On Description of Isomorphism Classes of filiform Leibniz algebras in dimensions 7 and 8
aa r X i v : . [ m a t h . R A ] A p r On Description of Isomorphism Classes offiliform Leibniz algebras in dimensions 7 and 8 Isamiddin S. Rakhimov and Munther A. Hassan , Institute for Mathematical Research (INSPEM) & Department of Mathematics, FS, Universiti Putra Malaysia43400, UPM, Serdang, Selangor Darul Ehsan, Malaysia, [email protected] munther [email protected] Abstract
The paper concerns the classification problem of a subclass of complex filiform Leibniz algebrasin dimensions 7 and 8. This subclass arises from naturally graded filiform Lie algebras. We give acomplete list of isomorphism classes of algebras including Lie case. In parametric families cases, thecorresponding orbit functions (invariants) are given. In discrete orbits case, we show representativesof the orbits.
Leibniz algebras were introduced by J. -L. Loday [5],[6]. A skew-symmetric Leibniz algebra is aLie algebra. The main motivation of J. -L. Loday to introduce this class of algebras was the searchof an “obstruction” to the periodicity in algebraic K − theory. Besides this purely algebraic motivation,some relationships with classical geometry, non-commutative geometry and physics have been recentlydiscovered. The present paper deals with the low-dimensional case of subclass of filiform Leibniz algebras.This subclass, arises from naturally graded filiform Lie algebras. Definition 1.1.
An algebra L over a field K is called a Leibniz algebra, if its bilinear operation [ · , · ] satisfies the following Leibniz identity: [ x, [ y, z ]] = [[ x, y ] , z ] − [[ x, z ] , y ] , for any x, y, z ∈ L. Onward, all algebras are assumed to be over the fields of complex numbers C . Let L be a Leibniz algebra. We put: L = L, L k +1 = [ L k , L ] , k ≥ . Definition 1.2.
A Leibniz algebra L is said to be nilpotent, if there exists s ∈ N , such that L ⊃ L ⊃ ... ⊃ L s = { } . Definition 1.3.
A Leibniz algebra L is said to be filiform, if dimL i = n − i, where n = dimL, and ≤ i ≤ n. We denote by
Leib n the set of all n − dimensional filiform Leibniz algebras.The following theorem from [4] splits the set of fixed dimension filiform Liebniz algebras in to threedisjoint subsets. Theorem 1.1.
Any ( n + 1) − dimensional complex filiform Leibniz algebra L admits a basis { e , e , ..., e n } called adapted, such that the table of multiplication of L has one of the following forms, where non definedproducts are zero: F Leib n +1 = [ e , e ] = e , [ e i , e ] = e i +1 , ≤ i ≤ n − , [ e , e ] = α e + α e + ... + α n − e n − + θe n , [ e j , e ] = α e j +2 + α e j +3 + ... + α n +1 − j e n , ≤ j ≤ n − ,α , α , ..., α n , θ ∈ C . On low-dimensional filiform Leibniz algebras and their invariants
SLeib n +1 = [ e , e ] = e , [ e i , e ] = e i +1 , ≤ i ≤ n − , [ e , e ] = β e + β e + ... + β n e n , [ e , e ] = γe n , [ e j , e ] = β e j +2 + β e j +3 + ... + β n +1 − j e n , ≤ j ≤ n − ,β , β , ..., β n , γ ∈ C .T Leib n +1 = [ e i , e ] = e i +1 , ≤ i ≤ n − , [ e , e i ] = − e i +1 , ≤ i ≤ n − , [ e , e ] = b , e n , [ e , e ] = − e + b , e n , [ e , e ] = b , e n , [ e i , e j ] = a i,j e i + j +1 + · · · + a n − ( i + j +1) i,j e n − + b i,j e n , ≤ i < j ≤ n − , [ e i , e j ] = − [ e j , e i ] , ≤ i < j ≤ n − , [ e i , e n − i ] = − [ e n − i , e i ] = ( − i b i,n − i e n , where a ki,j , b i,j ∈ C , b i,n − i = b whenever 1 ≤ i ≤ n − , and b = 0 for even n. It should be mentioned that in the theorem above the structure constants α , α , ..., α n , θ and β , β , ..., β n , γ in the first two cases are free, however, in the third case there are relations among a ki,j , b i,j that are different in each fixed dimension (see Lemma 2.1 and Lemma 2.2). According to the theoremeach class has the so-called adapted base change sending an adapted basis to adapted and they can bestudied autonomously. The classes F Leib n , SLeib n in low dimensional cases, have been considered in [8],[9]. The general methods of classification for Leib n has been given in [1], [2] and [7]. This paper dealswith the classification problem of low-dimensional cases of T Leib n . Note that the class
T Leib n contains all n − dimensional filiform Lie algebras.The outline of the paper is as follows. Section 1 is an introduction to a subclass of Leibniz algebrasthat we are going to investigate. Section 2 presents the main results of the paper consisting of a completeclassification of a subclass of low dimensional filiform Leibniz algebras. Here, for 7- and 8-dimensionalcases we give complete classification. For parametric family cases the corresponding invariant functionsare presented. Definition 1.4.
Let { e , e , ..., e n } be an adapted basis of L from T Leib n +1 . Then a nonsingular lineartransformation f : L → L is said to be adapted if the basis { f ( e ) , f ( e ) , ..., f ( e n ) } is adapted. The subgroup of GL n +1 consisting of all adapted transformations is denoted by G ad . The followingproposition specifies elements of G ad . Proposition 1.1.
Any adapted transformation f in T Leib n +1 can be represented as follows: f ( e ) = e ′ = n X i =0 A i e i , f ( e ) = e ′ = n X i =1 B i e i , f ( e i ) = e ′ i = [ f ( e i − ) , f ( e )] , ≤ i ≤ n,A , A i , B j , ( i, j = 1 , ..., n ) are complex numbers and A B ( A + A b ) = 0 . Proof.
Since a filiform Leibniz algebra is 2-generated (see Theorem 1.1.) it is sufficient to consider theadapted action of f on the generators e , e : f ( e ) = e ′ = n X i =0 A i e i , f ( e ) = e ′ = n X i =0 B i e i . Then f ( e i ) = [ f ( e i − ) , f ( e )] = A i − ( A B − A B ) e i + n X j = i +1 ( ∗ ) e j , ≤ i ≤ n. Note that A = 0 , ( A B − A B ) = 0 , otherwise f ( e n ) = 0 . The condition A B ( A + A b ) = 0 appearsnaturally since f is not singular. samiddin S. Rakhimov, Munther A. Hassan f ( e ) , f ( e )] = B ( A B − A B ) e + n X j =4 ( ∗ ) e j . Then for the basis { f ( e ) , f ( e ) , ..., f ( e n ) } to be adapted B ( A B − A B ) = 0 . But according to the observation above( A B − A B ) = 0 . Therefore B = 0 . In G ad we specify the following transformations called elementary: τ ( a, b, c ) = τ ( e ) = a e + b e ,τ ( e ) = c e , a c = 0 ,τ ( e i +1 ) = [ τ ( e i ) , τ ( e )] , ≤ i ≤ n − ,σ ( a, k ) = σ ( e ) = e + a e k , ≤ k ≤ n,σ ( e ) = e ,σ ( e i +1 ) = [ σ ( e i ) , σ ( e )] , ≤ i ≤ n − ,φ ( c, k ) = φ ( e ) = e ,φ ( e ) = e + c e k , ≤ k ≤ n,φ ( e i +1 ) = [ φ ( e i ) , φ ( e )] , ≤ i ≤ n − , where a, b, c ∈ C . Proposition 1.2.
Let L be an algebra from T Leib n +1 , then any adapted transformation f can be repre-sented as the composition: f = φ ( B n , n ) ◦ φ ( B n − , n − ◦ ... ◦ φ ( B , ◦ σ ( A n , n ) ◦ σ ( A n − , n − ◦ ... ◦ σ ( A , ◦ τ ( A , A , B ) . Proof.
