# On the Cohomology and Extensions ofn-ary Multiplicative Hom-Nambu-Lie Superalgebras

Authors: Lijun Tian; Baoling Guan; Yao Ma
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we discuss the representations ofn-ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notion of representations for n-ary multiplicative Hom-Nambu-Lie algebras. We also give the cohomology of an n-ary multiplicative Hom-Nambu-Lie superalgebra and obtain a relation between extensions of an n-ary multiplicative Hom-Nambu-Lie superalgebra b by an abelian one a and Z1b,a0¯. We also introduce the notion of T∗-extensions of n-ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric n-ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not 2 in the case α is a surjection is isometric to a suitable T∗-extension.


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## Body

## 1. Introduction

In 1996, the concept ofn-Lie superalgebras was firstly introduced by Daletskii and Kushnirevich in [1]. Moreover, Cantarini and Kac gave a more general concept of n-Lie superalgebras again in 2010 in [2]. n-Lie superalgebras are more general structures including n-Lie algebras (n-ary Nambu-Lie algebras), n-ary Nambu-Lie superalgebras, and Lie superalgebras [3].The general Hom-algebra structures arose first in connection with quasideformation and discretizations of Lie algebras of vector fields. These quasideformations lead to quasi-Lie algebras, a generalized Lie algebra structure in which the skew symmetry and Jacobi conditions are twisted. Hom-Lie algebras, Hom-Lie superalgebras, Hom-Lie bialgebras, Hom-Lie2-algebras, and quasi-Hom-Lie algebras are discussed in [4–14]. The n-ary Hom-Nambu-Lie algebras have been introduced in [15]. It is the generalization of n-ary algebras of Lie type by twisting the identities using linear maps. It includes n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type such as n-ary Nambu algebras, n-ary Nambu-Lie algebras, and n-ary Lie algebras [16].Cohomologies are powerful tools in mathematics, which can be applied to algebras and topologies as well as the theory of smooth manifolds or of holomorphic functions. The cohomology of Lie algebras was defined by Chevalley and Eilenberg in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie groups in [17]. The cohomology of Lie superalgebras was introduced by Scheunert and Zhang in [18] and was used in mathematics and theoretical physics: the theory of cobordisms, invariant differential operators, central extensions, and deformations. The theory of cohomology for n-ary Hom-Nambu-Lie algebras and n-Lie superalgebras can be found in [19, 20]. This paper generalizes it to n-ary multiplicative Hom-Nambu-Lie superalgebras.The extension is an important way to find a larger algebra and there are many extensions such as general extensions, abelian extensions, nonabelian extensions, double extensions, and Kac-Moody extensions. Abelian extensions and nonabelian extensions of Hom-Lie algebras are, respectively, researched in [21, 22]. The general extensions of n-Hom-Lie algebras is researched in [23]. In 1997, Bordemann introduced the notion of T∗-extensions of Lie algebras in [24]. The method of T∗-extension was used in [25] and was generalized to many other algebras recently in [26, 27]. This paper researches general extensions and T∗-extensions of n-ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the paper also discusses representations of n-ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notions of representations for n-ary multiplicative Hom-Nambu-Lie algebras.This paper is organized as follows. In Section2, we give the representation and the cohomology for an n-ary multiplicative Hom-Nambu-Lie superalgebra. In Section 3, we give a one-to-one correspondence between extensions of an n-ary multiplicative Hom-Nambu-Lie superalgebras b by an abelian one a and Z1b,a0¯. In Section 4, we introduce the notion of T∗-extensions of n-ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric n-ary multiplicative Hom-Nambu-Lie superalgebra g,·,⋯,·,α,,g over an algebraically closed field of characteristic not 2 such that αg=g is isometric to (a nondegenerate ideal of codimension 1 of) a T∗-extension of a nilpotent n-ary multiplicative Hom-Nambu-Lie superalgebra whose nilpotent length is at most a half of the nilpotent length of g.


## 2.n-ary Hom-Nambu-Lie Superalgebras

In the paper, letg=g0¯⊕g1¯ be a finite-dimensional ℤ2-graded vector space. The degree of an element x in g will be denoted by ∣x∣ and in what follows appearance of ∣x∣ will mean that x is a homogeneous element and ∣x∣ stands for its degree, where ∣x∣∈ℤ2 and ℤ2=0¯,1¯.Definition 1.
Ann-ary Hom-Nambu-Lie superalgebra is a triple g,·,⋯,·,αii=1n−1 consisting of a ℤ2-graded vector space g=g0¯⊕g1¯, a multilinear mapping
(1)·,⋯,·:g×⋯×g⏟n⟶gand a family αii=1n−1 of even linear maps αi:g⟶g, satisfying
(2)x1,⋯,xn=x1+⋯+xn,(3)x1,⋯,xi,xi+1,⋯,xn=−−1xixi+1x1,⋯,xi+1,xi,⋯,xn,(4)α1x1,⋯,αn−1xn−1,y1,⋯,yn=∑i=1n−1x1+⋯+xn−1y1+⋯+yi−1·α1y1,⋯,αi−1yi−1,x1,⋯,xn−1,yi,αiyi+1,⋯,αn−1yn,where ∣x∣∈ℤ2 denotes the degree of a homogeneous element x∈g.Ann-ary Hom-Nambu-Lie superalgebra g,·,⋯,·,αii=1n−1 is multiplicative, if α1=⋯=αn−1=α and the following equality is satisfied:
(5)αx1,⋯,xn=αx1,⋯,αxn,∀x1,x2,⋯,xn∈g.Moreover, the multiplicativen-ary Hom-Nambu-Lie superalgebra g,·,⋯,·,αii=1n−1 is also denoted by g,·,⋯,·,α.For a multiplicativen-ary Hom-Nambu-Lie superalgebra g,·,⋯,·,α, equation (4) can be read:
(6)αx1,⋯,αxn−1,y1,⋯,yn=∑i=1n−1∣x1∣+⋯+∣xn−1∣∣y1∣+⋯+∣yi−1∣·αy1,⋯,αyi−1,x1,⋯,xn−1,yi,αyi+1,⋯,αyn.It is clear thatn-ary Hom-Nambu-Lie algebras and Hom-Lie superalgebras are particular cases of n-ary Hom-Nambu-Lie superalgebras.Definition 2.
Letg,·,⋯,·,αii=1n−1 and g′,·,⋯,·′,α′ii=1n−1 be two n-ary Hom-Nambu-Lie superalgebras. A linear map f:g⟶g′ is an n-ary Hom-Nambu-Lie superalgebra homomorphism if it satisfies
(7)fx1,⋯,xn=fx1,⋯,fxn′,f∘αi=αi′∘f,∀i=1,⋯,n−1.Example 3.
Letg,·,⋯,· 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 4.
Letg,·,⋯,·g,αii=1n−1 be an n-ary Hom-Nambu-Lie superalgebra. A graded subspace H⊆g is a Hom-subalgebra of g,·,⋯,·g,αii=1n−1 if αiH⊆H and H is closed under the bracket operation ·,⋯,·g, i.e., u1,⋯,ung∈H, for all u1,⋯,un∈H. A graded subspace H⊆g is a Hom-ideal of g,·,⋯,·g,αii=1n−1 if αiH⊆H and u1,u2,⋯,ung∈H, for all u1∈H and u2,⋯,un∈g.Definition 5.
