Twisted cocycles of Lie algebras and corresponding invariant functions
aa r X i v : . [ m a t h - ph ] M a y Twisted Cocycles of Lie Algebras and Corresponding InvariantFunctions
Jiˇr´ı Hrivn´ak , Petr Novotn´y Department of Physics, Faculty of Nuclear sciences and Physical Engineering, Czech TechnicalUniversity, Bˇrehov´a 7, 115 19 Prague 1, Czech Republic
ABSTRACT. We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalizethe concept of cohomology cocycles of Lie algebras. A special case is generalization of 1–cocycles with respect to theadjoint representation – so called ( α, β, γ )–derivations. Parametric sets of spaces of cocycles allow us to define complexfunctions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants – weshow how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensionalone-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of1–parametric continuous contraction. Introduction
The search for a new concept of invariant characteristics of Lie algebras led to the definition of( α, β, γ )–derivations in [1]. Consider a complex Lie algebra L and its derivation D ∈ der L , i.e. alinear operator D ∈ End L satisfying for all x, y ∈ L the equation D [ x, y ] = [ Dx, y ] + [ x, Dy ]. Forfixed α, β, γ ∈ C is a linear operator A ∈ End L called an ( α, β, γ ) –derivation if for all x, y ∈ L theequation(1) αA [ x, y ] = β [ Ax, y ] + γ [ x, Ay ]holds [1]. In this article we denote the set of all operators satisfying (1) by der ( α,β,γ ) L . Investigatingthe spaces der ( α,β,γ ) L , various Jordan and Lie operator algebras were obtained. In [1] was also shownthat the dimensions of these operator algebras and, in fact, the dimensions of all spaces der ( α,β,γ ) L as well as their intersections form invariant characteristics of Lie algebras. The invariance of thesedimensions enabled us to define so called invariant function ψ L by the relation(2) ψ L ( α ) ≡ dim der ( α, , L . This complex function ψ L turned out to be very valuable: it was used to classify all complex three–dimensional indecomposable Lie algebras. More significantly, it resolved some parametric continua ofnilpotent Lie algebras.Prior to invention of the invariant function ψ L , the isomorphism problem for algebras insidenilpotent parametric continuum had to be solved explicitly. If algebras inside a nilpotent parametriccontinuum are considered then all well known characteristics such as dimensions of derived , lowercentral and upper central sequences , Lie algebra of derivations , radical , nilradical , ideals , subalgebras [2, 3] and megaideals [4] naturally coincide. Moreover, due to Engel’s theorem none of the ’trace’invariants based on the adjoint representation such as C pq [5] or χ i [6] exists in this case. Thenilpotent parametric continua appeared for example in [7] where all graded contractions [8, 9] of thePauli graded sl(3 , C ) [10] were found. The invariant function ψ L was successfully applied in [1] tosome of the nilpotent parametric continua from [7]. This invariant function did not , however, resolve all nilpotent continua which were discussed in [7].Often, the search of new invariant characteristics of Lie algebras is motivated by the classificationof degenerations [5, 11, 12] or 1-parametric continuous contractions [13]. The applicability of theinvariant function ψ L (or its possible generalizations) in this field remained an open problem. Corresponding author: Tel.: +420 2 24358351; fax: +420 2 22320861
