0967 some extensions from a quadratic lie algebra

SOME EXTENSIONS FROM A QUADRATIC LIE ALGEBRA DUONG MINH THANH* ABSTRACT In this paper, we give some extensions from a quadratic Lie algebra to quadratic Lie superalgebras and odd quadratic Lie superal[.]

Duong Minh Thanh Tạp chí KHOA HỌC ĐHSP TPHCM _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ SOME EXTENSIONS FROM A QUADRATIC LIE ALGEBRA DUONG MINH THANH* ABSTRACT In this paper, we give some extensions from a quadratic Lie algebra to quadratic Lie superalgebras and odd-quadratic Lie superalgebras Moreover, we use the cohomology to recover some results obtained by the method of double extension Keywords: Quadratic Lie algebras, Quadratic Lie superalgebras, Odd-quadratic Lie superalgebras, Extensions, Symplectic structure TÓM TẮT Một số mở rộng từ đại số Lie toàn phương Trong báo này, đưa số mở rộng từ đại số Lie toàn phương lên siêu đại số Lie toàn phương siêu đại số Lie tồn phương lẻ Bên cạnh chúng tơi sử dụng công cụ đối đồng điều để chứng minh lại số kết thu từ phương pháp mở rộng kép Từ khóa: đại số Lie tồn phương, siêu đại số Lie toàn phương, siêu đại số Lie toàn phương lẻ, mở rộng, cấu trúc symplectic ad * g* its dual space Denote by ad and Introduction Let g be a complex Lie algebra and the adjoint and coadjoint representations of g , respectively It is known that the semidirect product given by: g = g⊕ g* of g and g* by ad * is a Lie algebra with the bracket   X + f ,Y + g   =   X,Y   + ad * (X ) (g) − ad * (Y ) ( f ) , More particularly, we have   X,Y   =   g X,Y   , g   f , g  for all X ,Y ∈ g =0 g, ∀ X,Y ∈ g, f , g ∈ g*   X, f   = − f g oa d(X ) and f , g ∈ g Remark that g is also a quadratic Lie algebra * with invariant symmetric bilinear form B defined by: B(X + f ,Y + g) = f (Y ) + g(X ) ∀ X,Y ∈ g, f , g ∈ g* , In 1985, A Medina and P Revoy gave the notion of double extension to completely characterize all quadratic Lie algebras [9] This notion is regarded as a generalization of the Tạp chí KHOA HỌC ĐHSP TPHCM Số 51 năm 2013 _ of _ _semidirect _ _ _ _ product _ _ _ by _ _the _ coadjoint _ _ _ _ representation _ _ _ _ _ Another _ _ _ _ generalization _ _ _ _ _ _ is_ called _ _ T*definition extension given by M Bordemann that is suffictent * Ph.D., HCMC University of Education to describe solvable quadratic Lie algebras [4] We will recall them and some basic results in Section Sections and are devoted to give an expansion of these two notions for Lie superalgebras In particular, we present a way to obtain a quadratic Lie superalgebra since a Lie algebra and a symplectic vector space It is regarded as a rather special case of the notion of generalized double extension in [1] In a slight change of the notion of T*-extension, we give a manner of how to get an odd-quadratic Lie superalgebra from a Lie algebra In the last section, we introduce an approach to quadratic Lie algebras by the cohomology given in [9] and [10] From this, we give an explanation of the structure of double extension as well as it allows us to construct new quadratic Lie algebra structures from a given quadratic Lie algebra Quadratic Lie algebras Definintion 2.1 Let g be a Lie algebra A bilinear form B : g× g → is called: (i) £ symmetric if B(X,Y) = B(Y, X) for all X,Y ∈g , (ii) non-degenerate if B(X,Y) = for all Y ∈g implies X = , (iii) invariant if B([X,Y], Z) = B(X,[Y, Z]) for all X,Y, Z ∈g A Lie algebra g is called quadratic if there exists a bilinear form B on g such that B is symmetric, non-degenerate and invariant Definition 2.2 Let (g, B) be a quadratic Lie algebra and D be a derivation of g We say D a skew-symmetric derivation of g if it satisfies B(D(X),Y) = −B(X, D(Y )), ∀X,Y ∈g Denote by Der a (g, B) the vector space of skew-symmetric derivations of (g, B) then Der (g, is a subalgebra of Der(g) , the Lie algebra of derivations of g The a B) notion of double extension is defined as follows (see also in [9]) Definition 2.3 Let g be a Lie algebra, Lie algebra Let φ : g →D er (h, a B) g* be its dual space and (h, B) be a quadratic be a Lie algebra endomorphism Denote by ϕ : h×h →g* the linear mapping defined by: ϕ(X,Y )Z = B(φ(Z) (X),Y ), ∀X,Y ∈h, Z ∈g Consider the vector space h = g ⊕ h⊕ g* and define a product on h:   X + F + f ,Y + G + g  =   X,Y h   +   F,G h + ad * (X)(g) − ad * (Y ) ( f ) g + φ( X)(G) − φ (Y)(F) + ϕ(F, G) Duong Minh Thanh Tạp chí KHOA HỌC ĐHSP TPHCM _ _ _ _ _ for all X,Y ∈g, _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ f , g ∈ and F,G ∈h Then h becomes a quadratic Lie algebra with g* the bilinear form B given by: B( X + F + f ,Y + G + g) = f (Y ) + g( X) + B(F,G) fo r X,Y al ∈g, l f , a F,G ∈h The Lie g n ∈g* d algebra h, B ( ) is called the double extension of (h, B) by g by means of φ Note that when h = then this definition is reduced to {0 the notion of the } semidirect product of g and g* by the coadjoint representation Proposition 2.4 ([8], 2.11, [9], Theorem I) Let g be an indecomposable quadratic Lie algebra such that it is not simple nor one-dimensional Then g is the double extension of a quadratic Lie algebra by a simple or one-dimensional algebra Sometimes, we use a particular case of the notion of double extension, that is a double extension by a skew-symmetric derivation It is explicitly defined as follows Definition 2.5 be a quadratic Lie C ∈ Der a (g, Let (g, B) algebra and B) On the vector space g = we define the product: g⊕ £ e ⊕ £ f   X,Y   =   X,Y   + g g B(C(X ) ,Y ) f , [e, X] = C(X) and  f , g = fo X,Y ∈g Then g is a quadratic Lie algebra with r invariant bilinear form B all defined by: Số 51 năm 2013 Tạp chí KHOA HỌC ĐHSP TPHCM _ _ _ B ( e , e ) _ _ _ _ _ _ _ _ 0g double extens , t ion of Bh g by ( e C or a oneX dim ensi , = Y onal dou B ble ) ( exte = nsio f n, for B , shor ( X t f , T ) Y h e = ) o n B a en di ( d m e , e B n ( si g ) e o , n = al f d B o ( ) u bl f = e e , xt e gforX,Y n ) all∈g In si this o = case, n we call _ _ s a r e s u f f i c i e n t f o r s t u d y i n g s o l v a b l e q u a d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ratic Lie algebras by the following proposition (see [6] or [8]) Proposition 2.6 Let (g, B) be a solvable quadratic Lie algebra of dimension n , n ≥ Assume g nonAbelian Then g is a one-dimensional double extension of a solvable quadratic Lie algebra of dimension n − We give now another generalization given by M Bordemann as follows Definition 2.7 Let g be a Lie algebra and θ : g× g →g* be a 2-cocycle of g , that is a skew-symmetric bilinear map satisfying: Duong Minh