Letg,·,⋯,·g,αii=1n−1 be an n-ary Hom-Nambu-Lie superalgebra. A Hom-ideal H of g is abelian if u1,u2,u3,⋯,ung=0 for all u1,u2∈H and u3,⋯,un∈g.


## 3. Cohomology forn-ary Multiplicative Hom-Nambu-Lie Superalgebras

Definition 6.
Letg,·,⋯,·,α be an n-ary multiplicative Hom-Nambu-Lie superalgebra. X=x1∧⋯∧xn−1∈∧n−1g is called a fundamental object of g. For all z∈g,X·z≔x1,⋯,xn−1,z. It is clear that ∣X∣=∣x1∣+⋯+∣xn−1∣.
LetX=x1∧⋯∧xn−1 and Y=y1∧⋯∧yn−1 be two fundamental objects of g. A bilinear map ·,·α:∧n−1g×∧n−1g⟶∧n−1g defined by
(8)X,Yα=∑i=1n−1−1∣X∣∣y1∣+⋯+∣yi−1∣αy1∧⋯∧αyi−1∧X·yi∧αyi+1∧⋯∧αyn−1.
A linear mapα:∧n−1g⟶∧n−1g defined by αX=αx1∧⋯∧αxn−1. Then, αX,Yα=αX,αYα.Proposition 7.
Letg,·,⋯,·,α be an n-ary multiplicative Hom-Nambu-Lie superalgebra. Suppose that X=x1∧⋯∧xn−1, Y=y1∧⋯∧yn−1 and Z=z1∧⋯∧zn−1 are fundamental objects of g and z is an arbitrary element in g. Then,
(9)αX·Y·z=−1XYαY·X·z+X,Yα·αz,(10)X,Yα·αz=−−1XYY,Xα·αz.Proof.
It is easy to see that (9) is equivalent to (6). Using (9), by exchanging X and Y, we have
(11)αY·X·z=Y,Xα·αz+−1XYαX·Y·z.
Comparing (9) with (11), we obtain (10).Definition 8.
Letg,·,⋯,·,α be an n-ary multiplicative Hom-Nambu-Lie superalgebra and V=V0¯⊕V1¯ be a ℤ2-graded vector space over a field K and ν∈EndV. A graded representation ρ of g on V is a linear map ρ:∧n−1g⟶EndV,X↦ρX=ρx1,⋯,xn−1 such that
(12)ρXVβ⊆Vβ+∣X∣,∀β∈ℤ2,(13)ραXρY=−1XYραYρX+ρX,Yα∘ν,(14)ραx1,⋯,αxn−2,y1,⋯,yn∘ν=∑i=1n−1n−i−1∣x1∣+⋯+∣xn−2∣∣y1∣+⋯+∣yi∣∧+⋯+∣yn∣·−1∣yi∣∣yi+1∣+⋯+∣yn∣ραy1,⋯,αyi^,⋯,αynρx1,⋯,xn−2,yi,for X,Y∈∧n−1g and x1,⋯,xn−2,y1,⋯,yn∈g, where the sign ∧ indicates that the element below must be omitted. The ℤ2-graded representation space V,ν is said to be a graded g-module.We use a supersymmetric notationx1,⋯,xn−1,v(like (3)) to denote ρX·v. Set x1,⋯,xn−2,v1,v2=0 and α+νx+v=αx+νv for all x∈g and v∈V, then g⊕V,·,⋯,·,α+ν becomes an n-ary multiplicative Hom-Nambu-Lie superalgebra such that V is a ℤ2-graded abelian ideal of g, that is,
(15)V,g,⋯,g⏟n−1⊆VandV,V,g,⋯,g⏟n−2=0.In the sequel, we will usually abbreviateρX·v with X·v.Example 9.
Letg,·,⋯,·,α be an n-ary multiplicative Hom-Nambu-Lie superalgebra. Then, ad:∧n−1g⟶Endg,X↦adX defined by
(16)adXz=X·zis a graded representation of g, it is also called the adjoint graded representation of g.Definition 10.
Letg,·,⋯,·,α be an n-ary multiplicative Hom-Nambu-Lie superalgebra and V,ν be a graded g-module. An m-cochain is an m+1-linear map
(17)f:∧n−1g⊗⋯⊗∧n−1g⏟m∧g⟶Vsuch that
(18)ν∘fX1,X2,⋯,Xm,z=fαX1,αX2,⋯,αXm,αzfor all X1,X2,⋯,Xm∈∧n−1g and z∈g. We denote the set of m-cochain by Cmg,V.Definition 11.
Form≥1, we define an m-coboundary operator δm of the n-ary multiplicative Hom-Nambu-Lie superalgebra g,·,⋯,·,α, which is an even linear map, by
(19)δmfX1,⋯,Xm,Xm+1,z=∑i<j−1i−1∣Xi∣∣Xi+1∣+⋯+∣Xj−1∣fαX1,⋯,αXi^,⋯,Xi,Xjα,⋯,αXm+1,αz+∑i=1m+1−1i−1∣Xi∣∣Xi+1∣+⋯+∣Xm+1∣fαX1,⋯,αXi^,⋯,αXm+1,Xi·z+∑i=1m+1−1i+1−1∣Xi∣∣f∣+∣X1∣+⋯+∣Xi−1∣αmXi·fX1,⋯,Xi^,⋯,Xm+1,z+−1mfX1,⋯,Xm,·Xm+1•ααmz,where Xi∈∧n−1g,i=1,⋯,m+1,z∈g and the last term is defined by
(20)fX1,⋯,Xm,·Xm+1•ααmz=∑i=1n−1−1∣f∣+∣X1∣+⋯+∣Xm∣∣Xm+11∣+⋯+∣Xm+1i−1∣·αmXm+11,⋯,fX1,⋯,Xm,Xm+1i,⋯,αmXm+1n−1,αmz,where Xm+1=Xm+11∧⋯∧Xm+1n−1∈∧n−1g.Theorem 12.
Letf∈Cmg,V be an m-cochain. Then, δm+1∘δmf=0.Proof.
See the appendix.Remark 13.
Them-coboundary operator δm as above is a generalization of the one defined for n-ary multiplicative Hom-Nambu-Lie algebras in [16] and for first-class n-Lie superalgebras in [20].The mapf∈Cmg,V is called an m-supercocycle if δmf=0. We denote by Zmg,Vthe graded subspace spanned by m-supercocycles. Since δm+1∘δmf=0 for all f∈Cmg,V, δm−1Cm−1g,V is a graded subspace of Zmg,V. Therefore, we can define a graded cohomology space Hmg,V of gas the graded space Zmg,V/δm−1Cm−1g,V.


## 4. Extensions ofn-ary Multiplicative Hom-Nambu-Lie Superalgebras

Definition 14.
Letgi,·,⋯,·i,αii=1,2,⋯ be a family of n-ary multiplicative Hom-Nambu-Lie superalgebras over K. fi:gi→gi+1 is a morphism of n-ary multiplicative Hom-Nambu-Lie superalgebras. The sequence
(21)g1⟶f1g2⟶f2⋯⟶gi⟶figi+1⟶fi+1⋯is called an exact sequence of n-ary multiplicative Hom-Nambu-Lie superalgebras, if it satisfies kerfi+1=figii=1,2,⋯.Definition 15.
