aa r X i v : . [ m a t h . R A ] J a n Cohomology and Deformation of Leibniz Superalgebras
RB Yadav , ∗ Sikkim University, Gangtok, Sikkim, 737102, I NDIA
Abstract
In this article, we introduce a deformation cohomology of Leibniz superalgebras. Also,we introduce formal deformation theory of Leibniz superalgebras. Using deformationcohomology we study the formal deformation theory of Leibniz superalgebras.
Keywords:
Leibniz superalgebra, cohomology, Extension,formal deformations, Equivalent formal deformations
1. Introduction
Leibniz algebras were introduced by J.L Loday in [1] as a noncommutative gen-eralization of Lie algebras. Lie superalgebras were studied and a classification wasgiven by Kac [2]. Leits [3] introduced a cohomology for Lie superalgebras. Leibnizsuperalgebras [4] are a noncommutative generalizations of Lie superalgebras. Leibnizsuperalgebras were studied in [5], [6].The deformation is a tool to study a mathematical object by deforming it into afamily of the same kind of objects depending on a certain parameter. Algebraic defor-mation theory was introduced by Gerstenhaber for rings and algebras [7],[8],[9], [10],[11]. Deformation theory of Lie superalgebras was introduced and studied by Binegar[12]. Recently, algebraic deformation theory has been studied by several authors [13],[14], [15] etc. ∗ Corresponding author
Email address: [email protected] (RB Yadav)
Preprint submitted to January 20, 2021 urpose of this paper is to introduce deformation cohomology and formal defor-mation theory of Leibniz superalgebras. Organization of the paper is as follows. InSection 2, we recall definition of Leibniz superalgebra and give some examples. InSection 3, we introduce deformation complex and deformation cohomology of Leib-niz superalgebras. In Section 4, we compute cohomology of Leibniz superalgebras indegree and dimension , and . In Section 5, we introduce deformation theory ofLeibniz superalgebras. In this section we see that infinitesimals of deformations arecocycles . Also, in this section we give an example of a formal deformation of a Leib-niz superalgebras. In Section 6, we study equivalence of two formal deformations andprove that infinitesimals of any two equivalent deformations are cohomologous.
2. Leibniz Superalgebras
In this section, we recall definitions of Leibniz superalgebra and module over aLeibniz superalgebras. We give some examples of Leibniz superalgebras. Throughoutthe paper we denote a fixed field by K . Also, we denote the ring of formal power serieswith coefficients in K by K [[ t ]] . In any Z -graded vector space V we use a notation inwhich we replace degree deg ( a ) of an element a ∈ V by ’a’ whenever deg ( a ) appearsin an exponent; thus, for example ( − ab = ( − deg ( a ) deg ( b ) . Definition 2.1.
Let V = V ⊕ V and W = W ⊕ W be Z graded vector spacesover a field K . A linear map f : V → W is said to be homogeneous of degree α if deg( f ( a )) − deg( a ) = α , for all a ∈ V β , β ∈ { , } . We write ( − deg( f ) = ( − f .Elements of V β are called homogeneous of degree β. Definition 2.2. [5] A (left) Leibniz superalgebra is a Z -graded K -vector space L = L ⊕ L equipped with a bilinear map [ − , − ] : L × L → L satisfying the followingconditions: [ a, b ] ∈ L α + β , [[ a, b ] , c ] = [ a, [ b, c ]] − ( − αβ [ b, [ a, c ]] , (Leibniz identity)for all a ∈ L α , b ∈ L β and c ∈ L γ . If second condition in the Definition 2 isreplaced by [ x, [ y, z ]] = [[ x, y ] , z ] − ( − yz [[ x, z ] , y ] , then L is called right Leibniz uperalgebra. In this paper we consider only left Leibniz superalgebra. Let L and L be two Leibniz superalgebras. A homomorphism f : L → L is a K -linear map suchthat f ([ a, b ]) = [ f ( a ) , f ( b )] . Given a Leibniz superalgebra L we denote by [ L, L ] thevector subspace of L spanned by the set { [ x, y ] : x, y ∈ L } . A Leibniz superalgebra L is called abelian if [ L, L ] = 0 . Example 2.1.
Clearly, every Lie superalgebra is a Leibniz superalgebra. A Leibnizsuperalgebra L = L ⊕ L is a Lie superalgebra if [ a, b ] = − ( − ab [ b, a ] for all a ∈ L α , b ∈ L β . Example 2.2.