E-mail address: jiri.hrivnak@fjfi.cvut.cz (J. Hrivn´ak) Precisely these facts motivated the work undertaken in this article. The idea of finding newinvariant characteristics of Lie algebras via generalizing standard Chevalley cohomology cocycles isfollowed. New invariant functions are defined and applied not only to some cases from [7] but toall four-dimensional complex Lie algebras. Application of these generalized cocycles to 1-parametriccontinuous contractions is discussed. It is shown that these new invariants of Lie algebras constitutean essential tool for resolving parametric continua of nilpotent Lie algebras and provide powerfulnecessary contraction criteria.In Section 2, twisted cocycles of Lie algebras are introduced and two–dimensional twisted cocyclesof the adjoint representation are investigated in detail. It is shown that there are sixteen cases ofthese two-dimensional cocycles which can be described by four complex parameters.In Section 3, the invariant functions ϕ and ϕ are defined and their behavior on low–dimensionalLie algebras demonstrated. The invariant functions ψ and ϕ are used to classify all three and four–dimensional complex Lie algebras. New algorithm for the identification of a four–dimensional complexLie algebra is also formulated.In Section 4, possible application of the invariant functions ψ, ϕ, ϕ to contractions is considered.Necessary criterion for existence of a 1-parametric continuous contraction is formulated. The in-variant function ψ is used to classify continuous contractions among three–dimensional Lie algebras.Application of the invariant functions on nilpotent parametric continua of Lie algebras resulting fromcontractions of the Pauli graded sl(3 , C ) is demonstrated.In Appendix A, the tables of the invariant functions ψ, ϕ, ϕ for two, three and four–dimensionalLie algebras are listed.In Appendix B, proof of the classification Theorem 3.6 is located.2. Twisted Cocycles of Lie Algebras
Let L be an arbitrary complex Lie algebra and ( V, f ) its representation. We denote by C q ( L , V )the vector space of all V –cochains of dimension q for q ∈ N and C ( L , V ) = V . We generalize thenotions of cocycles analogously to ( α, β, γ )–derivations. Let κ = ( κ ij ) be a ( q + 1) × ( q + 1) complexsymmetric matrix. We call c ∈ C q ( L , V ), q ∈ N for which0 = q +1 X i =1 ( − i +1 κ ii f ( x i ) c ( x , . . . , ˆ x i , . . . , x q +1 )++ q +1 X i,j =1 i The definition of( α, β, γ )–derivations is now included in the definition of twisted cocycles. Considering the adjointrepresentation and its one-dimensional twisted cocycles, we immediately have:(5) Z (cid:0) L , ad L , (cid:0) β αα γ (cid:1)(cid:1) = der ( α,β,γ ) L . In this section we investigate in detail the space Z ( L , ad L , κ ). For this purpose it may be moreconvenient to use different notation, analogous to that of derivations, defined by(6) coc ( α ,α ,α ,β ,β ,β ) L = Z (cid:18) L , ad L , (cid:18) β α α α β α α α β (cid:19)(cid:19) , i. e. in the space coc ( α ,α ,α ,β ,β ,β ) L are such B ∈ C ( L , L ) which for all x, y, z ∈ L satisfy0 = α B ( x, [ y, z ]) + α B ( z, [ x, y ]) + α B ( y, [ z, x ])+ β [ x, B ( y, z )] + β [ z, B ( x, y )] + β [ y, B ( z, x )] . (7)Six permutations of the variables x, y, z ∈ L in the defining equation (7) give Lemma 2.1. Let L be a complex Lie algebra. Then for any α , α , α , β , β , β ∈ C are all thefollowing six spaces equal:(1) coc ( α ,α ,α ,β ,β ,β ) L (2) coc ( α ,α ,α ,β ,β ,β ) L (3) coc ( α ,α ,α ,β ,β ,β ) L (4) coc ( α ,α ,α ,β ,β ,β ) L (5) coc ( α ,α ,α ,β ,β ,β ) L (6) coc ( α ,α ,α ,β ,β ,β ) L Lemma 2.2. Let L be a complex Lie algebra. Then for any α , α , α , β , β , β ∈ C is the spacecoc ( α ,α ,α ,β ,β ,β ) L equal to all of the following:(1) coc ( α + α ,α + α ,α + α ,β + β ,β + β ,β + β ) L ∩ coc ( α − α ,α − α ,α − α ,β − β ,β − β ,β − β ) L (2) coc (0 ,α − α ,α − α , ,β − β ,β − β ) L ∩ coc (2 α ,α + α ,α + α , β ,β + β ,β + β ) L (3) coc (0 ,α − α ,α − α , ,β − β ,β − β ) L ∩ coc (2 α ,α + α ,α + α , β ,β + β ,β + β ) L (4) coc (0 ,α − α ,α − α , ,β − β ,β − β ) L ∩ coc (2 α ,α + α ,α + α , β ,β + β ,β + β ) L