Letg,·,⋯,·g,αg,a,·,⋯,·a,αa and b,·,⋯,·b,αb be n-ary multiplicative Hom-Nambu-Lie superalgebras over K. g is called an extension of b by a if there is an exact sequence of n-ary multiplicative Hom-Nambu-Lie superalgebras:
(22)0⟶a⟶ιg⟶πb⟶0.Letg,·,⋯,·g,αand b,·,⋯,·b,β be two n-ary multiplicative Hom-Nambu-Lie superalgebras over K. Suppose that a is an abelian graded ideal of g, i.e., a is a graded ideal such that
(23)a,a,g,⋯,g⏟n−2=0We consider the case that gis an extension of b by an abelian graded ideal a of g. Let τ:b⟶g be a homogeneous even linear map with π∘τ=idb and α∘τ=τ∘β. Let B=b1∧⋯∧bn−1∈∧n−1b and let ρ:∧n−1b⟶Enda,B↦τB=τb1∧⋯∧τbn−1. Then, a becomes a graded b-module. Let us write τb=0,b and then denote the elements of g by a,b for all a∈a and b∈b. Then, the bracket in g is defined by
(24)a1,b1,⋯,an,bn=∑i=1nτb1,⋯,ai,⋯,τbn+fB,bn,B·bn,where fB,bn=τB·τbn−τB·bn and ∣ai,bi∣=∣ai∣=∣bi∣,∀1≤i≤n. It is easy to see that f∈C1b,a0¯. Let A=a1∧⋯∧an−1,A,B=a1,b1∧⋯∧an−1,bn−1 and αA,βB=αa1,βb1∧⋯∧αan−1,βbn−1. Then,
(25)αA,βB·A′,B′·an′,bn′−∑i=1n−1∣A∣∣a1′∣+⋯+∣ai−1′∣αa1′,βb1′,⋯,A,B·ai′,bi′,⋯,αan′,βbn′=αA,βB·∑i=1nτb1′,⋯,ai′,⋯,τbn′+fB′,bn′,B′·bn′−∑i=1n−1∣A∣∣a1′∣+⋯+∣ai−1′∣·αa1′,βb1′,⋯,∑j=1n−1τb1,⋯,aj,⋯,τbn−1,τbi′+τB·ai′+fB,bi′,B·bi′,⋯,αan′,βbn′=τβB·∑i=1nτb1′,⋯,ai′,⋯,τbn′+∑j=1n−1τβb1,⋯,αaj,⋯,τβbn−1,τB′·bn′+τβB·fB′,bn′+fβB,B′·bn′,βB·B′·bn−∑i=1n−1∣A∣∣a1′∣+⋯+∣ai−1′∣·∑j=1n−1τβb1′,⋯,τb1,⋯,aj,⋯,τbn−1,τbi′,⋯,τβbn′+τβb1′,⋯,τB·ai′,⋯,τβbn′+τβb1′,⋯,fB,bi′,⋯,τβbn′+∑j≠iτβb1′,⋯,αaj′,⋯,τB·bi′,⋯,τβbn′+fβb1′,⋯,B·bi′,⋯,βbn′,βb1′,⋯,B·bi′,⋯,βbn′=δ1fB,B′,bn′,0.Therefore,f∈Z1b,a0¯.Conversely, suppose that an abeliann-ary multiplicative Hom-Nambu-Lie superalgebras a is a graded b-module, ρB·a≔τB·a, and f∈Z1b,a0¯. Let g≔a,b=x,y∣x∈a,y∈b,α′≔α+β, where α+βx,y=αx,βy,x∈a,y∈b. Then g,α′ is an n-ary multiplicative Hom-Nambu-Lie superalgebra with the bracket defined by (24). Then we can define an exact sequence
(26)0⟶a⟶ιg⟶πb⟶0,where ιa=a,0,πa,b=b. Thus g is an extension of b by a and ιa is an abelian graded ideal of g.Therefore, we get the following theorem.Theorem 16.
Suppose thata,·,⋯,·a,α and b,·,⋯,·b,β are two n-ary multiplicative Hom-Nambu-Lie superalgebras over K and a is abelian. Then, there is a one-to-one correspondence between extensions of b by a and Z1b,a0¯.


## 5.T∗-Extensions of n-ary Multiplicative Hom-Nambu-Lie Superalgebras

Letg,·,⋯,·,α be an n-ary multiplicative Hom-Nambu-Lie superalgebra and g∗ be its dual space. Since g=g0¯⊕g1¯ and g∗=g0¯∗⊕g1¯∗ are ℤ2-graded vector space, the direct sum g⊕g∗=g0¯⊕g0¯∗⊕g1¯⊕g1¯∗ is a ℤ2-graded vector space. In the sequel, whenever x+f∈g⊕g∗ appears, it means that x+f is homogeneous and ∣x+f∣=∣x∣=∣f∣.Lemma 17.
Letg∗ be the dual ℤ2-graded vector space of an n-ary multiplicative Hom-Nambu-Lie superalgebra g,·,⋯,·,α. Let us consider the even linear map ad∗:∧n−1g⟶Endg∗ defined by
(27)ad∗Xfz=−−1XffadXz,for all X∈∧n−1g,f∈g∗ and z∈g. Then, ad∗ is a representation of g on g∗ if and only if the following conditions hold:
(28)adXadαY−−1XYadYadαX=α∘adX,Yα;adx1,⋯,xn−2,yiadαy1,⋯,αyi^,⋯,αyn=−1∣x1∣+⋯+∣xn−2∣∣y1∣+⋯+∣yi∣∧+⋯+∣yn∣·−adαy1,⋯,αyi^,⋯,αynadx1,⋯,xn−2,yifor all i=1,2,⋯,n.Proof.
⇒ We firstly prove that the necessity holds. Then, by the definition of ad∗, one gets
(29)ad∗αXad∗Yfz=−−1∣X∣∣Y∣+∣f∣ad∗YfadαXz=−−1∣X∣∣Y∣+∣f∣−−1YffadYadαXz=−1∣X∣∣Y∣+∣f∣+YffadYadαXzand
(30)−1XYad∗αYad∗Xfz=−1XY−−1YX+fad∗XfadαYz=−−1Yf−−1xffadXadαYz=−1∣X∣+∣Y∣∣f∣fadXadαYz.
Moreover, we have(31)ad∗X,Yα∘νfz=−−1X+YfνfadX,Yαz=−−1∣X∣+∣Y∣∣f∣fα∘adX,Yαz.
By (13), we have adXadαY−−1XYadYadαX=α∘adX,Yα.(32)ad∗αx1,⋯,αxn−2,y1,⋯,ynνfz=−−1x1+⋯+xn−2+y1+⋯+ynffαx1,⋯,αxn−2,y1,⋯,yn,αz=−−1x1+⋯+xn−2+y1+⋯+ynf−−1zy1+⋯+yn·fαx1,⋯,αxn−2,αz,y1,⋯,yn=−−1x1+⋯+xn−2+y1+⋯+ynf−−1zy1+⋯+yn·∑i=1n−1x1+⋯+xn−2+zy1+⋯+yi−1·fαy1,⋯,αyi−1,x1,⋯,xn−2,z,yi,αyi+1,⋯,αyn=−∑i=1n−1n−i−1yiyi+1+⋯+yn+x1+⋯+xn−2+y1+⋯+ynf·−1x1+⋯+xn−2y1+⋯+yi∧+⋯+ynfαy1,⋯,αyi^,⋯,αyn,x1,⋯,xn−2,yi,z=−∑i=1n−1n−i−1yiyi+1+⋯+yn+x1+⋯+xn−2+y1+⋯+ynf·−1x1+⋯+xn−2y1+⋯+yi∧+⋯+yn·fadαy1,⋯,αyi^,⋯,αynadx1,⋯,xn−2,yiz,(33)∑i=1n−1n−i−1∣x1∣+⋯+∣xn−2∣∣y1∣+⋯+∣yi∣∧+⋯+∣yn∣+∣yi∣∣yi+1∣+⋯+∣yn∣·ad∗αy1,⋯,αyi^,⋯,αynad∗x1,⋯,xn−2,yifz=∑i=1n−1n−i−1∣x1∣+⋯+∣xn−2∣∣y1∣+⋯+∣yi∣∧+⋯+∣yn∣+∣yi∣∣yi+1∣+⋯+∣yn∣·−−1∣y1∣+⋯+∣yi∣∧+⋯+∣yn∣∣x1∣+⋯+∣xn−2∣+∣yi∣+∣f∣·ad∗x1,⋯,xn−2,yifadαy1,⋯,αyi^,⋯,αynz=∑i=1n−1n−i−1∣yi∣∣y1∣+⋯+∣yi−1∣+∣f∣∣y1∣+⋯+∣yn∣+∣x1∣+⋯+∣xn−2∣·fadx1,⋯,xn−2,yiadαy1,⋯,αyi^,⋯,αynz.