The proof is straightforward.
Proposition 1.3.
The transformations g = φ ( B n , n ) ◦ φ ( B n − , n − ◦ φ ( B n − , n − ◦ σ ( A n , n ) ◦ σ ( A n − , n − ◦ σ ( A n − , n − ◦ σ ( A n − , n − , if n even, and g = φ ( B n , n ) ◦ φ ( B n − , n − ◦ σ ( A n , n ) ◦ σ ( A n − , n − ◦ σ ( A n − , n − , for odd n does not change the structure constants of algebras from T Leib n +1 . Proof.
Let us prove the assertion when n is even. • Consider the transformation, σ ( A n − , n −
3) = σ ( e ) = e + A n − e n − ,σ ( e ) = e ,σ ( e i +1 ) = [ σ ( e i ) , σ ( e )] , ≤ i ≤ n − . To show that the transformation σ ( A n − , n − , does not change the structure constants, note that σ ( e ) = e + ( ∗ ) A n − e n − + ( ∗∗ ) A n − e n , σ ( e ) = e + ( ⋆ ) A n − e n , and σ ( e i ) = e i , ∀ i ≥ , andby a simple computation one can see that the transformation σ ( A n − , n −
3) does not change thestructure constants. • The transformation σ ( A n − , n − , does not change the structure constants, because σ ( e ) = e − A n − ( ∗ ) e n , and σ ( e i ) = e i , ∀ i ≥ . • For the transformation σ ( A n − , n − , it is enough show that σ ( e ) = e , note that σ ( e ) =[ σ ( e ) , σ ( e )] = e + A n − [ e n − , e ] , since n is even then [ e n − , e ] = 0 , and hence σ ( e ) = e It is easy to see that σ ( A n , n ) , does not change the structure constants. On low-dimensional filiform Leibniz algebras and their invariants
Analogously, we check that φ ( B k , k ) does not change the structure constants, when n − ≤ k ≤ n. Consider the transformation φ ( B k , k ) = φ ( e ) = e ,φ ( e ) = e + B k e k , n − ≤ k ≤ n,φ ( e i +1 ) = [ φ ( e i ) , φ ( e )] , ≤ i ≤ n − , • If k = n − , then φ ( e ) = e + B n − e n − , φ ( e ) = e + B n − e n , φ ( e i ) = e i , where i ≥ . A simple computation shows that φ ( B n − , n −
2) does not change the structure constants.Note that [ φ ( e ) , φ ( e )] = [ e + B n − e n − , e + B n − e n − ]= [ e , e ] + B n − [ e , e n − ] + B n − [ e n − , e ] + B n − [ e n − , e n − ] = [ e , e ] , (here [ e , e n − ] = [ e n − , e ] = 0 , since n is even). • If k = n − , then we get φ ( e ) = e + B n − e n , φ ( e i ) = e i , where i ≥ . Consider the bracket [ φ ( e ) , φ ( e )] = − φ ( e ) + b ′ , e ′ n , and then [ e , e + B n − e n − ] = − e − B n − e n + b ′ , e n , implies that − e + b , e n − B n − e n = − e − B n − e n + b ′ , e n , therefore b ′ , = b , . The chain of equalities [ φ ( e ) , φ ( e )] = b ′ , e ′ n , [ e + B n − e n − , e + B n − e n − ] = b ′ , e n , [ e , e ] + B n − [ e , e n − ] + B n − [ e n − , e ] = b ′ , e n , show that b ′ , = b , . One easily can see that[ φ ( e ) , φ ( e )] = [ e + B n − e n − , e + B n − e n ] = [ e , e ] + B n − [ e , e n ] + B n − [ e n − , e ] = [ e , e ] . • If k = n, it is obvious.The following lemma from [10] keeps track the behavior some of the structure constants under theadapted base change. Lemma 1.1.
Let { e , e , ..., e n } −→ { e ′ , e ′ , ..., e ′ n } be an adapted base change, b , , b , , b , , ... and b ′ , , b ′ , , b ′ , , ... be the respective structure constants. Then for b ′ , , b ′ , and b ′ , one has b ′ , = A b , + A A b , + A b , A n − B ( A + A b ) , b ′ , = A b , + 2 A b , A n − ( A + A b ) , b ′ , = B b , A n − ( A + A b ) . The next sections deal with the classification problem of
T Leib n in dimensions 7 and 8. Here, toclassify algebras from T Leib and T Leib we represent them as a disjoint union of their subsets. Some ofthese subsets turn out to be single orbits, and the others contain infinitely many orbits. In the last case,we give invariant functions to discern the orbits. samiddin S. Rakhimov, Munther A. Hassan T Leib n , n = 7 , . T Leib Any algebra L from T Leib can be represented as one dimensional central extension ( C ( L ) = < e > ) of6-dimensional filiform Lie algebra with adapted basis { e , e , ..., e } (see Theorem 1.1) and on the adaptedbasis { e , e , ..., e } the class T Leib can be represented as follows: T Leib = [ e i , e ] = e i +1 , ≤ i ≤ , [ e , e i ] = − e i +1 , ≤ i ≤ , [ e , e ] = b , e , [ e , e ] = − e + b , e , [ e , e ] = b , e , [ e , e ] = − [ e , e ] = a , e + a , e + b , e , [ e , e ] = − [ e , e ] = a , e + b , e , [ e , e ] = − [ e , e ] = − a , e + b , e , [ e , e ] = − [ e , e ] = a , e + b , e , [ e , e ] = − [ e , e ] = b , e , [ e , e ] = − [ e , e ] = b , e . The next lemma specifies the set of structure constants of algebras from
T Leib . Lemma 2.1.
The structure constants of algebras from
T Leib satisfy the following constraints: . b , = a , , . b , = a , − b , , and . b , = b , = a , = 0 . Proof.