Thanh Tạp chí KHOA HỌC ĐHSP TPHCM _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ θfollo ( wing ) prod ouct: Z + ([ Y Z cycl e Y Z _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ θ for X, all Y, Z ∈ g De fin e on th e ve ct or sp ac e T * (g ): = g ⊕ g* th e   X + f ,Y + g   =   X,Y   + ad * (X) (g) − ad * (Y ) ( f ) + θ ( X,Y ) for all X ,Y ∈ f , g ∈ g* T becomes a Lie algebra and it is called the T** Then g, (g) θ extension of g by means of θ In addition, if θ satisfies the cyclic condition, i.e θ (X,Y)Z = θ(Y, T for all X,Y, Z ∈g is quadratic with the bilinear *θ Z)X then (g) form: B(X + f ,Y + g) = f (Y ) + g(X ) , ∀ X,Y ∈ g, f , g ∈ g* Note that in the case of θ = then this notion is exactly the semidirect product by the coadjoint representation Proposition 2.8 [4] Let (g, B) be an even-dimensional quadratic Lie algebra over £ If g is solvable then it is i-isomorphic to a T*-extension T θ * (h) quotient algebra of g by a totally isotropic ideal of h where h is the Quadratic Lie superalgebras Definition 3.1 Let g = g0 ⊕ be a Lie superalgebra If there is a non-degenerate g1 supersymmetric bilinear form B on g such that B is even and invariant then the pair (g, B) is called a quadratic Lie superalgebra Note that if (g, B) is a quadratic Lie superalgebra then g0 is a quadratic Lie algebra and g is a symplectic vector space with the restriction of the bilinear form B on each part Lemma 3.2 Let g be a Lie algebra and ) ( h, B h a symplectic vector space with symplectic form Bh Let ψ : g→End( h) be a Lie algebra endomorphism satisfying: Bh ( ψ ( X )(Y ), Z ) = −Bh ( Y ,ψ ( X )(Z )) , ∀X ∈g, Y , Z ∈h Denote by φ :h × h→g* the bilinear map defined by: φ( X ,Y )Z = Bh ( ψ (Z )( X ),Y ) , ∀X ,Y ∈h, Z ∈g Then φ is symmetric, i.e φ (X,Y) = φ(Y, X) for all X ,Y ∈h Proof For all X ,Y ∈h, Z ∈g , φ( X ,Y )Z = Bh ( ψ (Z )( X ),Y ) = −Bh ( X ,ψ (Z )(Y )) = Bh ( ψ (Z )(Y ), X ) = φ (Y , X )Z Then one has φ (X,Y) = φ(Y, X) Số 51 năm 2013 Tạp chí KHOA HỌC ĐHSP TPHCM _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Theorem 3.3 Keep notions as in the above lemma and define on the vector space g = g⊕ g* ⊕h the following bracket:   X + f + F ,Y + g + G   g =   X ,Y + ad*( X )(g) − ad*(Y )( f ) g + ψ ( X )(G) − ψ (Y )(F ) + φ (F, G) for all X ,Y ∈g, f,g an F,G ∈h Then g becomes a quadratic Lie ∈g* d * = g⊕ g , = h and the bilinear form B defined by: superalgebra with g0 g B(X + f + F,Y + g + G) = f (Y ) + g(X ) + Bh(F,G) for all X ,Y ∈g, f , g ∈g* F,G ∈h and Proof We check first that the bracket satisfies the super-antisymmetric property X + f ,Y + g ∈g0 then   Y , X   = −(−1)xy[Y , X ] , ∀X ∈gx ,Y ∈ gy Indeed, if   X + f ,Y + g   =   X ,Y   + ad*( X )(g) − ad*(Y )( f ) = −   Y + g, X + f g  If X + f ∈g ,Y ∈ g1 then   X + f ,Y   = ψ ( X )(Y ) and   Y , X + f   = − ψ ( X )(Y ) And if X ∈ g1,Y ∈ g1 then   X ,Y   = φ (X ,Y ) = φ(Y , X ) =[ X ,Y ] Therefore, one has   Y , X   = −(−1)xy[Y , X ] , ∀X ∈ gx ,Y ∈gy Next, we check the Jacobi identity: ( −1) zx   +  X,Y,Z    X ,Y     = is right for all X ∈gx , Y∈ gy and ( −1) xy  Y, Z,X  +   Z ∈gz Indeed, if identity is clear Let X + f ,Y + g ∈g0 and Z ∈ g1 then and 10 X ,Y , Z ∈g0 ( −1) yz  Z,  then the Jacobi Duong Minh Thanh Tạp chí KHOA HỌC ĐHSP TPHCM  _  X + f, Y + g, _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Z     =   X + f ,ψ (Y )(Z )   = ψ ( X ) ( ψ (Y )(Z ) ) = ψ ( X ) oψ (Y )(Z ),  Y + g,   Z , X + f    =   Y + g, − ψ ( X )(Z )   = − ψ ( Y ) ( ψ ( X )(Z )   ) = − ψ ( Y ) oψ ( X )(Z )  Z ,   X + f ,Y + g    =  Z ,   X ,Y   + ad* ( X )(g) − ad* (Y )( f ) =     − ψ (   X ,Y   Therefore, ) ( Z)       X + f ,   Y + g, Z    +  Y + g,   Z , X + f    +  Z ,   X + f ,Y + g     = If X + f ∈ g0 and Y , Z ∈ g1 then   = + + φ = φ = − φ (Y  X f ,   Y , Z      X f , (Y , Z )   ad*( X ) ( (Y , Z ) ) , Z ) oad( X ),  Y ,   Z, X + f    =   Y , − ψ (X )(Z )   = − φ (Y ,ψ (X )(Z ))   1 and −   Z ,   X + f ,Y     = −   Z ,ψ ( X )(Y )  = − φ (Z ,ψ ( X )(Y )) To prove − φ (Y , Z ) oad( X ) − φ(Y ,ψ ( X )(Z )) − φ (Z ,ψ ( X )(Y )) T ∈g we = , let have: − φ (Y , Z ) oad( X )(T ) = − φ (Y , Z ) (   X ,T   ) ) ( = − Bh ψ (   X ,T   ) (Y ), Z , and − φ (Y ,ψ ( X )(Z ))(T ) = − Bh ( ψ ( T ) (Y ),ψ ( X )(Z ) ) = Bh ( ψ ( X ) oψ ( T ) (Y ), Z ) − φ (Z ,ψ ( X )(Y ))(T ) = − Bh ( ψ ( T ) (Z ),ψ ( X )(Y ) ) = Bh ( ψ ( X ) (Y ),ψ ( T ) (Z )) = − Bh ( ψ ( T ) oψ ( X )(Y ), Z ) Therefore − φ(Y , Z ) oad( X ) − φ(Y ,ψ ( X )(Z )) − φ (Z ,ψ ( X )(Y )) = and then       X + f ,   Y , Z    +  Y ,   Z , X + f    −  Z ,   X + f ,Y   = If X ,Y , Z ∈ g1 then the Jacobi identity is obviously satisfied since    X ,   Y , Z    =   X , φ (Y , Z )   = In final, we shall check B invariant This is a straightforward computation It is easy to see that B is symmetric on g0 ×g0 , skew-symmetric on g1 × g1 and vanish on g0 × g1 Hence, we can conclude that g is a quadratic Lie superalgebra Now we combine Definition 2.7 and Theorem 3.3 to get a more general result as follows θ : g × g →g* a 2-cocycle of g Assume Theorem 3.4 Let g be a Lie algebra and ( h, B ) h a symplectic vector space with symplectic form Bh Let ψ : g→ End( h) be a Lie algebra endomorphism satisfying: Bh ( ψ ( X )(Y ), Z ) = −Bh ( Y ,ψ ( X )(Z )) , ∀X ∈g , Y , Z ∈h Denote by φ : h × h →g* the bilinear map defined by: φ( X ,Y )Z = Bh ( ψ (Z )( X ),Y ) , ∀X ,Y ∈h, Z ∈g and define on the vector space g = g⊕ g* ⊕h the following bracket: * *   X + f + F ,Y + g + G   g =   X ,Y   g + ad ( X )(g) − ad (Y )( f ) + θ (X ,Y ) + ψ ( X )(G) − ψ (Y )(F ) + φ (F, G) Số 51 năm 2013 Tạp chí KHOA HỌC ĐHSP TPHCM _ for all _ _ _ _ _ _ X ,Y ∈g , _ _ _ _ f,g∈ g* _ _ _ _ an d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ F ,G ∈h Then g becomes a quadratic Lie superalge =*g⊕ = h bra with g , g g0 As a consequence of Definition 2.7 and Theorem 3.4, we also have the following corollary Corollary 3.5 If θ is cyclic then g is a quadratic Lie superalgebra with the bilinear form: B ( X + F + f ,Y + G + g ) = f (Y ) + g( X ) + Bh(F , G) f X ,Y f , g o ∈g , ∈g* r and a ll F ,G ∈h Odd-quadratic Lie superalgebras Definitio g = be a Lie superalgebra If n 4.1 g0 ⊕ there is a non-degenerate Let g1 supersymmetric bilinear form B on g such that B is odd and invariant then the pair (g, B) is called an odd-quadratic Lie superalgebra Let g be a Lie algebra and ϕ : g* × g* →g be a bilinear map We define on the vector space g = g⊕ g* the following bracket: *   X + f ,Y + g   =   X ,Y   + ad ( X ) (g) − ad*(Y )( f ) + ϕ( f , g) * f X ,Y f , g ∈g We search some condition o ∈g, such that g