By (14), we obtain
(34)adx1,⋯,xn−2,yiadαy1,⋯,αyi^,⋯,αyn=−1∣x1∣+⋯+∣xn−2∣∣y1∣+⋯+∣yi∣∧+⋯+∣yn∣·−adαy1,⋯,αyi^,⋯,αyiadx1,⋯,xn−2,yi.
⇐ It is easy to see that the sufficiency holds. The proof is complete.The representationad∗ as defined in Lemma 17 is called the coadjoint representation of g. Let θ be a homogeneous n-linear map from ∧ng into g∗ of degree 0. Now, we define a bracket on g⊕g∗:
(35)x1+f1,⋯,xn+fnθ=x1,⋯,xng+θx1,⋯,xn+∑i=1n−1n−i−1∣xi∣∣xi+1∣+⋯+∣xn∣ad∗x1,⋯,xi^,⋯,xn·fi.Theorem 18.
Letg,·,⋯,·,α be an n-ary multiplicative Hom-Nambu-Lie superalgebra. Assume that the coadjoint representation exists. Then, g⊕g∗,·,⋯,·θ,α′ is an n-ary multiplicative Hom-Nambu-Lie superalgebra if and only if θ∈Z1g,g∗0¯, where α′x+f=αx+f∘α,∀x∈g,y∈g∗.Proof.
It is clear that·,⋯,·θ satisfies (3) if and only if θ∈C1g,g∗0¯. Let X+F=x1+f1∧⋯∧xn−1+fn−1 and Y+G=y1+g1∧⋯∧yn−1+gn−1. Then, we have
(36)α′X+F·Y+G·yn+gn=αX+F∘α·∑i=1n−1n−i−1yiyi+1+⋯+ynad∗y1,⋯,yi^,⋯,yn·gi+Y·yn+θY,yn=αX·Y·yn+θαX,Y·yn+ad∗αX·θY,yn+∑j=1n−1−1n−j−1xjxj+1+⋯+xn−1+Y+ynad∗αx1,⋯,αxj^,⋯,αxn−1,Y·yn·fj∘α+∑i=1n−1n−i−1yiyi+1+⋯+ynad∗αX·ad∗y1,⋯,yi^,⋯,yn·gi,(37)∑i=1n−1∣X∣∣y1∣+⋯+∣yi−1∣αy1+g1∘α,⋯,X+F·yi+gi,⋯,αyn+gn∘αθ=∑i=1n−1∣X∣∣y1∣+⋯+∣yi−1∣αy1+g1∘α,⋯,X·yi+θX,yi+ad∗X·gi+∑j=1n−1−1n−j−1∣xj∣∣xj+1∣+⋯+∣xn−1∣+∣yi∣ad∗x1,⋯,xj^,⋯,xn−1,yi·fj,⋯,αyn+gn∘αθ=∑i=1n−1∣X∣∣y1∣+⋯+∣yi−1∣αy1,⋯,X·yi,⋯,αyn+θαy1,⋯,X·yi,⋯,αyn+∑k<i−1n−k−1∣yk∣∣yk+1∣+⋯+∣yn∣+∣X∣ad∗αy1,⋯,αyk^,⋯,X·yi,⋯,αyn·gk∘α+∑i<k−1n−k−1∣yk∣∣yk+1∣+⋯+∣yn∣ad∗αy1,⋯,X·yi,⋯,αyk^,⋯,αyn·gk∘α+−1n−i−1∣X∣+∣yi∣∣yi+1∣+⋯+∣yn∣ad∗αy1,⋯,αyi^,⋯,αyn·θX,yi+ad∗X·gi+∑j=1n−1−1n−j−1∣xj∣∣xj+1∣+⋯+∣xn−1∣+∣yi∣ad∗x1,⋯,xj^,⋯,xn−1,yi·fj.
Since,⋯,g satisfies (4) and ad∗X satisfies (14), it can be concluded that ,⋯,θ satisfies (4) if and only if
(38)0=θαX,Y·yn+ad∗αX·θY,yn−∑i=1n−1∣X∣∣y1∣+⋯+∣yi−1∣·θαy1,⋯,X·yi,⋯,αyn−∑i=1n−1∣X∣∣y1∣+⋯+∣yi−1∣·−1n−i−1∣X∣+∣yi∣∣yi+1∣+⋯+∣yn∣·ad∗αy1,⋯,αyi^,⋯,αyn·θX,yi=δθX,Y,yn,i.e., θ∈Z1g,g∗0¯.Definition 19.
Letg,·,⋯,·,α be an n-ary multiplicative Hom-Nambu-Lie superalgebra. A bilinear form ,g on g is said to be nondegenerate if
(39)g⊥=x∈g∣x,yg=0,∀y∈g=0;invariant if
(40)x1,⋯,xn−1,yg,zg=−−1∣x1∣+⋯+∣xn−1∣∣y∣y,x1,⋯,xn−1,zgg,∀x1,⋯,xn−1,y,z∈g;supersymmetric if
(41)x,yg=−1xyy,xg;consistent if
(42)x,yg=0,∀x,y∈g,x≠y;α is called ,g-symmetric, if
(43)αx,yg=αy,xg,∀x,y∈g;a subspace I of g is called isotropic if I⊆I⊥.In this section, we only consider consistent bilinear forms. Ifg admits a nondegenerate invariant supersymmetric bilinear form ,g such that α is ,g-symmetric, then we call g,·,⋯,·g,α,,g a metric n-ary multiplicative Hom-Nambu-Lie superalgebra. In particular, a metric vector space is a pair V,α consisting of a ℤ2-graded vector space V=V0¯⊕V1¯ and an endomorphism α of V admitting a nondegenerate invariant supersymmetric bilinear form ,g such that α is ,g-symmetric.Lemma 20.
With notations of Theorem18, define a bilinear form ,θ:g⊕g∗×g⊕g∗⟶K by
(44)x+f,y+gθ=fy+−1xygx.
Then,y+g,x+fθ=−1xyx+f,y+gθ,,θ is nondegenerate and α′ is ,θ-symmetric, where α′x+f=αx+f∘α,x∈g,f∈g∗. Moreover, g⊕g∗,·,⋯,·θ,α′,,θ is metric if and only if the following identity holds
(45)θX,yz+−1yzθX,zy=0.Proof.