The relations easily can be found by applying the Leibniz identity to the triples of the basis vectors { e , e , e } , { e , e , e } { e , e , e } , { e , e , e } , and { e , e , e } . Further unifying the above table of multiplication we rewrite it via new parameters c , , c , , c , , c , , c , , c , , c , as follows: T Leib = [ e i , e ] = e i +1 , ≤ i ≤ , [ e , e i ] = − e i +1 , ≤ i ≤ , [ e , e ] = c , e , [ e , e ] = − e + c , e , [ e , e ] = c , e , [ e , e ] = − [ e , e ] = c , e + c , e + c , e , [ e , e ] = − [ e , e ] = c , e + c , e , [ e , e ] = − [ e , e ] = ( c , − c , ) e , [ e , e ] = − [ e , e ] = c , e . An algebra from
T Leib with parameters c , , c , , c , , c , , c , , c , , c , is denoted by L ( C ) , where C = ( c , , c , , c , , c , , c , , c , , c , ) . The next theorem represents the action of the adapted base change to the parameters c , , c , , c , , c , , c , , c , , c , of an algebra from T Leib . Theorem 2.1. ( Isomorphism criterion for
T Leib ) Two filiform Leibniz algebras L ( C ) and L ( C ′ ) , where C = ( c , , c , , c , , c , , c , , c , , c , ) and C ′ = ( c ′ , , c ′ , , c ′ , , c ′ , , c ′ , , c ′ , , c ′ , ) , from T Leib are isomorphic if and only if there exist A , A , B , B , B ∈ C : such that A B = 0 and the following equalities hold: On low-dimensional filiform Leibniz algebras and their invariants c ′ , = A c , + A A c , + A c , A B , (1) c ′ , = 2 A c , + A c , A , (2) c ′ , = B c , A , (3) c ′ , = B c , A , (4) c ′ , = B (cid:0) A c , + 2 A c , (cid:1) A , (5) c ′ , = A B c , + A ( B − B B ) c , + A B (5 c , − c , ) ( A c , − B c , ) A B , (6) c ′ , = B c , A . (7) Proof. “If” part. The equations (1)–(3) occur due to Lemma 1.1 (remind that in this case n is eventherefore b = 0).Notice that according to the Proposition 1.2 and 1.3 the adapted transformation in T Leib can betaken in the form e ′ = f ( e ) = A e + A e ,e ′ = f ( e ) = B e + B e + B e ,e ′ i +1 = f ( e i +1 ) = [ f ( e i ) , f ( e )] , ≤ i ≤ n − , where A B = 0 or more precisely as follows: e ′ = A e + A e ,e ′ = B e + B e + B e ,e ′ = A B e + A B e + ( A B − A B c , ) e − A ( B c , + B c , ) e + A ( B c , − B c , − B c , ) e ,e ′ = A B e + (cid:0) A B − A A B c , (cid:1) e + ( A B − A A B c , − A A B c , ) e +( − A (cid:0) − A B c , + A B c , c , + A B c , + 2 A B c , + 2 A B c , − A B c , (cid:1) ) e , (8) e ′ = A B e + ( A B − A A B c , ) e +( A B − A A B c , + A A B b , − A A B c , c , − A A B c , + A A B c , ) e ,e ′ = A B e + ( A B − A A B c , + A A B b , ) e ,e ′ = A B e . Consider[ e ′ , e ′ ] = A B ( c , e + c , e + c , e ) + A B B ( c , e + c , e ) + B ( A B c , − A B c , + A B ) ( c , − c , ) e + A B c , e − A B B c , e = c ′ , e ′ + c ′ , e ′ + c ′ , e ′ . Now bearing in mind (8) and equating the coefficients of e , e and e we get the equalities (4),(5) and (6), respectively. The last equality follows from[ f ( e ) , f ( e )] = c ′ , f ( e ) = ⇒ A B c , e = c ′ , A B e = ⇒ c ′ , = B c , A . samiddin S. Rakhimov, Munther A. Hassan L ( C ) to L ( C ′ ) . Indeed,[ e ′ , e ′ ] = [ A e + A e , A e + A e ]= A [ e , e ] + A A [ e , e ] + A A [ e , e ] + A [ e , e ]= (cid:0) A c , + A A c , + A c , (cid:1) e = c ′ , A B e = c ′ , e ′ . [ e ′ , e ′ ] = [ A e + A e , B e + B e + B e ]= A B ( − e + c , e ) − A B e − A B e + A B c , e + A B ( c , e + c , e + c , e ) + A B ( c , e + c , e )= − ( A B e + A B e + ( A B − A B c , ) e − A ( B c , + B c , ) e + A ( B c , − B c , − B c , ) e ) + B ( A c , + 2 A c , ) e = − e ′ + A B c ′ , e = − e ′ + c ′ , e ′ . [ e ′ , e ′ ] = [ B e + B e + B e , B e + B e + B e ] = [ B e , B e ] = B c , e = A B c ′ , e = c ′ , e ′ . The brackets [ e ′ , e ′ ] , [ e ′ , e ′ ] , [ e ′ , e ′ ] and [ e ′ , e ′ ] can be gotten similarly.The next section deals with the classification problem of T Leib . T Leib In this subsection we give a list of all algebras from
T Leib . Represent
T Leib as a union of the following subsets: U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = c , = c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = c , = c , = 0 , c , c , − c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = c , = c , = 4 c , c , − c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , = c , = 0 , c , c , − c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , = c , = 4 c , c , − c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = 0 , c , = 0 , c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; On low-dimensional filiform Leibniz algebras and their invariants U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = c , = c , = 0 } . Proposition 2.1.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , ! (4 c ′ , c ′ , − c ′ , ) = (cid:18) c , c , (cid:19) (4 c , c , − c , ) ,c ′ , c ′ , = c , c , and (cid:16) c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , (cid:1) c , c , .
2. For any λ , λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:18) c , c , (cid:19) (4 c , c , − c , ) = λ , c , c , = λ , (cid:0) c , c , − c , c , (cid:1) c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , λ , λ , , , λ , λ , λ ∈ C . Proof.
1. “If” part due to Theorem 2.1 if one substitutes the expressions for c ′ , , c ′ , , c ′ , , c ′ , , c ′ , ,c ′ , , c ′ , : c ′ , c ′ , ! (4 c ′ , c ′ , − c ′ , ) = (cid:18) c , c , (cid:19) (4 c , c , − c , ) ,c ′ , c ′ , = c , c , , (cid:16) c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , (cid:1) c , c , . “ Only if ” part. Let the equalities c ′ , c ′ , ! (4 c ′ , c ′ , − c ′ , ) = (cid:18) c , c , (cid:19) (4 c , c , − c , ) ,c ′ , c ′ , = c , c , , (cid:16) c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , (cid:1) c , c , hold.Consider the base change (8) in Theorem 2.1 with A = − A c , c , , B = A c , , and B = 18 A c , c , (4 c , (cid:0) A c , + 4 B c , (cid:1) + A (5 c , − c , ) (cid:0) c , c , − c , c , c , (cid:1) ) . This changing leads L ( C ) into L (cid:18) c , c , (cid:19) (4 c , c , − c , ) , , , c , c , , (cid:0) c , c , − c , c , (cid:1) c , c , , , ! . samiddin S. Rakhimov, Munther A. Hassan L ( C ′ ) into L c ′ , c ′ , ! (4 c ′ , c ′ , − c ′ , ) , , , c ′ , c ′ , , (cid:16) c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , , , . Since c ′ , c ′ , ! (4 c ′ , c ′ , − c ′ , ) = (cid:18) c , c , (cid:19) (4 c , c , − c , ) , c ′ , c ′ , = c , c , and (cid:16) c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , (cid:1) c , c , , the algebras are isomorphic.2. Obvious. Proposition 2.2.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , = c , c , , (cid:16) c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , (cid:1) c , c , .
2. For any λ , λ ∈ C , there exists L ( C ) ∈ U : c , c , = λ , (cid:0) c , c , − c , c , (cid:1) c , c , = λ . The orbits in U can be parameterized as L (0 , , , λ , λ , , , λ , λ ∈ C .Proof. The proof is similar that of Proposition 2.1, where we put A = − A c , c , , B = A c , , and B = ( c , (cid:0) A c , + B c , (cid:1) + A ( c , (cid:0) c , c , − c , c , (cid:1) (5 c , − c , )))2 A c , c , . Since the proving of the next coming Propositions 2.3 – 2.13 are similar those of Propositions 2.1and 2.2 we decided to omit the details of them. “If” parts of them follow from Theorem 1.2, for “Onlyif” part we just give the respective values of the coefficients A , A , B , B and B in the base change(8). Note that if no value is given then it is considered as an arbitrary. Proposition 2.3.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , = c , c , .
2. For any λ ∈ C ∗ , there exists L ( C ) from U : c , c , = λ. On low-dimensional filiform Leibniz algebras and their invariants
Then orbits in U can be parameterized as L (1 , , , λ, , , , λ ∈ C ∗ .Proof. The respective values of coefficients are: A = − A c , c , , B = A c , and B = A (cid:0) c , c , − c , c , + c , c , (cid:1) + 4 B c , c , A c , c , . Proposition 2.4.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , = c , c , .
2. For any λ ∈ C ∗ , there exists L ( C ) from U : c , c , = λ. Then orbits in U can be parameterized as L (0 , , , λ, , , , λ ∈ C ∗ .Proof. Here A = − A c , c , , B = A c , and B = A (cid:0) c , c , − c , c , + c , c , (cid:1) + 4 B c , c , A c , c , . Proposition 2.5.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , c ′ , = c , c , c , .