Given any Z -graded vector space V = V ⊕ V we can define a mul-tiplication on V by [ x, y ] = 0 , for all x ∈ V α , y ∈ V β . This gives an abelian Leibnizsuperalgebra structure on V. Example 2.3.
Let A = A ⊕ A be an associative K -superalgebra equipped with ahomogeneous linear map T : A → A of degree and satisfying the condition T ( a ( T b )) = (
T a )( T b ) = T (( T a ) b ) , (1) for all a, b ∈ A . Define a bilinear map on A by [ a, b ] = ( T a ) b − ( − ab b ( T a ) , for all a ∈ A α , b ∈ A β . One can easily verify that [ − , − ] satisfies the two conditionsfor a Leibniz superalgebra. This makes A a Leibniz superalgebra that we denote by A SL . If T = Id, then A SL turns out to be a Lie superalgebra. If T is an algebra mapon A which is idempotent ( T = T ), then condition 1 is satisfied. If T is a square-zero derivation, that is, T ( ab ) = ( T a ) b + a ( T b ) and T = 0 , then the condition 1 issatisfied. Example 2.4.
Let V be a Z -graded K -vector space. The free Leibniz superalgebra L (V ) is the universal Leibniz superalgebra for maps from V to Leibniz superalgebras.Let T ( V ) := ⊕ n ≥ V ⊗ n be the reduced tensor module. [5] T ( V ) is the free Leibnizsuperalgebra over V with the multiplication defined inductively by [ v, x ] = v ⊗ x, for all x ∈ T ( V ) , v ∈ V [ y ⊗ v, x ] = [ y, v ⊗ x ] − ( − yv v ⊗ [ y, x ] , for all x ∈ T ( V ) , v ∈ V andhomogeneous y ∈ T ( V ) . Example 2.5.
Let L = L ⊕ L be a Z -graded K -vector space, where L is twodimensional subspace of L generated by { x, y } and L is 1-dimensional generated by { z } . Define a bilinear map [ − , − ] : L × L → L given by [ y, x ] = x, [ y, y ] = x, [ x, x ] = [ x, y ] = [ x, z ] = [ z, x ] = [ z, z ] = [ z, y ] = [ y, z ] = 0 . One can easily verify that L together with [ − , − ] is a Leibniz superalgebra. Definition 2.3. [5] Let L = L ⊕ L be a Leibniz superalgebra. A Z -graded vectorspace M = M ⊕ M over the field K is called a module over L if there exist twobilinear maps [ − , − ] : L × M → M and [ − , − ] : M × L → M (we use the samenotation for both the maps and differentiate them from context) such that followingconditions are satisfied [[ a, b ] , m ] = [ a, [ b, m ]] − ( − ab [ b, [ a, m ]] [[ a, m ] , b ] = [ a, [ m, b ]] − ( − am [ m, [ a, b ]] [[ m, a ] , b ] = [ m, [ a, b ]] − ( − ma [ a, [ m, b ]] ,for all a ∈ L α , b ∈ L β , m ∈ M γ , α, β, γ ∈ { , } . Clearly, every Leibniz superalgebra is a module over itself. In the next section weshall discuss some more examples of modules over Leibniz superalgebras.