Ifx+f is orthogonal to all elements of g⊕g∗, then for arbitrary element y+g∈g⊕g∗, we have fy=0 and −1xygx=0, which implies that x=0 and f=0, so ,θ is nondegenerate. Moreover, we have
(46)y+g,x+fθ=gx+−1yxfy=−1xyfy+−1xygx=−1xyx+f,y+gθ.
In addition, one gets(47)α′x+f,y+gθ=αx+f∘α,y+gθ=f∘αy+−1xygαx,(48)x+f,α′y+gθ=x+f,αy+g∘αθ=fαy+−1xyg∘αx.
Hence,α′x+f,y+gθ=x+f,α′y+gθ.
Furthermore,g⊕g∗,,θ is metric if and only if
(49)0=X+F·y+g,z+hθ+−1Xyy+g,X+F·z+hθ=X·y+θX,y+ad∗X·g,z+hθ+∑i=1n−1−1n−i−1∣xi∣∣xi+1∣+⋯+∣xn−1∣+∣y∣ad∗x1,⋯,xi^,⋯,xn−1,y·fi,z+hθ+−1Xyy+g,X·z+θX,z+ad∗X·hθ+−1Xyy+g,∑i=1n−1−1n−i−1∣xi∣∣xi+1∣+⋯+∣xn−1∣+∣z∣ad∗x1,⋯,xi^,⋯,xn−1,z·fiθ=θX,yz+−1yzθX,zy,i.e., (45) holds.Now, we give the definition ofT∗-extensions.Definition 21.
For a1-supercocycle θ satisfying (45) we shall call the metric n-ary multiplicative Hom-Nambu-Lie superalgebra g⊕g∗,·,⋯,·θ,α′,,θ the T∗-extension of g,·,⋯,·,α (by θ) and denote it by Tθ∗g,·,⋯,·θ,α′.Theorem 22.
Letg,·,⋯,·,α be an n-ary multiplicative Hom-Nambu-Lie superalgebra over a field K. Let
(50)g0=g,gm+1=gm,⋯,gmgandg1=g,gm+1=gm,g,⋯,gg,∀m≥0.
g is called solvable (nilpotent) of length k if and only if there is a smallest integer k such that gk=0 (gk=0). Then
(1)
Ifg is solvable of length k, then Tθ∗g is solvable of length k or k+1.(2)
Ifg is nilpotent of length k, then Tθ∗g is nilpotent of length at least k and at most 2k−1. In particular, the nilpotent length of T0∗g is k.(3)
Ifg can be decomposed into a direct sum of two Hom-ideals of g, then T0∗g can be too.Proof.
(1) Suppose thatg is solvable of length k. Since Tθ∗gm/g∗≅gm and gk=0, we have Tθ∗gk⊆g∗, which implies Tθ∗gk+1=0 because g∗ is abelian, and it follows that Tθ∗g is solvable of length k or k+1.
(2) Suppose thatg is nilpotent of length k. Since Tθ∗gm/g∗≅gm and gk=0, we have Tθ∗gk⊆g∗. Let f∈Tθ∗gk⊆g∗,y∈g, Xj+Fj=Xj1+Fj1∧⋯∧Xjn−1+Fjn−1∈Tθ∗g∧n−1, j=1,⋯,k−1. Then
(51)X1+F1⋯Xk−1+Fk−1·fy=ad∗X1⋯ad∗Xk−1·fy∈fgk=0.
This proves thatTθ∗g2k−1=0. Hence Tθ∗g is nilpotent of length at least k and at most 2k−1.
Now consider the case of trivialT∗-extension T0∗g of g. Note that
(52)X1+F1⋯Xk−1+Fk−1·y+g=adX1⋯adXk−1·y+ad∗X1⋯ad∗Xk−1·g+∑j=1k−1∑i=1n−1−1n−i−1∣Xji∣∣Xji+1∣+⋯+∣Xjn−1∣+∣y∣+∣Xj+1∣+⋯+∣Xk−1∣·ad∗X1⋯ad∗Xj−1ad∗Xj1,⋯,Xji^,⋯,Xjn−1,adXj+1⋯adXk−1·y·Fji=0.
Then,Tθ∗gk=0, as required.
(3) Suppose that0≠g=I⊕J, where I and J are two nonzero Hom-ideals of g. Let I∗=f∈g∗∣fJ=0 and J∗=f∈g∗∣fI=0. Then, I∗(resp. J∗) can canonically be identified with the dual space of I(resp. J) and g∗≅I∗⊕J∗.
Note that(53)T0∗I,T0∗g,⋯,T0∗g0=I⊕I∗,g⊕g∗,⋯,g⊕g∗0=I,g,⋯,gg+I∗,g,⋯,g0+I,g,⋯,g,g∗0⊆I⊕I∗=T0∗I,since
(54)I∗,g,⋯,g0J=I∗J,g,⋯,gg⊆I∗J=0andI,g,⋯,g,g∗0J=g∗I,J,g,⋯,gg=g∗0=0.
Moreover, forx+f∈T0∗I=I⊕I∗, we have α′x+f=αx+f∘α∈I⊕I∗ since f∘α∈g∗ and f∘αJ∈fJ=0, that is, α′T0∗I⊆T0∗I. Then, T0∗I is a Hom-ideal of T0∗g and so is T0∗J in the same way. Hence, T0∗g can be decomposed into the direct sum T0∗I⊕T0∗J of two nonzero Hom-ideals of T0∗g.Lemma 23.
Letg,,·,⋯,·,α,,θ be a metric n-ary multiplicative Hom-Nambu-Lie superalgebra of even dimension m over a field K and I be an isotropic m/2-dimensional Hom-ideal of g. Then, I is abelian.Proof.
Since dimI+dimI⊥=m/2+dimI⊥=m and I⊆I⊥, we have I=I⊥.
ByI is a Hom-ideal of g, one gets
(55)g,g,⋯,g,I,Igθ=g,⋯,g,Ig,Iθ⊆I,Iθ=0,which implies g,⋯,g,I,Ig⊆g⊥=0.Definition 24.
Letg,·,⋯,·g,α and g′,·,⋯,·g′,β be two n-ary Hom-Nambu-Lie superalgebras. A bijective homomorphism ϕ:g⟶g′ is called an isomorphism of n-ary Hom-Nambu-Lie superalgebras.Definition 25.
Two metricn-ary multiplicative Hom-Nambu-Lie superalgebras g,·,⋯,·g,α,,g and g′,·,⋯,·g′,β,,g′ is said to be isometric if there exists an n-ary multiplicative Hom-Nambu-Lie superalgebra isomorphism ϕ:g⟶g′ such that x,yg=ϕx,ϕyg′,∀x,y∈g.Theorem 26.
Letg,·,⋯,·g,β,,g be a metric n-ary multiplicative Hom-Nambu-Lie superalgebra of dimension m over a field K of characteristic not 2. Suppose that Tθ∗g1,·,⋯,·θ,α′,,θ is a T∗-extension of g1,·,⋯,·g1,α. Then, g,·,⋯,·g,β,,g is isometric to Tθ∗g1,·,⋯,·θ,α′,,θ if and only if m is even and g contains an isotropic Hom-ideal I of dimension m/2. In particular, g1≅g/I.Proof.
(⇒) Since dimg1 = dimg1∗, dimg = dimTθ∗g1=m is even. Moreover, α′f=f∘α∈g1∗ for all f∈g1∗. It is clear that g1∗ is a Hom-ideal of dimension m/2 and by the definition of ,θ, we have g1∗,g1∗θ=0, i.e., g1∗ is isotropic.