2. For any λ ∈ C , there exists L ( C ) ∈ U : c , c , c , = λ. Then orbits in U can be parameterized as L ( λ, , , , , , , λ ∈ C .Proof. For this case: A = c , c , , B = c , c , and B = c , c , + B c , − A c , c , c , c , . Proposition 2.6.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , c , , (cid:16) c ′ , c ′ , − c , c , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , (cid:1) c , c , and (cid:16) c ′ , c ′ , − c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , (cid:1) c , c , . samiddin S. Rakhimov, Munther A. Hassan
2. For any λ , λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , c , = λ , (cid:0) c , c , − c , c , (cid:1) c , c , = λ , (cid:0) c , c , − c , (cid:1) c , c , = λ . Then orbits in U can be parameterized as L ( λ , λ , , , , λ , , λ , λ , λ ∈ C . Proof.
We put here, A = − A c , c , , B = A c , . Proposition 2.7.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , b , − c , c , (cid:1) c , c , and (cid:16) c ′ , c ′ , − c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , (cid:1) c , c , .
2. For any λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , b , − c , c , (cid:1) c , c , = λ and (cid:0) c , c , − c , (cid:1) c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , , , λ , , λ , λ ∈ C . Proof.
For this case we put A = − A c , c , , and B = A c , . Proposition 2.8.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , (cid:1) c , c , .
2. For any λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , (cid:1) c , c , = λ. Then orbits in U can be parameterized as L (1 , , , , , λ, , λ ∈ C .Proof. Here, A = − A c , c , and B = A c , . Proposition 2.9.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , ! (cid:16) c ′ , c ′ , − c ′ , (cid:17) = (cid:18) c , c , (cid:19) (cid:0) c , c , − c , (cid:1) , c ′ , c ′ , c ′ , = c , c , c , . On low-dimensional filiform Leibniz algebras and their invariants
2. For any λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:18) c , c , (cid:19) (cid:0) c , c , − c , (cid:1) = λ , c , c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , , λ , , , λ , λ ∈ C .Proof. We put A = c , c , , A = − c , c , , and B = c , c , . Proposition 2.10.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:18) c ′ , c ′ , (cid:19) (cid:16) c ′ , c ′ , − c ′ , (cid:17) = (cid:18) c , c , (cid:19) (cid:0) c , c , − c , (cid:1) .
2. For any λ ∈ C , there exists L ( C ) ∈ U : (cid:18) c , c , (cid:19) (cid:0) c , c , − c , (cid:1) = λ. Then orbits in U can be parameterized as L ( λ, , , , , , , λ ∈ C .Proof. Here, we take A = − A c , c , , and B = A c , . Proposition 2.11.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , ! c ′ , = (cid:18) c , c , (cid:19) c , .
2. For any λ ∈ C , there exists L ( C ) ∈ U : (cid:18) c , c , (cid:19) c , = λ. Then orbits in U can be parameterized as L (0 , , , , λ, , , λ ∈ C . Proof.
Take A = − A c , c , , and B = A c , . Proposition 2.12.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , c ′ , = c , c , c , .
2. For any λ ∈ C , there exists L ( C ) ∈ U : c , c , c , = λ .Then orbits in U can be parameterized as L (1 , , , , λ, , , λ ∈ C .Proof. Here, B = c , A . samiddin S. Rakhimov, Munther A. Hassan Proposition 2.13.
The subsets U , U , U , U , U , U , U , U , U , U , U , U , U and U are single orbits with representatives L (1 , , , , , , , , L (0 , , , , , , , ,L (0 , , , , , , , L (0 , , , , , , , L (1 , , , , , , , L (0 , , , , , , , L (0 , , , , , , ,L (0 , , , , , , , L (1 , , , , , , , L (1 , , , , , , , L (0 , , , , , , , L (0 , , , , , , ,L (0 , , , , , , and L (0 , , , , , , , respectively.Proof. To prove it, we give the respective values of A , A , B , B and B in the base change (8) leadingto the appropriate representatives.For U and U : B = A c , and B = A c , + B c , A c , . For U and U : A = − A c , c , and B = A c , . For U and U : A = − A c , c , and B = A c , . For U : A = − A c , c , and B = A c , . For U : A = − A c , c , . For U and U : B = c , A . For U and U : B = A c , . Note that the orbits U and U can be included in the parametric family of orbits U and U , respectively at λ = 0 . T Leib Here, we have n = 7 , means odd case, and any algebra L from T Leib can be represented as one di-mensional central extension ( C ( L ) = < e > ) of 7-dimensional filiform Lie algebra with adapted basis { e , e , ..., e } (see Theorem 1.1) and on the adapted basis { e , e , ..., e } the class T Leib can be repre-sented as follows:4 On low-dimensional filiform Leibniz algebras and their invariants
T Leib = [ e i , e ] = e i +1 , ≤ i ≤ , [ e , e i ] = − e i +1 , ≤ i ≤ , [ e , e ] = b , e , [ e , e ] = − e + b , e , [ e , e ] = b , e , [ e , e ] = − [ e , e ] = a , e + a , e + a , e + b , e , [ e , e ] = − [ e , e ] = a , e + a , e + b , e , [ e , e ] = − [ e , e ] = ( a , − a , ) e + b , e , [ e , e ] = − [ e , e ] = b , e , [ e , e ] = − [ e , e ] = a , e + b , e , [ e , e ] = − [ e , e ] = b , e , [ e , e ] = − [ e , e ] = b , e , [ e i , e − i ] = − [ e − i , e i ] = ( − i b , e , ≤ i ≤ n − . The next lemma specifies the set of structure constants of algebras from
T Leib . Lemma 2.2.
The structure constants of algebras from
T Leib satisfy the following constraints: b , = a , , b , = a , − b , , b , = a , , b , = a , − a , , and b , ( a , + 2 a , ) = 0 . Proof.