Proof. Suppose α , α , α , β , β , β ∈ C are given. We demonstrate the proof on the case 1. – theproof of the other cases is analogous. Let B ∈ coc ( α ,α ,α ,β ,β ,β ) L ; then for arbitrary x, y, z ∈ L wehave 0 = α B ( x, [ y, z ]) + α B ( z, [ x, y ]) + α B ( y, [ z, x ])+ β [ x, B ( y, z )] + β [ z, B ( x, y )] + β [ y, B ( z, x )](8) 0 = α B ( z, [ x, y ]) + α B ( y, [ z, x ]) + α B ( x, [ y, z ])+ β [ z, B ( x, y )] + β [ y, B ( z, x )] + β [ x, B ( y, z )] . (9)By adding and subtracting equations (8) and (9) we obtain0 = ( α + α ) B ( x, [ y, z ]) + ( α + α ) B ( z, [ x, y ]) + ( α + α ) B ( y, [ z, x ])+( β + β )[ x, B ( y, z )] + ( β + β )[ z, B ( x, y )] + ( β + β )[ y, B ( z, x )](10) 0 = ( α − α ) B ( x, [ y, z ]) + ( α − α ) B ( z, [ x, y ]) + ( α − α ) B ( y, [ z, x ])+( β − β )[ x, B ( y, z )] + ( β − β )[ z, B ( x, y )] + ( β − β )[ y, B ( z, x )](11)and thus coc ( α ,α ,α ,β ,β ,β ) L is the subset of the intersection 1. Similarly, starting with equations(10), (11) we obtain equations (8), (9) and the remaining inclusion is proven. (cid:3) We show in the following theorem that four parameters are sufficient for the description of allspaces of two–dimensional twisted cocycles. Theorem 2.3. Let L be a Lie algebra. Then for any α , α , α , β , β , β ∈ C there exist α, β, γ, δ ∈ C such that the subspace coc ( α ,α ,α ,β ,β ,β ) L ⊂ C ( L , L ) is equal to some of the following sixteensubspaces:(1) coc ( α, , ,β, , L ; coc ( α, , ,β, , − L ; coc ( α, , − ,β, , L ; coc ( α,β, − β,γ, , − L (2) coc ( α, , ,β, , L ; coc ( α, , ,β, , L ; coc ( α,β, − β,γ, , L ; coc ( α, , − ,β, , L (3) coc ( α, , ,β, , L ; coc ( α, , ,β, , L ; coc ( α, , ,β,γ, − γ ) L ; coc ( α, , ,β, , − L (4) coc ( α,β,γ,δ, , L ; coc ( α,β +1 ,β − ,γ, , L ; coc ( α, , ,β,γ +1 ,γ − L ; coc ( α,β,β,γ, , L Proof. Suppose α + α = 0 and β + β = 0. Then the following four cases are possible:(1) α = − α = 0 and β = − β = 0. In this case we havecoc ( α ,α ,α ,β ,β ,β ) L = coc ( α , , ,β , , L . (2) α = − α = 0 and β = − β = 0. In this case we have:coc ( α ,α ,α ,β ,β ,β ) L = coc ( α , , ,β ,β ,β ) L = coc (0 , , , ,β − β ,β − β ) L ∩ coc (2 α , , , β , , L = coc (0 , , , , , − L ∩ coc (2 α , , , β , , L = coc ( α , , ,β , , − L (3) α = − α = 0 and β = − β = 0. In this case we have:coc ( α ,α ,α ,β ,β ,β ) L = coc ( α ,α ,α ,β , , L = coc (0 ,α − α ,α − α , , , L ∩ coc (2 α , , , β , , L = coc (0 , , − , , , L ∩ coc (2 α , , , β , , L = coc ( α , , − ,β , , L (4) α = − α = 0 and β = − β = 0. In this case we have:coc ( α ,α ,α ,β ,β ,β ) L = coc (0 ,α − α ,α − α , ,β − β ,β − β ) L ∩ coc (2 α , , , β , , L = coc (0 , α − α β − β , − α − α β − β , , , − L ∩ coc (2 α , , , β , , L = coc ( α , α − α β − β , − α − α β − β ,β , , − L Discussion of the three remaining cases of the values of α + α and β + β is similar to the previouscase and we omit it. (cid:3) It may be more convenient, sometimes, to use different distribution of the cocycle spaces than inTheorem 2.3. Henceforth, we investigate mainly the cocycle space coc (1 , , ,λ,λ,λ ) L which for λ = 0fits in the class coc ( α,β,β,γ, , L , with α = β = 1 /λ, γ = 1. For λ = 0, the space coc (1 , , , , , L is aspecial case of the space coc ( α, , ,β, , L , with α = 1 , β = 0. We also put α = 0 , β = 1 , γ = λ intococ ( α,β,β,γ, , L and investigate the space coc (0 , , ,λ, , L .3. Invariant Functions Theorem 3.1. Let g : L → e L be an isomorphism of Lie algebras L and e L . Then the mapping ̺ : C q ( L , L ) → C q ( e L , e L ) , q ∈ N defined for all c ∈ C q ( L , L ) and all x , . . . , x q ∈ e L by( ̺c )( x , . . . , x q ) = gc ( g − x , . . . , g − x q )is an isomorphism of vector spaces C q ( L , L ) and C q ( e L , e L ). For any complex symmetric ( q + 1)–squarematrix κ ̺ ( Z q ( L , ad L , κ )) = Z q ( e L , ad