(⇐) Suppose that I is an m/2-dimensional isotropic graded ideal of g. By Lemma 23, I is abelian. Let g1=g/I and π:g⟶g1 be the canonical projection. Since chK≠2, we can choose a complement graded subspace g0⊆g such that g=g0∔I and g0⊆g0⊥. Then, g0⊥=g0 since dimg0=m/2.
Denote byp0 (resp. p1) the projection g⟶g0 (resp. g⟶I) and let f1∗ denote the homogeneous linear map I⟶g1∗:z↦f1∗z, where f1∗zπx≔z,xg,∀x∈g,∀z∈I.
Ifπx=πy, then x−y∈I, hence z,x−yg∈z,Ig=0 and so z,xg=z,yg, which implies f1∗ is well defined. Moreover, f1∗ is bijective and ∣f1∗z∣=∣z∣ for all z∈I.
In addition,f1∗ has the following property:
(56)f1∗x1,⋯,zk,⋯,xngπy=−1n−k−1∣zk∣∣xk+1∣+⋯+∣xn∣ad∗πx1,⋯,πxk^,⋯,πxn·f1∗zkπy,where x1,⋯,xk−1,xk+1,⋯,xn∈g, zk∈I.
Define a homogeneousn-linear map
(57)θ:g1×⋯×g1⟶g1∗πx1,⋯,πxn↦f1∗p1x1,⋯,xng,where x1,⋯,xn∈g0. Then, θ is well defined since πg0:g0⟶g0/I≅g/I=g1 is a linear isomorphism and θ∈C1g1,g1∗0¯.
Now, define the bracket ong1⊕g1∗ by (35), then, g1⊕g1∗,α′ is a metric n-ary multiplicative Hom-Nambu-Lie superalgebra. Let φ be a linear map g⟶g1⊕g1∗ defined by φx+z=πx+f1∗z,∀x+z∈g=g0∔I. Since πg0 and f1∗ are linear isomorphisms, φ is also a linear isomorphism. Note that
(58)φx1+z1,⋯,xn+zng=φx1,⋯,xng+∑k=1nx1,⋯,zk,⋯,xng=φp0x1,⋯,xng+p1x1,⋯,xng+∑k=1nx1,⋯,zk,⋯,xng=πx1,⋯,xng+f1∗p1x1,⋯,xng+∑k=1nx1,⋯,zk,⋯,xng=πx1,⋯,πxng1+θπx1,⋯,πxn+∑k=1n−1n−k−1∣zk∣∣xk+1∣+⋯+∣xn∣ad∗πx1,⋯,πxk^,⋯,πxn·f1∗zk=πx1+f1∗z1,⋯,πxn+f1∗znθ=φx1+z1,⋯,φxn+znθ,where we use the definitions of φ and θ and (56). Moreover, φ∘α=α′∘φ. In fact, for x+z∈g=g0∔I, then,
(59)φ∘αx+z=φαx+αz=παx+f1∗αzα′∘φx+z=α′πx+f1∗z=απx+f1∗z∘α.
Moreover,(60)f1∗z∘απx=f1∗zπαx=z,αxg=αz,xg=f1∗αzπx.
Therefore,f1∗z∘α=f1∗αz, one gets φ∘α=α′∘φ. Then, φ is an isomorphism of n-ary multiplicative Hom-Nambu-Lie superalgebras, hence, g1⊕g1∗ is an n-ary multiplicative Hom-Nambu-Lie superalgebra. Furthermore, we have
(61)φx0+z,φx0′+z′θ=πx0+f1∗z,πx0′+f1∗z′θ=f1∗zπx0′+−1x0x0′f1∗z′πx0=z,x0′g+−1x0x0′z′,x0g=x0+z,x0′+z′g,then, φ is isometric. The relation
(62)φx1+z1,⋯,φxn+znθ,φxn+1+zn+1θ=φx1+z1,⋯,xn+zng,φxn+1+zn+1θ=x1+z1,⋯,xn+zng,xn+1+zn+1g=−−1x1+⋯+xn−1xnxn+zn,x1+z1,⋯,xn−1+zn−1,xn+1+zn+1gg=−−1x1+⋯+xn−1xnφxn+zn,φx1+z1,⋯,φxn−1+zn−1,φxn+1+zn+1θθ,implies that ,θ on g1⊕g1∗ is invariant.
Forx+f,y+g∈g1⊕g1∗, then, there exist x′+z1,y′+z2∈g0∔I such that φx′+z1=x+f and φy′+z2=y+g. Hence, we have
(63)α′x+f,y+gθ=α′φx′+z1,φy′+z2θ=φαx′+z1,φy′+z2θ=αx′+z1,y′+z2θ=αy′+z2,x′+z1θ=φαy′+z2,φx′+z1θ=α′φy′+z2,φx′+z1θ=α′y+g,x+fθ.
Therefore,g1⊕g1∗,·,⋯,·θ,α′,,θ is a metric n-ary multiplicative Hom-Nambu-Lie superalgebra. In this way, we get a T∗-extension Tθ∗g1,·,⋯,·θ,α′,,θ of g1,·,⋯,·g1,α and consequently, g,·,⋯,·g,β,,g and Tθ∗g1,·,⋯,·θ,α′,,θ are isometric as required.Suppose thatg,·,⋯,·,α is an n-ary multiplicative Hom-Nambu-Lie superalgebra and θ1, θ2∈Z1g,g∗0¯ satisfying (45). Tθ1∗g and Tθ2∗g are said to be equivalent if there exists an isomorphism of n-ary multiplicative Hom-Nambu-Lie superalgebras ϕ:Tθ1∗g⟶Tθ2∗g such that ϕg∗=idg∗ and the induced map ϕ¯:Tθ1∗g/g∗⟶Tθ2∗g/g∗ is the identity, i.e., ϕx−x∈g∗. Moreover, if ϕ is also an isometry, then, Tθ1∗g and Tθ2∗g are said to be isometrically equivalent.Proposition 27.
Suppose thatg,·,⋯,·,α is an n-ary multiplicative Hom-Nambu-Lie superalgebra over a field K of characteristic not 2 and θ1, θ2∈Z1g,g∗0¯ satisfying (45). Then, we have
(1)
Tθ1∗g,·,⋯,·θ1,α′,,θ1 is equivalent to Tθ2∗g,·,⋯,·θ2,α′,,θ2 if and only if θ1−θ2≔δθ′∈δC0g,g∗0¯ and θ′xα=θ′αx for all x∈g. Moreover,(64)x,yθ′≔12θ′xy+−1xyθ′yxbecomes a supersymmetric invariant bilinear form on g and α is ,θ′-symmetric.
(2)
Tθ1∗g is isometrically equivalent to Tθ2∗g if and only if there is θ′∈C0g,g∗0¯ such that θ1−θ2=δθ′ and the bilinear form induced by θ′ in (64) vanishesProof.