These relations come out from sequentially applications of the Leibniz identity to the triples ofthe basis vectors { e , e , e } , { e , e , e } , { e , e , e } , { e , e , e } and { e , e , e } respectively.Further unifying the T Leib table of multiplication we rewrite it via parameters c , , c , , c , , c , , c , , c , , c , , c , , c , as follows: T Leib = [ e i , e ] = e i +1 , ≤ i ≤ , [ e , e i ] = − e i +1 , ≤ i ≤ , [ e , e ] = c , e , [ e , e ] = − e + c , e , [ e , e ] = c , e , [ e , e ] = − [ e , e ] = c , e + c , e + c , e + c , e , [ e , e ] = − [ e , e ] = c , e + c , e + c , e , [ e , e ] = − [ e , e ] = ( c , − c , ) e + ( c , − c , ) e , [ e , e ] = − [ e , e ] = ( c , − c , ) e , [ e , e ] = − [ e , e ] = c , e + c , e , [ e , e ] = − [ e , e ] = c , e , [ e i , e − i ] = − [ e − i , e i ] = ( − i c , e , ≤ i ≤ n − . So an algebra from
T Leib with parameters c , , c , , c , , c , , c , , c , , c , , c , , c , , c , is denoted by L ( C ) , where C = ( c , , c , , c , , c , , c , , c , , c , , c , , c , , c , ) . Note that L ( C ) ∈ T Leib if and only if c , ( c , + 2 c , ) = 0 . Notice that according to the Proposition 1.2 and 1.3 the adapted transformation in
T Leib can betaken in the form f ( e ) = A e + A e + A e + A e + A e ,f ( e ) = B e + B e + B e + B e + B e ,f ( e i +1 ) = [ f ( e i ) , f ( e )] , ≤ i ≤ n − , where A B ( A + A c , ) = 0 . The next theorem represents the action of the adapted base change to the parameters c , , c , , c , , c , , c , , c , , c , , c , , c , , c , of an algebra from T Leib . samiddin S. Rakhimov, Munther A. Hassan Theorem 2.2. (Isomorphism criterion for
T Leib ) Two filiform Leibniz algebras L ( C ) and L ( C ′ ) , where C = ( c , , c , , c , , c , , c , , c , , c , , c , , c , , c , ) and C ′ = ( c ′ , , c ′ , , c ′ , , c ′ , , c ′ , , c ′ , , c ′ , , c ′ , , c ′ , , c ′ , ) , from T Leib are isomorphic, if and only if there exist A , A , A , A , A , B , B , B , B , B ∈ C , such that A B ( A + A c , ) = 0 and the following equalities hold: c ′ , = A c , + A A c , + A c , c A ( A + A c , ) ,c ′ , = A c , + 2 A c , A ( A + A c , ) ,c ′ , = B c , A ( A + A c , ) ,c ′ , = B c , A ,c ′ , = B (cid:0) A c , + 2 A c , (cid:1) A ,c ′ , = A B c , + A ( B − B B ) c , + A B (5 c , − c , ) ( A c , − B c , ) A B ,c ′ , = − A B ( A + c , A ) ( A A B c , c , − A A B B c , c , − A A B B c , c , + A A B B c , c , − A A B c , c , − A A B c , c , − A B B B c , − A A B c , c , − A A B c , c , c , + 4 A A B B c , c , c , +12 A A B c , c , c , − A A B c , c , − A A B c , c , + 2 A A B c , c , +6 A A B B c , c , + 2 A A B B c , c , + 2 A A A B B c , c , − A A B B c , c , − A A B B c , c , + 2 A B c , c , c , − A A B B c , c , c , + 3 A A B c , c , + 2 A A B c , c , +4 A A B B B c , c , + 9 A A B B c , c , + 3 A A B c , c , − A A B c , c , + A A B c , c , + A B B c , − A A B c , − A A B c , − A B c , c , − A A B B c , c , − A A B B c , c , +3 A A B B c , − A A B B c , c , − A A B B c , − A B B B c , − A B c , + A B c , − A B B c , + 3 A B B c , + 2 A B B c , +2 A B B c , − A A B c , c , − A A B c , c , c , +2 A A B c , c , c , c , + 9 A A B c , c , ) ,c ′ , = B c , A ,c ′ , = 1 A B ( A + c , A ) ( A B c , + (cid:0) − A B + 2 A B B − A A B c , − A B c , (cid:1) c , +3 A A B c , c , − A A B c , ) ,c ′ , = B c , A + c , A . On low-dimensional filiform Leibniz algebras and their invariants
The following base change is adapted and it transforms L ( C ) to L ( C ′ ) e ′ = A e + A e + A e + A e + A e ,e ′ = B e + B e + B e + B e + B e ,e ′ = A B e + A B e + ( A B + ( A B − A B ) c , ) e + ( A B + ( A B − A B ) c , +( A B − A B ) c , ) e + (( A B − A B − A B + A B ) c , + ( A B − A B ) c , +( A B − A B ) c , + ( − A B + A B ) c , + A B ) e + ( ∗ ) e ,e ′ = A B e + (cid:0) A B − A A B c , (cid:1) e + ( A B − A A B c , + A ( − A B + A B ) c , ) e +( (cid:0) − A A B + A B (cid:1) c , + (cid:0)(cid:0) A A B − A B (cid:1) c , + A A B − A A B (cid:1) c , + (9)( − A A B + A A B ) c , + ( A A B − A A B + A A B ) c , + A A B c , + A B ) e + ( ∗ ) e ,e ′ = A B e + ( A B − A A B c , ) e + A ( A B − A A B c , − A A B c , − A A B c , + A A B c , + A A B c , + A B c , − A B c , c , ) e + ( ∗ ) e ,e ′ = A B e + ( A B − A A B c , + A A B c , ) e + ( ∗ ) e ,e ′ = A B e + ( ∗ ) e ,e ′ = B A ( A + A c , ) e . The next section deals with the applications of the results of the previous section to the classificationproblem of
T Leib . T Leib In this section we give a list of all algebras from
T Leib ;Represent T Leib as a union of the following subsets: M = { L ( C ) ∈ T Leib : c , = 0 } ,U = { L ( C ) ∈ T Leib : c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = 0 , c , = 0 } ; M = { L ( C ) ∈ T Leib : c , = 0 } U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = 0 , c , = 0 , c , c , − c , c , c , + 3 c , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = 0 , c , = 0 , c , c , − c , c , c , + 3 c , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = 0 , c , = 0 , c , c , − c , c , c , + 3 c , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = c , = 0 , c , = 0 , c , = 0 } ; samiddin S. Rakhimov, Munther A. Hassan U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = c , = c , = c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = c , = c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = 0 , c , = 0 , c , = c , = c , = c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , c , − c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , c , − c , = 0 , − c , + 4 c , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , c , − c , = 0 , − c , + 4 c , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , c , − c , = 0 , − c , + 4 c , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , c , − c , = 0 , − c , + 4 c , c , = 0 , c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = 0 , c , = 0 , c , c , − c , = 0 , − c , + 4 c , c , = 0 , c , = c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = 0 , c , = 0 , c , = c , = c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = 0 , c , = 0 , c , = c , = 0 , c , c , − c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = 0 , c , = 0 , c , = c , = 0 , c , c , − c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = 0 , c , = 0 , c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = c , = 0 , c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = c , = 0 , c , = 0 , c , = c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = c , = c , = 0 , c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = c , = c , = c , = 0 , c , = 0 } ; U = { L ( C ) ∈ T Leib : c , = c , = c , = c , = c , = c , = c , = c , = c , = 0 } . Proposition 2.14.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , (cid:17) c ′ , (cid:0) − c ′ , + c ′ , c ′ , (cid:1) = (cid:0) c , c , − c , (cid:1) c , ( − c , + c , c , ) , (cid:0) − c ′ , + c ′ , c ′ , (cid:1) c ′ , c ′ , c ′ , = ( − c , + c , c , ) c , c , c , , (cid:0) − c ′ , + c ′ , c ′ , (cid:1) (cid:16) − c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , c ′ , = ( − c , + c , c , ) (cid:0) − c , c , + c , c , (cid:1) c , c , , (cid:0) − c ′ , + c ′ , c ′ , (cid:1) c ′ , c ′ , (27 c ′ , c ′ , c ′ , − c ′ , c ′ , c ′ , c ′ , c ′ , + 4 c ′ , c ′ , c ′ , − c ′ , c ′ , c ′ , − c ′ , c ′ , c ′ , ) = ( − c , + c , c , ) c , c , (27 c , c , c , − c , c , c , c , c , + 4 c , c , c , − c , c , c , − c , c , c , ) . On low-dimensional filiform Leibniz algebras and their invariants
2. For any λ , λ , λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , (cid:1) c , ( − c , + c , c , ) = λ , (cid:0) − c ′ , + c ′ , c ′ , (cid:1) c ′ , c ′ , c ′ , = λ , ( − c , + c , c , ) (cid:0) − c , c , + c , c , (cid:1) c , c , = λ , ( − c , + c , c , ) c , c , (27 c , c , c , − c , c , c , c , c , + 4 c , c , c , − c , c , c , − c , c , c , ) = λ . Then orbits in U can be parameterized as L ( λ , , , λ , λ , λ , , − λ , , , λ , λ , λ , λ ∈ C . Proof.
