D′](xi),αk+k′(xi+1),…,αk+k′(xn)],
which yields that [D,D′]∈Derαk+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 adk-1(X)(y)=[x1,…,xn-1,αk-1(y)] be an inner αk-derivation on g and D∈Derαk′(g) for k≥1 and k′≥0. Then [D,adk-1(X)]∈Derαk+k′(g) and for any y∈g(23)[D,adk-1(X)](y)=D[x1,…,xn-1,αk-1(y)]-(-1)|D|(|x1|+⋯+|xn-1|)[x1,…,xn-1,αk-1(D(y))]=D[αk(x1),…,αk(xn-1),αk-1(y)]-(-1)|D|(|x1|+⋯+|xn-1|)·[αk+k′(x1),…,αk+k′(xn-1),αk-1(D(y))]=∑i≤n-1‍(-1)|D|(|x1|+⋯+|xi-1|)×[αk+k′(x1),…,D(αk(xi)),…,αk+k′(xn-1),αk+k′-1(y)(αk(xi))]=∑i≤n-1‍(-1)|D|(|x1|+⋯+|xi-1|)×[x1,…,D(xi),…,xn-1,αk+k′-1(y)]=∑i≤n-1‍(-1)|D|(|x1|+⋯+|xi-1|)adk+k′-1×(x1∧⋯∧D(xi)∧⋯∧xn-1)(y).
Therefore, [D,adk-1(X)]∈Innαk+k′-1(g).


## 3. Deformations ofn-Ary Multiplicative Hom-Nambu-Lie Superalgebras

Definition 14 (see [33]).
Form≥1, we call m-coboundary operator of the n-ary multiplicative Hom-Nambu-Lie superalgebra (g,[·,…,·],α) the even linear map δm:Cm(g,V)→Cm+1(g,V) by
(24)(δmf)(X1,…,Xm,Xm+1,z)=∑i<j‍(-1)i(-1)|Xi|(|Xi+1|+⋯+|Xj-1|)×f(α(X1),…,α(Xi)^,…,[Xi,Xj]α,…,α(Xm+1),α(z)α(Xi)^)+∑i=1m+1‍(-1)i(-1)|Xi|(|Xi+1|+⋯+|Xm+1|)×f(α(X1),…,α(Xi)^,…,α(Xm+1),Xi·z)+∑i=1m+1‍(-1)i+1(-1)|Xi|(|f|+|X1|+⋯+|Xi-1|)αm(Xi)·f(X1,…,Xi^,…,Xm+1,z)+(-1)m(f(X1,…,Xm,)·Xm+1)•ααm(z),
where Xi=Xi1∧⋯∧Xin-1∈g∧n-1, i=1,…,m+1,z∈g, and the last term is defined by
(25)(f(X1,…,Xm,)·Xm+1)•ααm(z)=∑i=1n-1‍(-1)(|f|+|X1|+⋯+|Xm|)(|Xm+11|+⋯+|Xm+1i-1|)·[αm(Xm+11),…,f(X1,…,Xm,Xm+1i),…,αm(Xm+1n-1),αm(z)].Theorem 15 (see [33]).
Letf∈Cm(g,V) be an m-cochain. Then δm+1∘δm(f)=0.In [33], it also points out that the map f∈Cm(g,V) is called an m-supercocycle if δmf=0. We denote by Zm(g,V) the graded subspace spanned by m-supercocycles. Since δm+1∘δm(f)=0 for all f∈Cm(g,V), δm-1Cm-1(g,V) is a graded subspace of Zm(g,V). Therefore, we can define a graded cohomology space Hm(g,V) of g as the graded factor space Zm(g,V)/δm-1Cm-1(g,V).We next will discuss the deformation ofn-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
(26)ft=∑p≥0‍fptp:g[[t]]×⋯×g[[t]]⟶g[[t]]
such that (g[[t]],ft,α) is also an n-ary multiplicative Hom-Nambu-Lie superalgebra. We call f1 the infinitesimal deformation of (g,[·,…,·],α).Since(g[[t]],ft,α) is an n-ary multiplicative Hom-Nambu-Lie superalgebra, ft satisfies
(27)α∘ft(x1,…,xn)=ft(α(x1),…,α(xn)),(28)|ft(x1,…,xn)|=|x1|+⋯+|xn|,(29)ft(α(x1),…,α(xn-1),ft(y1,…,yn))=∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)·ft(α(y1),…,α(yi-1),ft(x1,…,xn-1,yi),α(yi+1),…,α(yn)).