(1) Letϕ:Tθ1∗g⟶Tθ2∗g be an isomorphism of n-ary multiplicative Hom-Nambu-Lie superalgebras satisfying ϕg∗=idg∗ and ϕx−x∈g∗,∀x∈g. Set θ′x=ϕx−x. Then θ′∈C0g,g∗0¯ and
(65)0=ϕx1+f1,⋯,xn+fnθ1−ϕx1+f1,⋯,ϕxn+fnθ2=ϕx1,⋯,xng+θ1x1,⋯,xn−x1+θ′x1+f1,⋯,xn+θ′xn+fnθ2+∑i=1n−1n−i−1xixi+1+⋯+xnad∗x1,⋯,xi^,⋯,xn·fi=θ′x1,⋯,xng+θ1x1,⋯,xn−θ2x1,⋯,xn−∑i=1n−1n−i−1∣xi∣∣xi+1∣+⋯+∣xn∣ad∗x1,⋯,xi^,⋯,xn·θ′xi=θ1x1,⋯,xn−θ2x1,⋯,xn−δθ′x1,⋯,xn.
Byα′ϕ=ϕα′, we may obtain θ′xα=θ′αx for all x∈g.
For the converse, suppose thatθ′∈C0g,g∗0¯ satisfies θ1−θ2=δθ′ and θ′xα=θ′αx for all x∈g. Let ϕ:Tθ1∗g⟶Tθ2∗g be defined by ϕx+f=x+θ′x+f. Then ϕg∗=idg∗ and ϕx−x∈g∗,∀x∈g. Moreover, α′ϕ=ϕα′. In fact,
(66)α′ϕx+f=αx+θ′x+f=αx+θ′xα+fαϕα′x+f=ϕαx+fα=αx+θ′αx+fα.
Byθ′xα=θ′αx, one gets α′ϕ=ϕα′. Therefore, ϕ is an isomorphism of n-ary multiplicative Hom-Nambu-Lie superalgebras, that is, Tθ1∗g is equivalent to Tθ2∗g.
It is clear that,θ′ defined by (64) is supersymmetric. Note that
(67)X·y,zθ′+−1Xyy,X·zθ′=12θ′X·yz+−1∣X∣+∣y∣∣z∣θ′zX·y+12−1Xyθ′yX·z+−1∣X∣+∣z∣∣y∣θ′X·zy=12θ2X,yz−θ1X,yz+ad∗Xθ′yz+∑i=1n−1−1n−i−1∣xi∣∣xi+1∣+⋯+∣xn−1∣+∣y∣ad∗x1,⋯,xi^,⋯,xn−1,y·θ′xiz−12−1yzad∗X·θ′zy−12ad∗X·θ′yz+12−1yzθ2X,zy−θ1X,zy+ad∗Xθ′zy+∑i=1n−1−1n−i−1∣xi∣∣xi+1∣+⋯+∣xn−1∣+∣z∣ad∗x1,⋯,xi^,⋯,xn−1,z·θ′xiy=0,where we make use of 63=0 and θ1,θ2 satisfying (45). Then, ,θ′ is invariant. In addition,
(68)αx,yθ′=12θ′αxy+−1xyθ′yαx=12θ′αyx+−1xyθ′xαy=αy,xθ′since θ′xα=θ′αx for all x∈g. That is, α is x,yθ′-symmetric.
(2) Let the isomorphismϕ be defined as in (1). Then for all x+f,y+g∈Tθ1∗g, we have
(69)ϕx+f,ϕy+gθ2=x+θ′x+f,y+θ′y+gθ2=θ′xy+fy+−1xyθ′yx+−1xygx=2x,yθ′+x+f,y+gθ1.
Thus,ϕ is an isometry if and only if ,θ′=0.Lemma 28.
LetV,,V,α be a metric ℤ2-graded vector space of dimension m over an algebraically closed field K of characteristic not 2 and g⊆glV be a Lie superalgebra consisting of nilpotent homogeneous endomorphisms of V such that for each f∈g, the map f+:V⟶V defined by f+v,v′V=−1fvv,fv′V is contained in g, too. Suppose that W is an isotropic graded subspace of V which is stable under g and α, i.e., fW⊆W for all f∈g and αW⊆W, then W is contained in a maximally isotropic graded subspace Wmax of V which is also stable under g and α, moreover, dimWmax=m/2. If m is even, then Wmax=Wmax⊥. If m is odd, then Wmax⊂Wmax⊥,dimWmax⊥−dimWmax=1, and fWmax⊥⊆Wmax for all f∈g.Proof.
The proof is by induction onm. The base step m=0 is obviously true. For the inductive step, we consider the following two cases.Case 1.
W≠0 or there is a nonzero g-stable vector v∈V(that is, gv⊆Kv) such that v,vV=0.Case 2.
W=0 and every nonzero g-stable vector v∈V satisfies v,vV≠0.In the first case,Kv is a nonzero isotropic g-stable graded subspace, and W⊥ is also g-stable since w,fw⊥V=−1fwf+w,w⊥V=0. Now, consider the bilinear form ,V′ on the factor graded space V′=W⊥/W defined by x⊥+W,y⊥+WV′≔x⊥,y⊥V, then V′ is metric. Denote by π the canonical projection W⊥⟶V′ and define f′:V′⟶V′ by f′πw⊥=πfw⊥, then f′ is well defined since W and W⊥ are g-stable. Let g′≔f′∣f∈g. Then, g′ is a Lie superalgebra. For each f∈g, there is a positive integer k such that fk=0, which implies that f′k=0. Hence, g′ also consists of nilpotent homogeneous endomorphisms of V′. Note that g′ satisfies the same conditions of g. In fact, let x⊥ and y⊥ be two arbitrary elements in W⊥. Then, by the definition of ,V′, we have
(70)f′+πx⊥,πy⊥V′=−1f′x⊥πx⊥,f′πy⊥V′=−1fx⊥πx⊥,πfy⊥V′=−1fx⊥x⊥,fy⊥V=f+x⊥,y⊥V=πf+x⊥,πy⊥V′=f+′πx⊥,πy⊥V′,for arbitrary f∈g, which shows that f′+=f+′∈g′ for all f∈g.SincedimV′=dimW⊥−dimW=dimV−2dimW, we can use the inductive hypothesis to get a maximally isotropic g′-stable subspace W′max=Wmax/W in V′ and αW′max⊆W′max. Clearly, dimW′max = dimV′/2 = n−2dimW/2 = n/2−dimW. For all x⊥,y⊥∈Wmax, the relation x⊥,y⊥V=πx⊥,πy⊥V′=0 implies that Wmax is isotropic. Note that dimWmax=dimW′max+dimW=n/2, then, Wmax is maximally isotropic. Moreover, for all f∈g and w⊥∈Wmax, we have πfw⊥=f′πw⊥∈W′max, which implies fw⊥∈Wmax. It follows that Wmax is g-stable and αWmax⊆Wmax. This proves the first assertion of the lemma in this case.In the second case, by Engel’s Theorem of Lie superalgebras, there is a nonzerog-stable vector v∈V such that fv=0 for all f∈g. Clearly, Kv is a nondegenerate g-stable graded subspace of V, then V=Kve^Kv⊥ and Kv⊥ is also g-stable since fkv⊥,vV=−1fvkv⊥,f+vV=−1fvkv⊥,0V=0,∀f∈g. Now, if Kv⊥=0, then V=Kv and gV=0, hence g=0 and so 0 is the maximally isotropic g-stable subspace, then the lemma follows. If Kv⊥≠0, then again by Engel’s Theorem of Lie superalgebras, there is a nonzero g-stable vector w∈Kv⊥⊆V such that fw=0 for all f∈g. It follows that g vanishes on the two-dimensional nondegenerate subspace Kve^Kw of V. Without loss of generality, we can assume that v,vV=1=w,wV. Set c=v,wV, then it is easy to check that the nonzero vector v+−c+c2−1w is isotropic and g-stable. This contradicts the assumption of Case 2.Therefore, the existence of a maximally isotropicg-stable graded subspace Wmax containing W is proved. If m is even, then dimWmax=dimWmax⊥=m/2; if m is odd, then dimWmax⊥=m+1/2 and dimWmax=m−1/2. Since g′ is nilpotent, there exists a nonzero πw⊥∈V′ such that g′πw⊥=0. Note that dimV′=1, which implies g′V′=0, so gWmax⊥⊆Wmax.Theorem 29.