See proof of Proposition 2.1, where we put A = − A c , c , , B = − A ( − c , + c , c , )2 c , c , , and B = − A c , c , ( − c , + c , c , ) ( − A c , c , +8 A c , c , c , c , − A c , c , c , c , − A c , c , c , + 92 A c , c , c , c , − A c , c , c , c , − A c , c , c , c , +4 A c , c , c , c , − A c , c , c , c , + 8 B c , c , + 5 A c , c , c , ) , in base change (9). Proposition 2.15.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:0) − c ′ , + c ′ , c ′ , (cid:1) c ′ , c ′ , c ′ , = ( − c , + c , c , ) c , c , c , , (cid:0) − c ′ , + c ′ , c ′ , (cid:1) (cid:16) − c ′ , c ′ , + 2 c ′ , c ′ , (cid:17) c ′ , c ′ , = ( − c , + c , c , ) (cid:0) − c , c , + 2 c , c , (cid:1) c , c , , (cid:0) − c ′ , + c ′ , c ′ , (cid:1) c ′ , c ′ , (27 c ′ , c ′ , c ′ , − c ′ , c ′ , c ′ , c ′ , c ′ , + c ′ , c ′ , c ′ , − c ′ , c ′ , c ′ , − c ′ , c ′ , c ′ , ) = ( − c , + c , c , ) c , c , (27 c , c , c , − c , c , c , c , c , + c , c , c , − c , c , c , − c , c , c , ) .
2. For any λ , λ , λ ∈ C , there exists L ( C ) ∈ U : ( − c , + c , c , ) c , c , c , = λ , ( − c , + c , c , ) (cid:0) − c , c , + 2 c , c , (cid:1) c , c , = λ , ( − c , + c , c , ) c , c , (27 c , c , c , − c , c , c , c , c , + c , c , c , − c , c , c , − c , c , c , ) = λ . Then orbits in U can be parameterized as L (0 , , , λ , λ , λ , , − λ , , , λ , λ , λ ∈ C . Proof.
The respective value of coefficients in the base change (9) are: A = − A c , c , , B = − A ( − c , + c , c , ) c , c , , and B = − A c , c , ( − c , + c , c , ) ( − A c , c , + 2 A c , c , c , c , − A c , c , c , c , − A c , c , c , + 46 A c , c , c , c , − A c , c , c , c , − A c , c , c , c , +2 A c , c , c , c , − A c , c , c , c , + B c , c , + 10 A c , c , c , ) . samiddin S. Rakhimov, Munther A. Hassan Proposition 2.16.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , c ′ , (cid:0) − c ′ , + c ′ , c ′ , (cid:1) = c , c , c , (cid:0) − c , + c , c , (cid:1) ,c ′ , (cid:0) − b , + c ′ , c ′ , (cid:1) (cid:16) c ′ , c ′ , + 4 c ′ , c ′ , c ′ , − c ′ , c ′ , − c ′ , c ′ , (cid:17) = c , (cid:0) − c , + c , c , (cid:1) (cid:0) c , c , + 4 c , c , c , − c , c , − c , c , (cid:1) .
2. For any λ , λ ∈ C , there exists L ( C ) ∈ U : c , c , c , (cid:0) − c , + c , c , (cid:1) = λ ,c , (cid:0) − c , + c , c , (cid:1) (cid:0) c , c , + 4 c , c , c , − c , c , − c , c , (cid:1) = λ . Then orbits in U can be parameterized as L ( λ , , , , , λ , , − , , , λ , λ ∈ C . Proof.
Here, A = 2 c , − c , c , c , c , , A = − (cid:0) c , − c , c , (cid:1) c , c , c , , B = (cid:0) c , − c , c , (cid:1) c , c , , and B = 18 c , c , (2 c , − c , c , ) ( − c , c , + 32 c , c , c , c , − c , c , c , c , +8 c , c , c , c , − c , c , c , c , − c , c , + 320 c , c , c , − c , c , c , +120 c , c , c , − c , c , c , + 2 c , c , + 16 B c , c , ) . Proposition 2.17.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , c ′ , c ′ , = c , c , c , c , .
2. For any λ ∈ C , there exists L ( C ) ∈ U : c , c , c , c , = λ. Then orbits in U can be parameterized as L ( λ, , , , , , , , , , λ ∈ C . Proof.
For this case, we put in the base change (9) the following coefficients, A = c , c , , B = c , + A c , c , c , c , , and B = 12 c , c , c , ( c , + A c , c , ) ( − c , c , − A c , c , c , c , − A c , c , c , c , +18 A c , c , c , +46 A c , c , c , c , +38 A c , c , c , c , + A c , c , c , +2 A c , c , c , + A c , c , c , + B c , c , c , + 10 A c , c , c , ) . Proposition 2.18. On low-dimensional filiform Leibniz algebras and their invariants
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , (cid:1) c , c , ,c ′ , c ′ , = c , c , , (cid:16) − c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) − c , c , + c , c , (cid:1) c , c , , (cid:16) c ′ , c ′ , − c ′ , c ′ , c ′ , + 3 c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , c , + 3 c , c , (cid:1) c , c , .
2. For any λ , λ , λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , (cid:1) c , c , = λ , c , c , = λ , (cid:0) − c , c , + c , c , (cid:1) c , c , = λ , (cid:0) c , c , − c , c , c , + 3 c , c , (cid:1) c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , λ , λ , , , , λ , , λ , λ , λ , λ ∈ C . Proof.
For this case we put, A = − A c , c , , B = A c , , B = 18 A c , c , (4 B c , c , +4 A c , c , + 2 A c , c , c , c , − A c , c , c , c , + 5 A c , c , − A c , c , c , ) and B = 148 A c , c , (8 B c , c , − A c , c , c , c , + 16 A c , c , c , − A c , c , c , − A c , c , c , c , + 17 A c , c , c , + 12 B c , A c , c , c , − A A c , c , c , +48 B A A c , c , c , − A c , c , c , − A c , c , c , c , c , − A c , c , c , c , c , +30 A c , c , c , c , c , +40 A c , c , c , c , c , +30 B A c , c , c , c , +24 B A c , c , c , +12 B A c , c , c , c , − B A c , c , c , c , c , − B A c , c , c , c , ) . Proposition 2.19.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , = c , c , , (cid:16) − c ′ , c ′ , + 2 c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) − c , c , + 2 c , c , (cid:1) c , c , , (cid:16) c ′ , c ′ , − c ′ , c ′ , c ′ , + 3 c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , c , + 3 c , c , (cid:1) c , c , .
2. For any λ , λ , λ ∈ C , there exists L ( C ) ∈ U : c , c , = λ , (cid:0) − c , c , + 2 c , c , (cid:1) c , c , = λ , (cid:0) c , c , − c , c , c , + 3 c , c , (cid:1) c , c , = λ . Then orbits in U can be parameterized as L (0 , , , λ , λ , , , , λ , , λ , λ , λ ∈ C . samiddin S. Rakhimov, Munther A. Hassan Proof.
Here, A = − A c , c , , B = A c , , B = 12 A c , c , ( B c , c , + A c , c , + A c , c , c , c , − A c , c , c , c , + 5 A c , c , − A c , c , c , ) and B = 16 A c , c , ( B c , c , − A c , c , c , c , + 2 A c , c , c , − A c , c , c , − A c , c , c , c , + 17 A c , c , c , + 3 B c , A c , c , c , − A A c , c , c , +6 B A A c , c , c , − A c , c , c , − A c , c , c , c , c , − A c , c , c , c , c , +15 A c , c , c , c , c , +10 A c , c , c , c , c , +15 B A c , c , c , c , +3 B A c , c , c , +3 B A c , c , c , c , − B A c , c , c , c , c , − B A c , c , c , c , ) . Proposition 2.20.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , c ′ , (cid:0) − c ′ , c ′ , c ′ , + 3 c ′ , c ′ , + 2 c ′ , c ′ , (cid:1) = c , c , c , (cid:0) − c , c , c , + 3 c , c , + 2 c , c , (cid:1) , c ′ , c ′ , = c , c , .
2. For any λ ∈ C and λ ∈ C ∗ , there exists L ( C ) ∈ U : c , c , c , (cid:0) − c , c , c , + 3 c , c , + 2 c , c , (cid:1) = λ , c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , λ , , , , , , , λ ∈ C and λ ∈ C ∗ . Proof.