e L , κ )holds. Proof. Suppose we have g : L → e L such that for all x, y ∈ e L [ x, y ] e L = g [ g − x, g − y ] L holds. It is clear that the map ̺ : C q ( L , L ) → C q ( e L , e L ) , q ∈ N is linear and bijective, i. e. it is anisomorphism of these vector spaces. By putting f = ad L and rewriting definition (3) we have for c ∈ Z q ( L , ad L , κ ) and all x , . . . , x q ∈ e L q +1 X i =1 ( − i +1 κ ii [ g − x i , c ( g − x , . . . , \ g − x i , . . . , g − x q +1 )] L ++ q +1 X i,j =1 i For any q ∈ N and any complex symmetric ( q + 1)–square matrix κ is the dimensionof the vector space Z q ( L , ad L , κ ) an invariant characteristic of Lie algebras.Sixteen parametric spaces in Theorem 2.3 allow us to define sixteen invariant functions of up tofour variables. However, a complete analysis of possible outcome is beyond the scope of this work.Rather empirically, following calculations in dimension four and eight, we pick up two one–parametricsets of vector spaces to define two new invariant functions of a n –dimensional Lie algebra L . We callfunctions ϕ L , ϕ L : C → { , , . . . , n ( n − / } defined by the formulas( ϕ L )( α ) = dim coc (1 , , ,α,α,α ) L (12) ( ϕ L )( α ) = dim coc (0 , , α, , L (13)the invariant functions corresponding to two–dimensional twisted cocycles of the adjoint represen-tation of a Lie algebra L .From Theorem 3.1 follows immediately: Corollary 3.3. If two complex Lie algebras L , e L are isomorphic, L ∼ = e L , then it holds:(1) ψ L = ψ e L ,(2) ϕ L = ϕ e L ,(3) ϕ L = ϕ e L .3.1. Invariant Functions ψ , ϕ and ϕ of Low–dimensional Lie Algebras. We now investigatethe behaviour of the functions ϕ, ϕ in dimensions three and four. It was shown in [1] that theinvariant function ψ classifies indecomposable three–dimensional complex Lie algebras. Moreover,observing the tables in Appendix A, the following theorem holds. Theorem 3.4 (Classification of three–dimensional complex Lie algebras) . Two three–dimensional complex Lie algebras L , e L are isomorphic if and only if ψ L = ψ e L .Observing the tables of ϕ , we may also derive a quite interesting fact – the function ϕ alone alsoclassifies three–dimensional Lie algebras: Theorem 3.5 (Classification of three–dimensional complex Lie algebras II) . Two three–dimensional complex Lie algebras L and e L are isomorphic if and only if ϕ L = ϕ e L .Theorem 3.4 (or 3.5) provided complete classification of three–dimensional complex Lie algebras.We show in this section that combined power of the functions ψ and ϕ distinguishes among allcomplex four–dimensional Lie algebras. We define the number of occurrences of j ∈ C in acomplex function f . Let j be in the range of values of f . If there exist only finitely many mutuallydistinct numbers x , . . . , x m ∈ C for which f ( x ) = · · · = f ( x m ) = j holds then we write f : j m and say that j occurs in f m –times; otherwise we write f : j . Theorem 3.6 (Classification of four–dimensional complex Lie algebras) . Two four–dimensional com-plex Lie algebras L and e L are isomorphic if and only if ψ L = ψ e L and ϕ L = ϕ e L . Proof. See Appendix B. (cid:3) An efficient algorithm for the identification of four–dimensional Lie algebras was quite recentlypublished in [6]. We may now formulate an alternative algorithm : take a four–dimensional complexLie algebra L and(1) Calculate ψ L and ϕ L .(2) The range of values of the functions ψ and ϕ and the number of their occurrences determinesthe label (g- k ), k = 1 , . . . , 34 in Appendix A.(3) The algebra is now identified up to the exact value of parameter(s) of the parametric contin-uum. These parameters are determined in the following cases:(g-8) Pick any of the two values z ∈ C , z = 1, which satisfy ψ L ( z ) = 6, and put a = z . Then L ∼ = g , ( a ) ⊕ g holds.