Letg,·,⋯,·,α,,g be a nilpotent metric n-ary multiplicative Hom-Nambu-Lie superalgebra of dimension m over an algebraically closed field K of characteristic not 2. If J is an isotropic Hom-ideal of g, then g contains a maximally Hom-ideal I of dimension m/2 containing J. Moreover, if m is even, then g is isometric to some T∗-extension of g/I. If m is odd, then I⊥ is abelian and g is isometric to a nondegenerate graded ideal of codimension 1 in some T∗-extension of g/I.Proof.
Considerad∧n−1g=adX∣X∈∧n−1g. Then, ad∧n−1g is a Lie superalgebra. For any X∈∧n−1g, adX is nilpotent since g is nilpotent. Then, the following identity
(71)−adXy,zg=−1Xyy,adXzgimplies adX+≔−adX∈g. By J is an isotropic graded ideal of g, then, J is an isotropic ad∧n−1g-stable graded subspace and αJ⊆J, by Lemma 28 so there is a maximally isotropic ad∧n−1g-stable graded subspace I of g containing J such that αI⊆I and dimI=m/2,I is also an isotropic graded ideal of g. Moreover, if m is even, then, g is isometric to some T∗-extension of g/I by Theorem 26.
Ifm is odd, then dimI⊥−dimI=1 and ad∧n−1gI⊥⊆I by Lemma 28. Note that
(72)ZI=x∈g∣x,I,g,⋯,gg=0=x∈g∣g,x,I,g,⋯,ggg=0=x∈g∣I,g,⋯,gg,xg=0=I,g,⋯,gg⊥=ad∧n−1gI⊥,which implies that I⊥⊂ad∧n−1gI⊥⊥=ZI⊥, hence I⊥ is abelian.
Take any nonzero elementa∉g, we define α′ by
(73)α′x=aifx=a,αxifx∈g.
Then,Ka is a 1-dimensional abelian n-ary multiplicative Hom-Nambu-Lie superalgebra. Define a bilinear map ,a:Ka×Ka⟶K by a,aa=1. Then, ,a is a nondegenerate supersymmetric invariant bilinear form on Ka. Let g′=g∔Ka. Define
(74)x1+k1a,⋯,xn+knag′=x1,⋯,xng,(75)x+k1a,y+k2ag′=x,yg+k1a,k2aa.
Then,g′,·,⋯,·g′,α′,,g′ is a nilpotent metric n-ary multiplicative Hom-Nambu-Lie superalgebra since
(76)α′x+a,y+ag′=αx+αa,y+ag′=αx,yg+αa,aa=αy,xg+αa,aa=α′y+a,x+ag′for all x,y∈g and g is a nondegenerate Hom-ideal of codimension 1 of g′,α′. Since I⊥ is not isotropic and K is algebraically closed there exists z∈I⊥ and z,zg=−1. In addition, we have αI⊥⊆I⊥ since α′v⊥,vg=αv,v⊥g=0 for v∈I and v⊥∈I⊥. Let b=a+z and I=I∔Kb. Then, I′ is an m+1/2-dimensional isotropic graded ideal of g′.
In fact, for allx+k1a+k1z,y+k2a+k2z∈I′,
(77)x+k1a+k1z,y+k2a+k2zg′=x+k1z,y+k2zg+k1a,k2aa=x,yg+x,k2zg+k1z,yg+k1z,k2zg+k1k2=k1k2−k1k2=0.
In light of Theorem26, we conclude that g′ is isometric to some T∗-extension of g′/I′.
DefineΦ:g′⟶g/I,x+λa↦x−λz+I. Then
(78)Φx1+λ1a,⋯,Φxn+λnag/I=x1−λ1z+I,⋯,xn−λnz+Ig/I=x1,⋯,xng+I=Φx1,⋯,xng=Φx1+λ1a,⋯,xn+λnag′,where we use the fact that I⊥ is abelian and ad∧n−1gI⊥⊆I. Moreover, Φα=αΦ. In fact, for x+λa∈g′, we have
(79)Φαx+λa=Φαx+λαa=Φαx+λa=αx−λz+I=αx−λαz+I=αx−λz+I=αΦx+λa.
It’s clear thatΦ is surjective and kerΦ=I′, so g′/I′≅g/I, hence the theorem follows.Now, we show that there exists an isotropic Hom-ideal in every finite-dimensional metricn-ary multiplicative Hom-Nambu-Lie superalgebra and investigate the nilpotent length of g/I.Proposition 30.
Suppose thatg,·,⋯,·,α,,g is a finite-dimensional metric n-ary multiplicative Hom-Nambu-Lie superalgebra.
(1)
For any graded subspaceV⊆g, CV≔x∈g∣x,g,⋯,gg⊆V=g,⋯,g,V⊥g⊥(2)
gm=Cmg⊥, where C0g=0,Ci+1g=CCig(3)
Ifg is nilpotent of length k, then gi⊆Ck−igProof.
The relation(80)CV,g,⋯,g,V⊥gg=g,⋯,g,CVg,V⊥g⊆V,V⊥g=0shows that CV⊆g,⋯,g,V⊥g⊥. Notice that
(81)g,⋯,g,g,⋯,g,V⊥g⊥g,V⊥g=g,⋯,g,V⊥g⊥,g,⋯,g,V⊥gg=0,which implies g,⋯,g,g,⋯,g,V⊥g⊥g⊆V⊥⊥=V, i.e., g,⋯,g,V⊥g⊥⊆CV. Hence, (1) follows.By induction, (2) and (3) can be proved easily.Theorem 31.
Every finite-dimensional nilpotent metricn-ary multiplicative Hom-Nambu-Lie superalgebra g,·,⋯,·,α,,g over an algebraically closed field of characteristic not 2 such that αg=g is isometric to (a nondegenerate ideal of codimension 1 of) a T∗-extension of a nilpotent n-ary multiplicative Hom-Nambu-Lie superalgebra whose nilpotent length is at most a half of the nilpotent length of g.Proof.
DefineJ=∑i=0∞gi∩Cig. Since g is nilpotent, the sum is finite. Proposition 30 (2) says gi⊥=Cig, then gi∩Cig is isotropic for all i≥0. Since
(82)gi⊇gj⊇gj∩Cjg,ifi<j,we have
(83)gj∩Cjg⊥⊇gi⊥=Cig⊇Cig∩gi,ifi<j.
It follows that(84)gi∩Cig,gj∩Cjgg=0,∀i,j≥0.
Therefore,J is an isotropic graded ideal of g. Let k denote the nilpotent length of g. Using Proposition 30 (3) we can conclude that gk+1/2⊆Ck+1/2g. This implies that gk+1/2 is contained in J. By Theorem 29, there is a maximally isotropic graded ideal I of g containing J⊇gk+1/2. It means that g/I has nilpotent length at most k+1/2, and the theorem follows.Remark 32.
Most results concerningT∗-extensions in [20, 24, 26, 27] are contained in this section as special cases.



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