Here, we take A = − c , c , c , + 3 c , c , + 2 c , c , c , c , , A = − A c , c , , B = A c , ,B = 1 A c , c , (4 B c , c , + 4 A c , c , + A c , c , − A c , c , ) , and B = − A c , c , ( − B c , c , − A c , c , c , − A c , c , c , + 16 A c , c , c , +13 A c , c , c , − B c , A c , c , +48 A A c , c , − B A A c , c , − A c , c , +12 A c , c , c , c , + 30 B A c , c , c , − B A c , c , c , − B A c , c , c , ) . Proposition 2.21.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , = c , c , .
2. For any λ ∈ C ∗ , there exists L ( C ) ∈ U : c , c , = λ. Then orbits in U can be parameterized as L (1 , , , λ, , , , , , , λ ∈ C ∗ . Proof.
Here, A = − A c , c , , B = A c , , B = 4 B c , c , + 4 A c , c , + A c , c , − A c , c , A c , c , , and B = − A c , c , ( − B c , c , − A c , c , c , − A c , c , c , +16 A c , c , c , +13 A c , c , c , − B c , A c , c , +48 A A c , c , − B A A c , c , − A c , c , +12 A c , c , c , c , + 30 B A c , c , c , − B A c , c , c , − B A c , c , c , ) . Proposition 2.22.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , = c , c , .
2. For any λ ∈ C ∗ , there exists L ( C ) ∈ U : c , c , = λ. Then orbits in U can be parameterized as L (0 , , , λ, , , , , , , λ ∈ C ∗ . On low-dimensional filiform Leibniz algebras and their invariants
Proof.
Put, A = − A c , c , , B = A c , , B = 4 B c , c , + 4 A c , c , + A c , c , − A c , c , A c , c , , and B = − A c , c , ( − B c , c , − A c , c , c , − A c , c , c , +16 A c , c , c , +13 A c , c , c , − B c , A c , c , +48 A A c , c , − B A A c , c , − A c , c , +12 A c , c , c , c , + 30 B A c , c , c , − B A c , c , c , − B A c , c , c , ) . Proposition 2.23.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , c ′ , = c , c , c , .
2. For any λ ∈ C , there exists L ( C ) ∈ U : c , c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , , , , , , , , λ ∈ C . Proof.
For this case we put A = c , c , , A = c , c , c , , B = c , c , , B = 3 B c , + 3 c , c , c , − c , c , c , c , , and B = 16 c , c , ( B c , + 2 c , c , c , − c , c , c , c , + 2 c , c , + 3 B c , c , c , − B c , c , c , ) . Proposition 2.24.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , c , , (cid:16) − c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) − c , c , + c , c , (cid:1) c , c , , c ′ , c ′ , c ′ , = c , c , c , , c ′ , c ′ , − c ′ , c ′ , = 4 c , c , − c , c , .
2. For any λ , λ , λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , c , = λ , (cid:0) − c , c , + c , c , (cid:1) c , c , = λ , c , c , c , = λ , c , c , − c , c , = λ . Then orbits in U can be parameterized as L ( λ , λ , λ , , , λ , , , , , λ , λ , λ , λ ∈ C . Proof.
Put A = c , c , , A = − c , c , c , , B = c , c , , and B = 18 c , c , (8 c , c , + 4 B c , − c , c , c , c , + 4 c , c , c , + c , c , ) . samiddin S. Rakhimov, Munther A. Hassan Proposition 2.25.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , (cid:1) c , c , , c ′ , c ′ , = c , c , , c ′ , c ′ , c ′ , = c , c , c , .
2. For any λ , λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , (cid:1) c , c , = λ , c , c , = λ , c , c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , , λ , λ , , , , , λ , λ , λ ∈ C . Proof.
Here, A = − A c , c , , B = A c , , and B = 2 B c , c , − A c , c , + 2 A c , c , + A c , c , c , A c , c , . Proposition 2.26.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , = c , c , , c ′ , c ′ , c ′ , = c , c , c , .
2. For any λ , λ ∈ C , there exists L ( C ) ∈ U : c , c , = λ , c , c , c , = λ . Then orbits in U can be parameterized as L (0 , , , , , λ , λ , , , , , λ , λ ∈ C . Proof.
We put, A = − A c , c , , B = A c , , and B = B c , c , − A c , c , + A c , c , + A c , c , c , A c , c , . Proposition 2.27.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , c ′ , = c , c , c , , c ′ , c ′ , = c , c , .
2. For any λ , λ ∈ C there exists L ( C ) ∈ U : c , c , c , = λ , c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , , λ , , , , , , λ , λ ∈ C . Proof.
We put here, A = c , c , , B = c , c , , and B = B c , + 3 c , A c , c , + c , c , − c , A c , c , c , c , . On low-dimensional filiform Leibniz algebras and their invariants
Proposition 2.28.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , = c , c , .
2. For any λ ∈ C ∗ , there exists L ( C ) ∈ U : c , c , = λ. Then orbits in U can be parameterized as L (1 , , , , λ, , , , , , λ ∈ C ∗ . Proof.
Put B = A c , , and B = 3 A A c , + B c , + A c , − A A c , c , A c , . Proposition 2.29.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , c ′ , = c , c , .
2. For any λ ∈ C ∗ , there exists L ( C ) ∈ U : c , c , = λ. Then orbits in U can be parameterized as L (0 , , , , λ, , , , , , λ ∈ C ∗ . Proof.
Here, B = A c , , and B = 3 A A c , + B c , + A c , − A A c , c , A c , . Proposition 2.30.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , (cid:0) c ′ , c ′ , − c ′ , (cid:1) = (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , (cid:0) c , c , − c , (cid:1) , (cid:16) − c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , (cid:0) c ′ , c ′ , − c ′ , (cid:1) = (cid:0) − c , c , + c , c , (cid:1) c , (cid:0) c , c , − c , (cid:1) ,c ′ , c ′ , (cid:0) c ′ , c ′ , − c ′ , (cid:1) = c , c , (cid:0) c , c , − c , (cid:1) , (cid:16) c ′ , + c ′ , c ′ , − c ′ , c ′ , c ′ , (cid:17) (cid:0) c ′ , c ′ , − c ′ , (cid:1) = (cid:0) c , + c , c , − c , c , c , (cid:1) (cid:0) c , c , − c , (cid:1) .
2. For any λ , λ , λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , (cid:0) c , c , − c , (cid:1) = λ , (cid:0) − c , c , + c , c , (cid:1) c , (cid:0) c , c , − c , (cid:1) = λ ,c , c , (cid:0) c , c , − c , (cid:1) = λ , (cid:0) c , + c , c , − c , c , c , (cid:1) (cid:0) c , c , − c , (cid:1) = λ . Then orbits in U can be parameterized as L ( λ , λ , λ , , , , λ , , , , λ , λ , λ , λ ∈ C . samiddin S. Rakhimov, Munther A. Hassan Proof.
Here, A = − A c , c , , and B = A c , . Proposition 2.31.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , (cid:0) − c ′ , + 4 c ′ , c ′ , (cid:1) = (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , (cid:0) − c , + 4 c , c , (cid:1) , (cid:16) − c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , (cid:0) − c ′ , + 4 c ′ , c ′ , (cid:1) = (cid:0) − c , c , + c , c , (cid:1) c , (cid:0) − c , + 4 c , c , (cid:1) ,c ′ , c ′ , (cid:0) − c ′ , + 4 c ′ , c ′ , (cid:1) = c , c , (cid:0) − c , + 4 c , c , (cid:1) .
2. For any λ , λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , (cid:0) − c , + 4 c , c , (cid:1) = λ , (cid:0) − c , c , + c , c , (cid:1) c , (cid:0) − c , + 4 c , c , (cid:1) = λ , c , c , (cid:0) − c , + 4 c , c , (cid:1) = λ . Then orbits in U can be parameterized as L ( λ , λ , λ , , , , , , , , λ , λ , λ ∈ C . Proof.