3. Cohomology of Leibniz Superalgebras
Let L = L ⊕ L be a Leibniz superalgebra and M = M ⊕ M be a module over L . An n-linear map f : L × · · · × | {z } n times L → M is said to be homogeneous of degree α if deg( f ( x , · · · , x n )) − P ni =1 deg( x i )) = α , for homogeneous x i ∈ L , ≤ i ≤ n. We denote the degree of a homogeneous f by deg( f ) . We write ( − deg( f ) = ( − f .For each n ≥ , we define a K -vector space C n ( L ; M ) as follows: For n ≥ , C n ( L ; M ) consists of those f ∈ Hom K ( L ⊗ ( n ) , M ) which are homogeneous, and C ( L ; M ) = M . Clearly, C n ( L ; M ) = C n ( L ; M ) ⊕ C n ( L ; M ) , where C n ( L ; M ) C n ( L ; M ) are submodules of C n ( L ; M ) containing elements of degree 0 and 1,respectively. We define a K -linear map δ n : C n ( L ; M ) → C n +1 ( L ; M ) by δ n f ( x , · · · , x n +1 )= X i We use mathematical induction to prove this lemma. For f ∈ C ( L ; M ) = M ,by using Leibniz identity, we have d x d y f − ( − xy d y d x f = [ x, [ y, f ]] − ( − xy [ y, [ x, f ]]= [[ x, y ] , f ]= d [ x,y ] f. ( i ) holds for all f ∈ C m ( L ; M ) , ≤ m ≤ n . Let f ∈ C n +1 ( L ; M ) . Itis enough to prove that ( d x d y f − ( − xy d y d x f ) z = ( d [ x,y ] f ) z , for all homogeneous z ∈ L . We have ( d x d y f − ( − xy d y d x f ) z = ( d x d y f ) z − ( − xy ( d y d x f ) z = d x ( d y f ) z − ( − x ( f + y ) ( d y f ) [ x,z ] − ( − xy { d y ( d x f ) z − ( − y ( f + x ) ( d x f ) [ y,z ] } = d x d y ( f z ) − ( − yf d x f [ y,z ] − ( − x ( f + y ) { d y ( f [ x,z ] ) − ( − yf f [ y, [ x,z ]] }− ( − xy d y d x ( f z ) + ( − xy + xf d y f [ x,z ] + ( − fy { d x ( f [ y,z ] ) − ( − xf f [ x, [ y,z ]] } = d x d y ( f z ) − ( − xy d y d x ( f z ) + ( − yf + xf + xy f [ y, [ x,z ]] − ( − xf + yf f [ x, [ y,z ]] = d [ x,y ] f z − ( − ( x + y ) f f [[ x,y ] ,z ] = ( d [ x,y ] f ) z For f ∈ C ( L ; M ) , we have d x δf ( y ) = [ x, δf ( y )] − ( − xf δf ([ x, y ])= − [ x, [ f, y ]] + ( − xf [ f, [ x, y ]]= − [[ x, f ] , y ]= δd x f ( y ) . Suppose that ( ii ) holds for all f ∈ C m ( L ; M ) , ≤ m ≤ n . Let f ∈ C n +1 ( L ; M ) . It is enough to prove that ( d x δf − δd x f ) z = 0 , for all z ∈ L . We have ( δd x f ) z − ( d x δf ) z = ( − zx + zf d z d x f − δ (( d x f ) z ) − d x ( δf ) z + ( − xf ( δf ) [ x,z ] = ( − zx + zf d z d x f − δd x f z + ( − xf δf [ x,z ] − ( − zf d x d z f + d x δf z + ( − zf d [ x,z ] f − ( − xf δf [ x,z ] = 0 . Theorem 3.1. δ ◦ δ = 0 , that is, ( C ∗ ( L ; M ) , δ ) is a cochain complex. roof. For f ∈ C ( L ; M ) , we have δδf ( x , x ) = − δf ([ x , x ]) + ( − x f [ x , δf ( x )] + [ δf ( x ) , x ]= [ f, [ x , x ]] − ( − x f [ x , [ f, x ]] − [[ f, x ] , x ]= 0 . Assume that δ ◦ δf = 0 , for all f ∈ C q ( L ; M ) , ≤ q ≤ n , and let f ∈ C n +1 ( L ; M ) .Then for all x ∈ L , by using Lemmas 3.1,3.2, we have ( δδf ) x = ( − xf d x δf − δ (( δf ) x )= ( − xf d x δf − ( − xf δd x f + δδf x = 0 . From this we conclude that δδ = 0 . We denote ker( δ n ) by Z n ( L ; M ) and image of ( δ n − ) by B n ( L ; M ) . We callthe n -th cohomology Z n ( L ; M ) /B n ( L ; M ) of the cochain complex { C n ( L ; M ) , δ n } as the n -th deformation cohomology of L with coefficients in M and denote it by H n ( L ; M ) . Since L is a module over itself. So we can consider cohomology groups H n ( L ; L ) . We call H n ( L ; L ) as the n -th deformation cohomology group of L . Wehave Z n ( L ; M ) = Z n ( L ; M ) ⊕ Z n ( L ; M ) , B n ( L ; M ) = B n ( L ; M ) ⊕ B n ( L ; M ) , where Z ni ( L ; M ) and B ni ( L ; M ) are submodules of C ni ( L ; M ) , i = 0 , . Since bound-ary map δ n : C n ( L ; M ) → C n +1 ( L ; M ) is homogeneous of degree , we concludethat H n ( L ; M ) is Z -graded and H n ( L ; M ) = H n ( L ; M ) ⊕ H n ( L ; M ) , where H ni ( L ; M ) = Z ni ( L ; M ) /B ni ( L ; M ) , i = 0 , .We define two bilinear maps [ − , − ] : L × C n ( L ; M ) → C n ( L ; M ) and [ − , − ] : C n ( L ; M ) × L → C n ( L ; M ) [ a, f ]( a , · · · , a n ) = d a f ( a , · · · , a n )= [ a, f ( a , · · · , a n )] − n X i =1 ( − a ( a + ··· + a i − ) f ( a , · · · , [ a, a i ] , · · · , a n ) , (2) [ f, a ]( a , · · · , a n ) = n X i =1 ( − a ( a + ··· + a i − ) f ( a , · · · , [ a, a i ] , · · · , a n ) − [ a, f ( a , · · · , a n )] . (3)One can easily verify C n ( L ; M ) is a module over L with two actions given by 2 and 3.For each f ∈ C n ( L ; M ) , n > , we define f j ∈ C j ( L ; C n − j ( L ; M )) , ≤ j ≤ n by f j ( a , · · · , a j )( a j +1 , · · · , a n ) = f ( a , · · · , a n ) ,f = f n = f. We consider the cochain complex { C m ( L ; C n − j ( L, M )) , δ m } . As in [12], Onecan easily verify the following result. Lemma 3.3. ( δf j )( a , · · · , a j +1 ) = ( δf ) j +1 ( a , · · · , a j +1 ) + ( − j δ ( f j +1 ( a , · · · , a j +1 )) . 4. Cohomology of Leibniz Superalgebras in Low Degrees Let L = L ⊕ L be a Leibniz superalgebra and M = M ⊕ M be a module over L. For m ∈ M = C ( L ; M ) , f ∈ C ( L ; M ) and g ∈ C ( L ; M ) δ m ( x ) = − [ m, x ] , (4) δ f ( x , x ) = − f ([ x , x ]) + [ x , f ( x )] + [ f ( x ) , x ] , (5) δ g ( x , x , x ) = − g ([ x , x ] , x ) − ( − x x g ( x , [ x , x ]) + g ( x , [ x , x ])+( − x g [ x , g ( x , x )] − ( − x x + x g [ x , g ( x , x )] − [ g ( x , x ) , x ] . (6)9he set { m ∈ M | [ m, x ] = 0 , ∀ x ∈ L } is called annihilator of L in M and is denotedby ann M L . We have H ( L ; M ) = { m ∈ M | − [ m, x ] = 0 , for all x ∈ L } = ann M L. A homogeneous linear map f : L → M is called derivation from L to M if δ f = 0 . For every m ∈ M the map x [ m, x ] is called an inner derivation from L to M . We denote the vector spaces of derivations and inner derivations from L to M by Der ( L ; M ) and Der Inn ( L ; M ) respectively. By using 4, 5 we have H ( L ; M ) = Der ( L ; M ) /Der Inn ( L ; M ) . Let L be a Leibniz superalgebra and M be a module over L. We regard M as anabelian Leibniz superalgebra. An extension of L by M is an exact sequence / / M i / / E π / / L / / (*)of Leibniz superalgebras such that [ x, i ( m )] = [ π ( x ) , m ] , [ i ( m ) , x ] = [ m, π ( x )] . The exact sequence ( ∗ ) regarded as a sequence of K -vector spaces, splits. Thereforewithout any loss of generality we may assume that E as a K -vector space coincideswith the direct sum L ⊕ M and that i ( m ) = (0 , m ) , π ( x, m ) = x. Thus we have E = E ⊕ E , where E = L ⊕ M , E = L ⊕ M . The multiplication in E = L ⊕ M has then necessarily the form [(0 , m ) , (0 , m )] = 0 , [( x , , (0 , m )] = (0 , [ x , m ]) , [(0 , m ) , ( x , , [ m , x ]) , [( x , , ( x , x , x ] , h ( x , x )) , for some h ∈ C ( L ; M ) , for all homogeneous x , x ∈ L , m , m ∈ M. Thus, ingeneral, we have [( x, m ) , ( y, n )] = ([ x, y ] , [ x, n ] + [ m, y ] + h ( x, y )) , (7)10or all homogeneous ( x, m ) , ( y, n ) in E = L ⊕ M. Conversely, let h : L × L → M be a bilinear homogeneous map of degree . Forhomogeneous ( x, m ) , ( y, n ) in E we define multiplication in E = L ⊕ M by Equation7. For homogeneous ( x, m ) , ( y, n ) and ( z, p ) in E we have [[( x, m ) , ( y, n )] , ( z, p )]= ([[ x, y ] , z ] , [[ x, y ] , p ] + [[ x, n ] , z ] + [[ m, y ] , z ] + [ h ( x, y ) , z ] + h ([ x, y ] , z )) (8) [( x, m ) , [( y, n ) , ( z, p )]]= ([ x, [ y, z ]] , [ x, [ y, p ]] + [ x, [ n, z ]] + [ m, [ y, z ]] + [ x, h ( y, z )] + h ([ x, y ] , z ) (9) [( y, n ) , [( x, m ) , ( z, p )]]= ([ y, [ x, z ]] , [ y, [ x, p ]] + [ y, [ m, z ]] + [ n, [ x, z ]] + [ y, h ( x, z )] + h ( y, [ x, z ])) (10)From Equations 8, 9, 10 we conclude that E = L ⊕ M is a Leibniz superalgebra withproduct given by Equation 7 if and only if δ h = 0 . We denote the Leibniz superalgebragiven by Equation 7 using notation E h . Thus for every cocycle h ∈ C ( L ; M ) thereexists an extension E h : 0 / / M i / / E h π / / L / / of L by M , where i and π are inclusion and projection maps, that is, i ( m ) = (0 , m ) ,π ( x, m ) = x . We say that two extensions / / M / / E i / / L / / i = 1 , of L by M are equivalent if there is a Leibniz superalgebra isomorphism ψ : E → E such that following diagram commutes: / / M Id M (cid:15) (cid:15) / / E ψ (cid:15) (cid:15) / / L Id L (cid:15) (cid:15) / / / / M / / E / / L / / (**)11e use F ( L, M ) to denote the set of all equivalence classes of extensions of L by M .Equation 7 defines a mapping of Z ( L ; M ) onto F ( L, M ) . If for h, h ′ ∈ Z ( L ; M ) E h is equivalent to E h ′ , then commutativity of diagram ( ∗∗ ) is equivalent to ψ ( x, m ) = ( x, m + f ( x )) , for some f ∈ C ( L ; M ) . We have ψ ([( x , m ) , ( x , m )]) = ψ ([ x , x ] , [ x , m ] + [ m , x ] + h ( x , x ))= ([ x , x ] , [ x , m ] + [ m , x ] + h ( x , x ) + f ([ x , x ])) , (11) [ ψ ( x , m ) , ψ ( x , m )] = [( x , m + f ( x )) , ( x , m + f ( x ))]= ([ x , x ] , [ x , m + f ( x )] + [ m + f ( x ) , x ] + h ′ ( x , x )) . (12)Since ψ ([( x , m ) , ( x , m )]) = [ ψ ( x , m ) , ψ ( x , m )] , we have h ( x , x ) − h ′ ( x , x ) = − f ([ x , x ]) + [ x , f ( x )] + [ f ( x ) , x ]= δ ( f )( X , x ) (13)Thus two extensions E h and E h ′ are equivalent if and only if there exists some f ∈ C ( L ; M ) such that δ f = h − h ′ . We thus have following theorem: Theorem 4.1. The set F ( L, M ) of all equivalence classes of extensions of L by M is in one to one correspondence with the cohomology group H ( L ; M ) . This cor-respondence ω : H ( L ; M ) → F ( L, M ) is obtained by assigning to each cocycle h ∈ Z ( L ; M ) , the extension given by multiplication 7. 5. Deformation of Leibniz superalgebras Let L = L ⊕ L be a Leibniz superalgebra. We denote the ring of all formalpower series with coefficients in L by L [[ t ]] . Clearly, L [[ t ]] = L [[ t ]] ⊕ L [[ t ]] . Soevery a t ∈ L [[ t ]] is of the form a t = a t ⊕ a t , where a t ∈ L [[ t ]] and a t ∈ L [[ t ]] .12 efinition 5.1. Let L = L ⊕ L be a Leibniz superalgebra. A formal one-parameterdeformation of a Leibniz superalgebra L is a K [[ t ]] -bilinear map µ t : L [[ t ]] × L [[ t ]] → L [[ t ]] satisfying the following properties:(a) µ t ( a, b ) = P ∞ i =0 µ i ( a, b ) t i , for all a, b ∈ L , where µ i : L × L → L , i ≥ are bilinear homogeneous mappings of degree zero and µ ( a, b ) = [ a, b ] is theoriginal product on L.