becomes a Lie super r a ll g = g and g1 algebr a with 14 Duong Minh Thanh Tạp chí KHOA HỌC ĐHSP TPHCM _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = t ase, ϕ g h is * i obviou s sly I symme n c tric ( i )   g , X    X   , +  f =    g oad( X +  X ,   f , g     g ,  ϕ  (   X , ϕ ( f , g )   , − f ,   = n d f , g o a d ( X ) ) a   = ϕ ( X T e , m h ad( X ( ( g ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ and ( )  g,   h, f    = g oad ϕ ( h, f )   Then we hace the second condition: f oad( ϕ ( g, h) )) ) + g oad( ϕ ( h, f + h oad( ϕ ( f , g ) ) = For the bilinear form B is defined by: B( X + f ,Y + g) = f (Y ) + g( X ) , ∀ X f , g ∈ g* , ,Y ∈ g, one has: ( ( f oad( X = (( h g , , f h o a d ( _ _ gf h fo o( () 15 ( B   X + f ,Y + g   g , Z + h − f and B ( (   Z ,Y   ) ) (   X ,Y   + ϕ ( f , g ) ) + g (   Z , X   ) f = it must have h ( ϕ( f , g) ) = f =h (   Y , Z   + ϕ (g, h) ) ) − g(  X,Z ) X + f ,   Y + g, Z + h  g ) + h (   X ,Y   Hence ( ϕ (g, h)) Finally, we have the following result ϕ : g* × g* →g a symmetric bilinear map Theorem 4.2 Let g be a Lie algebra and satisfying two conditions: (i) ad( X )( ϕ ( f , g ) ) (ii) f oad( ϕ ( g, h) + g oad( ϕ ( h, f for all ) + ϕ ( f , g oad( X )) + ϕ ( g, f oad( X )) = 0, )) + h oad( ϕ ( f , g ) ) =0 f , g ∈g* X ,Y ∈g , Then the vector space g = g⊕ g* with the following bracket: * *   X + f ,Y + g   =   X ,Y   + ad ( X )(g) − ad (Y )( f ) + ϕ( f , g) for all f , g ∈g* is a Lie superalgebra and called the T * -extension of g by X ,Y ∈ g , s means of ϕ Moreover, if ϕ satisfies the condition h ( ϕ ( f , g )) = f an odd-quadratic with the bilinear form g, h)) ( ϕ( then g is B( X + f ,Y + g) = f (Y ) + g( X ) , ∀X ,Y ∈ g, f , g ∈g* Approach to quadratic Lie algebras by the structure equation 5.1 The associatied 3-form and the structure equation Given a finite dimensional complex vector space V , equipped with a nondegenerate symmetric bilinear form B In [10], G Pinczon and R Ushirobira ( introduced the notion of the super Poisson bracket on the exterior algebra Λ V n follows {X } n { Ω, Ω'} j = (−1) j =1 k +1 ∑ι j=1 X j (Ω) ∧ ι Xj (Ω') , ∀Ω ∈ Λk ) (V * ( Ω'∈ Λ V ) * * ) as with a fixed orthonorm al basis of V For a quadratic Lie algebra (g, B) , they defined a trilinear form I I (X ,Y , Z ) = B([X ,Y ] , Z ) X ,Y , Z ∈g then {I , I} = by for all Moreover, the quadratic Lie algebra structure of (g, is completely characterized by B) I and there is a one-to-one correspondence between the set of structures of quadratic Lie algebra and the set of I satisfying {I , I} = Then we call I the associated 3-form and {I , I} = the structure equation of (g, B) Số 51 năm 2013 Tạp chí KHOA HỌC ĐHSP TPHCM _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Recall that De (g, B) of skew-symmetric derivations of g is a Lie subalgebra of D er(g) of derivations of g Proposition 5.1 [9] There exists a natural isomorphism T between De (g, B) and the { Ω ∈ Λ ( g ) :{I , Ω} = 0} space * isomorphism that induces an from D era (g, B) / ad ( g ) onto the second cohomology group H2(g,£ ) Next, we shall use this isomorphism to construct a new structure of quadratic Lie algebra from g as follows Let Ig be the associated 3-form of g and assume Ω∈ Λ ( g ) such that * g = g⊕ £ e ⊕ £ f {I , Ω} = On the vector g we extend B space to B such that B(e, f ) =1 Ig = α ∧ Ω + Ig and B(e, e) = B( f , f ) = Set α = B( f , ) and define Theorem 5.2 (i) The element I (ii) The element I is the associated 3-form of the double extension of g by g g defines a quadratic Lie algebra structure on g T −1(Ω) Proof { (i) One has I g , I g (ii) For all } = { α ∧ Ω,α ∧ Ω} + ( { I ,α} ∧ Ω − α ∧ { I , Ω} ) + { I , I } = g X ,Y ∈ g , by [10], [X,Y] = ιX ∧Y f (I) g Also, I (X,Y , Z ) = I (X,Y , Z ) so g g g ( B [X,Y ] , Z g ) ( = B [X,Y]g , Z g Let C = T −1(Ω) then 18 ( e,[X,Y ]) g g then [e, X ]∈ g and [X,Y ] ∈ g⊕ £ [X,Y ]g = [X,Y ]g + Ω(X,Y ) f B g = I (e, X,Y ) = α ( e ) Ω(X,Y ) = B(C(X ) ,Y ) ) and then Duong Minh Thanh Tạp chí KHOA HỌC ĐHSP TPHCM _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ It means Ω(X ,Y ) = B(C(X ) ,Y ) By the invariance of B , one has [e, X ] = C( X ) So that I defines the double extension of C g by g Remark 5.3 (i) In the case g Abelian, i.e I = g then it is obviously {Ig, Ω} = for any 20 form Ω on g and therefore Ig = α ∧ Ω This case has been studied in [5] 19 If C = ad(X ) is an inner derivation of g then the double extension g of g (ii) by C has I = α ∧ ι (I ) + I In this case, ι (I ) g = It means e − X central and then we X g e− X recover a result in [7] that g is decomposable 5.2 Symplectic quadratic Lie algebras Definition 5.4 Given a Lie algebra g A non-degenerate skew-symmetric bilinear form ω : g× g →£ is called a symplectic structure on g if it satisfies ω ([X ,Y ] , Z ) + ω([Y , Z ] , X ) + ω ([Z , X ] ,Y ) = , ∀X ,Y , Z ∈ g A symplectic structure ω on a quadratic Lie algebra (g, is corresponding to a B) skew-symmetric invertible derivation D defined ω (X,Y ) = B(D(X ) for all by ,Y ) X ,Y ∈ g As above, a symplectic structure is exactly a non-degenerate 2-form ω satisfying {I ,ω} = In this case, we call (g, B,ω) a symplectic quadratic Lie algebra Let (g, be a symplectic quadratic Lie algebra Assume Ω is a nonB,ω) degenerate 2-form satisfying {I , Ω} = and Ω' is another 2-form satisfying {I , Ω ' } = The double extension g of g corresponding to Ω' has I g = α ∧ Ω'+ I We set a nondegenerate 2-form on g by Ω = Ω + λe* ∧ * ) ( f * + X with λ ≠ and some X ∈g We g search a condition of Ω , Ω ' , λ and X such that Ω define a skew-symmetric g invertible derivation on g and therefore it defines a symplectic structure on g By the condition {I , Ω} one has =0 { I , Ω } = { e ∧ Ω'+ I , Ω + λe ∧ ( f + X ) } = { e ∧ Ω', Ω} + { e ∧ Ω', λe ∧ ( f + X ) } + { I , λe ∧ ( f + X ) } Since { e ∧ Ω', Ω} = −e ∧ { Ω, Ω'} , { e ∧ Ω ' , λ e ∧ ( f + X ) } 0= * g * * * g * * * = − λ e* ∧ Ω' * * * * * * * * * * * and ... extension as well as it allows us to construct new quadratic Lie algebra structures from a given quadratic Lie algebra Quadratic Lie algebras Definintion 2.1 Let g be a Lie algebra A bilinear form... such that B is even and invariant then the pair (g, B) is called a quadratic Lie superalgebra Note that if (g, B) is a quadratic Lie superalgebra then g0 is a quadratic Lie algebra and g is a symplectic... give an expansion of these two notions for Lie superalgebras In particular, we present a way to obtain a quadratic Lie superalgebra since a Lie algebra and a symplectic vector space It is regarded