For this case, put in the base change (9), A = − A c , c , , and B = A c , . Proposition 2.32.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , c , , (cid:16) − c ′ , c ′ , + c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) − c , c , + c , c , (cid:1) c , c , .
2. For any λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , c , c , + c , c , (cid:1) c , c , = λ , (cid:0) − c , c , + c , c , (cid:1) c , c , = λ . Then orbits in U can be parameterized as L ( λ , λ , , , , , , , , , λ , λ ∈ C . On low-dimensional filiform Leibniz algebras and their invariants
Proof.
Put in the base change (9), A = − A c , c , , and B = A c , . Proposition 2.33.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , (cid:1) c , c , .
2. For any λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , c , (cid:1) c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , , , , , , , , λ ∈ C . Proof.
For this case, we put A = − A c , c , , and B = A c , . Proposition 2.34.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , (cid:16) − c ′ , c ′ , + 3 c ′ , c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , = c , (cid:0) − c , c , + 3 c , c , c , − c , c , (cid:1) c , ,c ′ , (cid:16) − c ′ , c ′ , + 3 c ′ , c ′ , (cid:17) c ′ , = c , (cid:0) − c , c , + 3 c , c , (cid:1) c , , c ′ , c ′ , c ′ , = c , c , c , .
2. For any λ , λ , λ ∈ C , there exists L ( C ) ∈ U : c , (cid:0) − c , c , + 3 c , c , c , − c , c , (cid:1) c , = λ ,c , (cid:0) − c , c , + 3 c , c , (cid:1) c , = λ , c , c , c , = λ . Then orbits in U can be parameterized as L ( λ , λ , λ , , , , , , , , λ , λ , λ ∈ C . Proof.
Here, we put in the base change (9) the following coefficients: A = c , c , , A = − c , c , c , , and B = c , c , . Proposition 2.35.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) − c ′ , c ′ , + 3 c ′ , c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) − c , c , + 3 c , c , c , − c , c , (cid:1) c , c , , (cid:16) − c ′ , c ′ , + 3 c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) − c , c , + 3 c , c , (cid:1) c , c , . samiddin S. Rakhimov, Munther A. Hassan
2. For any λ , λ ∈ C , there exists L ( C ) ∈ U : (cid:0) − c , c , + 3 c , c , c , − c , c , (cid:1) c , c , = λ , (cid:0) − c , c , + 3 c , c , (cid:1) c , c , = λ . Then orbits in U can be parameterized as L ( λ , λ , , , , , , , , , λ , λ ∈ C . Proof.
We put A = − A c , c , , and B = A c , . Proposition 2.36.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if (cid:16) c ′ , c ′ , − c ′ , c ′ , (cid:17) c ′ , c ′ , = (cid:0) c , c , − c , c , (cid:1) c , c , .
2. For any λ ∈ C , there exists L ( C ) ∈ U : (cid:0) c , c , − c , c , (cid:1) c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , , , , , , , , λ ∈ C . Proof.
Here, A = − A c , c , , and B = A c , . Proposition 2.37.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c ′ , (cid:16) c ′ , c ′ , − c ′ , (cid:17) c ′ , = c , (cid:0) c , c , − c , (cid:1) c , , c ′ , c ′ , c ′ , = c , c , c , .
2. For any λ , λ ∈ C , there exists L ( C ) ∈ U : c , (cid:0) c , c , − c , (cid:1) c , = λ , c , c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , , , λ , , , , , λ , λ ∈ C . Proof.
Her, we take A = c , c , , A = − c , c , , and B = c , c , . Proposition 2.38.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c , (cid:0) c , c , − c , (cid:1) c , = c , (cid:0) c , c , − c , (cid:1) c , . On low-dimensional filiform Leibniz algebras and their invariants
2. For any λ ∈ C , there exists L ( C ) ∈ U : c , (cid:0) c , c , − c , (cid:1) c , = λ . Then orbits in U can be parameterized as L ( λ , , , , , , , , , , λ ∈ C . Proof.
We put in the base change (9), A = r c , c , , A = − A c , c , , and B = c , c , . Proposition 2.39.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c , c , c , = c , c , c , .
2. For any λ ∈ C there exists L ( C ) ∈ U : c , c , c , = λ . Then orbits in U can be parameterized as L (0 , , , , , λ , , , , , λ ∈ C . Proof.
Here, A = − A c , c , , and B = c , c , . Proposition 2.40.
1. Two algebras L ( C ) and L ( C ′ ) from U are isomorphic, if and only if c , c , c , = c , c , c , .
2. For any λ ∈ C there exists L ( C ) ∈ U : c , c , c , = λ . Then orbits in U can be parameterized as L ( λ , , , , , , , , , , λ ∈ C . Proof.
Finally, put A = c , c , , and B = c , c , . Proposition 2.41.
The subsets U , U , U , U , U , U , U , U , U , U , U , U , U , U , U ,U , U , U , U , U , U and U are single orbits with representatives L (1 , , , , , , , , , ,L (0 , , , , , , , , , , L (1 , , , , , , , , , , L (0 , , , , , , , , , , L (1 , , , , , , , , , ,L (0 , , , , , , , , , , L (1 , , , , , , , , , , L (0 , , , , , , , , , , L (1 , , , , , , , , , ,L (0 , , , , , , , , , , L (1 , , , , , , , , , , L (0 , , , , , , , , , , L (1 , , , , , , , , , ,L (0 , , , , , , , , , , L (0 , , , , , , , , , , L (0 , , , , , , , , , , L (1 , , , , , , , , , ,L (0 , , , , , , , , , , L (1 , , , , , , , , , , L (0 , , , , , , , , , , L (1 , , , , , , , , , and L (0 , , , , , , , , , , respectively.Proof. To prove it, we give the appropriate values of A , A , A , A , B , B , B and B in the base change(9)(as for other A i , i = 4 , ..., , and B j , j = 5 , , U and U : B = A + A c , c , , and B = − A c , ( A + A c , ) ( A c , + 2 A c , A c , + A c , A c , − A c , A c , − A c , A c , − A c , c , − A B c , ) . samiddin S. Rakhimov, Munther A. Hassan U and U : B = A + A c , c , and B = − c , ( A + A c , ) ( A c , + 2 A c , A c , + c , A c , − B c , ) . For U and U : B = A + A c , c , and B = − A c , + 2 A c , A c , + c , A c , − B c , c , ( A + A c , ) . For U and U : A = A c , c , , B = A c , , B = B c , + A c , A c , , and B = − − B c , − A c , c , + 2 A c , c , − B A c , c , A c , . For U and U : A = − A c , c , , and B = A c , . For U and U : A = − A c , c , , and B = A c , . For U and U : A = − A c , c , , and B = A c , . For U and U : A = − A c , c , . For U and U : B = A c , . For U and U : B = A c , .
1. In
T Leib we distinguished 26 isomorphism classes (12 parametric family and 14 concrete) of sevendimensional Leibniz algebras and shown that they exhaust all possible cases.2. In the case of T Leib there are 49 isomorphism classes (27 parametric family and 22 concrete) andthey exhaust all possible cases. References [1] U. D. Bekbaev and I. S. Rakhimov, On classification of finite dimensional complex filiform Leibnizalgebras (part 1), (2006), http://front.math.ucdavis.edu/, ArXiv:math. RA/01612805.[2] U. D. Bekbaev and I. S. Rakhimov, On classification of finite dimensional complex filiform Leibnizalgebras (part 2), (2007), http://front.math.ucdavis.edu/, arXiv:0704.3885v1 [math.RA] .0
On low-dimensional filiform Leibniz algebras and their invariants [3] J. R. G´omez, A. Jimenez-Merchan and Y. Khakimdjanov, Low-dimensional filiform Lie algebras,