Equations (27)–(29) are, respectively, equivalent to
(27′)α∘fp(x1,…,xn)=fp(α(x1),…,α(xn)),(28′)|fp(x1,…,xn)|=|x1|+⋯+|xn|,(29′)∑p+q=l‍fp(α(x1),…,α(xn-1),fq(y1,…,yn))=∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)·(∑p+q=l‍fp(α(y1),…,α(yi-1),fq(x1,…,xn-1,yi),α(yi+1),…,α(yn))∑p+q=l‍α(yi-1)).
We call these the deformation equations for an n-ary multiplicative Hom-Nambu-Lie superalgebra.Equations(27′) and (28′) show that fp∈C1(g,g)0¯. In (29′), set l=1, and then
(30)[α(x1),…,α(xn-1),f1(y1,…,yn)]+f1(α(x1),…,α(xn-1),[y1,…,yn])-∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)×([x1,…,xn-1,yi],α(yi+1),…,α(yn))[α(y1),…,α(yi-1),f1(x1,…,xn-1,yi),α(yi+1),…,α(yn)]+f1(α(y1),…,α(yi-1),[x1,…,xn-1,yi],α(yi+1),…,α(yn)))=0;
that is, δ1f1(x1,…,xn-1,[y1,…,yn-1,yn])=0. Hence the infinitesimal deformation f1∈Z1(g,g)0¯.Definition 17.
Two deformationsft and ft′ 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 Φt:(g,ft,α)→(g,ft′,α), where Φt=∑i≥0‍φiti, φi:g→g is a linear map such that
(31)φ0=idg;φi∘α=α∘φi;Φt∘ft(x1,…,xn)=ft′(Φt(x1),…,Φt(xn))
and is denoted by ft~ft′. When f1=f2=⋯=0,ft=f0 is called the null deformation; if ft~f0, then ft is called the trivial deformation.Theorem 18.
Letft and ft′ be two equivalent deformations of the n-ary multiplicative Hom-Nambu-Lie superalgebra (g,[·,…,·],α). Then the infinitesimal deformations f1 and f1′ belong to the same cohomology class in the cohomology group H2(g,g).Proof.
PutB2(g,g):=δ1C1(g,g). It is enough to prove that f1-f1′∈B2(g,g). Let Φt:(g,ft,α)→(g,ft′,α) be an isomorphism of n-ary multiplicative Hom-Nambu-Lie superalgebras. Then φ1∈C1(g,g)0¯ and
(32)∑i≥0‍φi(∑j≥0‍fj(x1,…,xn))ti+j=∑i≥0‍fi′(∑j1≥0‍φj1(x1),…,∑jn≥0‍φjn(xn))ti+j1+⋯+jn,
and comparing with the coefficients of t1 for two sides of the above equation, we obtain
(33)f1(x1,…,xn)+φ1[x1,x2,…,xn]=[φ1(x1),x2,…,xn]+[x1,φ1(x2),x3,…,xn]+⋯+[x1,…,xn-1,φ1(xn)]+f1′(x1,…,xn).
Furthermore, one gets
(34)f1(x1,…,xn)-f1′(x1,…,xn)=-φ1[x1,x2,…,xn]+[φ1(x1),x2,…,xn]+[x1,φ1(x2),x3,…,xn]+⋯+[x1,…,xn-1,φ1(xn)]=-φ1[x1,…,xn]+∑i=1n‍(-1)n-i(-1)|xi|(|xi+1|+⋯+|xn|)×[x1,…,xi^,…,xn,φ1(xi)]=δ1φ1(x1,…,xn).