(b) µ t ( µ t ( a, b ) , c ) = µ t ( a, µ t ( b, c )) − ( − ab µ t ( b, µ t ( a, c )) , (14) for all homogeneous a, b, c ∈ L .The Equation 14 is equivalent to following equation: X i + j = r µ i ( µ j ( a, b ) , c )= X i + j = r { µ i ( a, µ j ( b, c )) − ( − ab µ i ( b, µ j ( a, c )) } , (15) for all homogeneous a, b, c ∈ L . Now we define a formal deformation of finite order of a Leibniz superalgebra L . Definition 5.2. Let L be a Leibniz superalgebra. A formal one-parameter deformationof order n of L is a K [[ t ]] -bilinear map µ t : L [[ t ]] × L [[ t ]] → L [[ t ]] satisfying the following properties:(a) µ t ( a, b ) = P ni =0 µ i ( a, b ) t i , ∀ a, b, c ∈ L , where µ i : L × L → T , ≤ i ≤ n , are K -bilinear homogeneous maps of degree , and µ ( a, b ) = [ a, b ] is the originalproduct on L .(b) µ t ( µ t ( a, b ) , c ) = µ t ( a, µ t ( b, c )) − ( − ab µ t ( b, µ t ( a, c )) , (16) for all homogeneous a, b, c ∈ L . emark 5.1. • For r = 0 , conditions 15 is equivalent to the fact that L is aLeibniz superalgebra. • For r = 1 , conditions 15 is equivalent to − µ ([ a, b ] , c ) − [ µ ( a, b ) , c ]+ µ ( a, [ b, c ]) − ( − abµ ( b, [ a, c ]) + [ a, µ ( b, c )] − ( − ab [ b, µ ( a, c )]= δ µ ( a, b, c ); for all homogeneous a, b, c ∈ L. Thus for r = 1 , 15 is equivalent to saying that µ ∈ C ( L ; L ) is a cocycle. Ingeneral, for r ≥ , µ r is just a 2-cochain, that is, in µ r ∈ C ( L ; L ) . Example 5.1. Consider the Leibniz superalgebra L = L ⊕ L in Example 2.5. Definea bilinear mapping µ : L × L → L by µ ( z, z ) = x, µ ( x, x ) = µ ( x, y ) = µ ( y, x ) = µ ( y, y ) = 0 ,µ ( z, x ) = µ ( x, z ) = µ ( y, z ) = µ ( z, y ) = 0 . Clearly, µ is homogeneous of degree . One can easily verify that µ t = µ + µ t ,where µ = [ − , − ] is the product in the Leibniz superalgebra L , is a formal oneparameter deformation of L. Definition 5.3. The cochain µ ∈ C ( L ; L ) is called infinitesimal of the deformation µ t . In general, if µ i = 0 , for ≤ i ≤ n − , and µ n is a nonzero cochain in C ( L ; L ) ,then µ n is called n-infinitesimal of the deformation µ t . Proposition 5.1. The infinitesimal µ ∈ C ( L ; L ) of the deformation µ t is a cocycle.In general, n-infinitesimal µ n is a cocycle in C ( L ; L ) . Proof. For n=1, proof is obvious from the Remark 5.1. For n > , proof is similar. 6. Equivalence of Formal Deformations and Cohomology Let µ t and ˜ µ t be two formal deformations of a Leibniz superalgebra L − L ⊕ L .A formal isomorphism from the deformation µ t to ˜ µ t is a K [[ t ]] -linear automorphism14 t : L [[ t ]] → L [[ t ]] of the form Ψ t = P ∞ i =0 ψ i t i , where each ψ i is a homogeneous K -linear map L → L of degree , ψ ( a ) = a , for all a ∈ T and ˜ µ t (Ψ t ( a ) , Ψ t ( b )) = Ψ t ◦ µ t ( a, b ) , for all a, b ∈ L. Definition 6.1. Two deformations µ t and ˜ µ t of a Leibniz superalgebra L are said tobe equivalent if there exists a formal isomorphism Ψ t from µ t to ˜ µ t . Formal isomorphism on the collection of all formal deformations of a Leibniz su-peralgebra L is an equivalence relation. Definition 6.2. Any formal deformation of T that is equivalent to the deformation µ is said to be a trivial deformation. Theorem 6.1.