Therefore, f1-f1′=δ1φ1∈δ1C1(g,g)0¯; that is, f1-f1′∈B2(g,g).Ann-ary multiplicative Hom-Nambu-Lie superalgebra (g,[·,⋯,·],α) isanalytically rigid if every deformation ft 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.
Letft be a deformation of the n-ary multiplicative Hom-Nambu-Lie superalgebra (g,[·,…,·],α) with ft=f0+frtr+fr+1tr+1+⋯; that is; f1=f2=⋯=fr-1=0. Then setting l=r in (29′), we have
(35)fr(α(x1),…,α(xn-1),[y1,…,yn])+[α(x1),…,α(xn-1),fr(y1,…,yn)]-∑i=1n‍(-1)(|x1|+⋯+|xn-1|)(|y1|+⋯+|yi-1|)·(α(yi+1),…,α(yn))[α(y1),…,α(yi-1),fr(x1,…,xn-1,yi),α(yi+1),…,α(yn)]+fr(α(y1),…,α(yi-1),[x1,…,xn-1,yi],α(yi+1),…,α(yn)))=0,
that is, δ2fr(x1,…,xn-1,[y1,…,yn])=0, δ2(fr)=0; that is, fr∈Z2(g,g)0¯. By our assumption H2(g,g)=0, one gets fr∈B2(g,g)0¯, and thus we can find hr∈C1(g,g)0¯ such that fr=δ1hr. Putting Φt=idg-hrtr, then
(36)Φt∘(idg+hrtr+hr2t2r+hr3t3r+⋯)=(idg-hrtr)∘(idg+hrtr+hr2t2r+hr3t3r+⋯)=(idg+hrtr+hr2t2r+hr3t3r+⋯)-(hrtr+hr2t2r+hr3t3r+⋯)=idg,
and moreover, (idg+hrtr+hr2t2r+hr3t3r+⋯)∘Φt=idg. Hence Φt:g→g is a linear isomorphism and Φt∘α=α∘Φt. Set ft′(x1,…,xn)=Φt-1ft(Φt(x1),…,Φt(xn)), and then ft′ is also a deformation of (g,[·,…,·],α) and ft~ft′. Note that Φtft′(x1,…,xn)=ft(Φt(x1),…,Φt(xn)). Let ft′=∑i≥0‍fi′ti. Then
(37)(idg-hrtr)∑i≥0‍fi′(x1,…,xn)ti=(f0+∑i≥r‍fiti)(x1-hr(x1)tr,…,xn-hr(xn)tr).
So
(38)∑i≥0‍fi′(x1,…,xn)ti-∑i≥0‍hr∘fi′(x1,…,xn)ti+r=f0(x1,…,xn)-∑i=1n‍f0(x1,…,hr(xi),…,xn)tr+∑1≤i<j≤n‍f0(x1,…,hr(xi),…,hr(xj),…,xn)t2r-∑1≤i<j<k≤n‍f0(x1,…,hr(xi),…,hr(xj),…,iiiiihr(xk),…,xnx1,…,hr(xi),…,hr(xj))t3r+⋯+(-1)nf0(hr(x1),hr(x2),…,hr(xn))tnr+∑i≥r‍fi(x1,…,xn)ti-∑i≥r‍∑j=1n‍fi(x1,…,hr(xj),…,xn)ti+r+∑i≥r‍∑1≤j≤k≤n‍fi(x1,…,hr(xj),…,hr(xk),…,iiiixn(xj))ti+2r+⋯.
By the above equation, one gets
(39)f0′(x1,…,xn)=f0(x1,…,xn)=[x1,…,xn];f1′(x1,…,xn)=⋯=fr-1′(x1,…,xn)=0;fr′(x1,…,xn)-hr[x1,…,xn]=-∑i=1n‍[x1,…,hr(xi),…,xn]+fr(x1,…,xn).
Furthermore, we have
(40)fr′(x1,…,xn)=-δ1hr(x1,…,xn)+fr(x1,…,xn)=0,
and hence, ft′=f0+∑i≥r+1‍fi′ti. By induction, one can prove ft~f0; that is, (g,[·,…,·],α) is analytically rigid.



---

*Source: 381683-2014-06-05.xml*