(g-11) There are two different complex numbers z , z = 0 which satisfy ϕ L ( z ) = ϕ L ( z ) = 13.If z − /z holds then put a = z − 1, otherwise put a = z − 1. Then L ∼ = g , ( a )holds.(g-17) There are three mutually different complex numbers z , z , z = 0 , − ϕ L ( z ) = ϕ L ( z ) = ϕ L ( z ) = 13. Put a = z +1 z +1 , b = z z − z +1 . Then L ∼ = g , ( a, b )holds.(g-18) Pick any of the six values z ∈ C , which satisfy ψ L ( z ) = 5, and put a = z . Then L ∼ = g , ( a, − − a ) holds.(g-19) Pick any of the two values z ∈ C , z = 1, which satisfy ψ L ( z ) = 6, and put a = z . Then L ∼ = g , ( a, a ) holds.(g-20) Take the value z ∈ C , which satisfies ϕ L ( z ) = 15, and put a = z − 1. Then L ∼ = g , ( a, z ∈ C , which satisfy ϕ L ( z ) = 13, and put a = z + 1. Then L ∼ = g , ( a, − 1) holds.(g-28) Pick any of the two values z ∈ C , z = 2 which satisfy ψ L ( z ) = 4, and put a = z . Then L ∼ = g , ( a ) holds.We demonstrate the above algorithm of identification on the following example. Example . In [6], a four–dimensional algebra L was introduced: L : [ e , e ] = − e − e + e , [ e , e ] = − e + 4 e , [ e , e ] = 2 e − e + e , [ e , e ] = 3 e − e + 5 e , [ e , e ] = 4 e − e + 2 e , [ e , e ] = 6 e − e + 3 e . (1) Computing the functions ψ L and ϕ L one obtains: α ψ L ( α ) 6 5 5 4 α ϕ L ( α ) 13 13 12 (2) The combination of occurrences ψ L : 6 , , ϕ L : 13 , 12 is unique for the case (g-11).(3) Since for z = 3, z = 1 the equality z − /z holds, one has a = z − L ∼ = g , (2). 4. Contractions of Lie Algebras Continuous Contractions of Lie Algebras. Suppose we have an arbitrary Lie algebra L =( V, [ , ]) and a continuous mapping U : (0 , i → GL ( V ), i. e. U ( ε ) ∈ GL ( V ) , < ε ≤ 1. If the limit(14) [ x, y ] = lim ε → U ( ε ) − [ U ( ε ) x, U ( ε ) y ]exists for all x, y ∈ V then we call the algebra L = ( V, [ , ] ) a one–parametric continuouscontraction (or simply a contraction ) of the algebra L and write L → L . We call the contraction L → L proper if L ≇ L . Contraction to the Abelian algebra is always possible via U ( ε ) = ε L → L is any one–parametric continuous contraction of a Lie algebra L then L is also a Lie algebra. Invariant characteristics of Lie algebras change after a contraction. Therelation among these characteristics before and after a contraction form useful necessary contractioncriteria. For example, such a set of these criteria, which provided the complete classification of contractions of three and four–dimensional Lie algebras, has been found in [13]. Our aim is to statenew necessary contraction criteria using ( α, β, γ )–derivations and twisted cocycles. Theorem 4.1. Let L be a complex Lie algebra, L → L and q ∈ N . Then for any ( q + 1) × ( q + 1)complex symmetric matrix κ (15) dim Z q ( L , ad L , κ ) ≤ dim Z q ( L , ad L , κ )holds. Proof. Suppose that the contraction L → L is performed by the mapping U , i. e. [ x, y ] =lim ε → [ x, y ] ε , where [ x, y ] ε = U ( ε ) − [ U ( ε ) x, U ( ε ) y ] , ∀ x, y ∈ L . Suppose L = ( V, [ , ]) and let us fix a basis { x , . . . , x n } of V . We denote the structural constants ofthe algebra L by c kij and the structural constants of the algebras L ε = ( V, [ , ] ε ) by c kij ( ε ). Then itholds(16) c kij (0) = lim ε → c kij ( ε ) , where c kij (0) are the structural constants of L . The dimension of the space Z q ( L , ad L , κ ) is determinedvia the relation(17) dim Z q ( L , ad L , κ ) = dim C q ( L , L ) − rank S q ( L , κ ) , where S q ( L , κ ) is the matrix corresponding to the linear system of equations generated from (3).We write the explicit form of this system for q = 1. Then we obtain from (3) that D = ( D ij ) ∈ Z (cid:0) L , ad L , (cid:0) β αα γ (cid:1)(cid:1) if and only if the linear system with the matrix S (cid:0) L , (cid:0) β αα γ (cid:1)(cid:1) is satisfied(18) S (cid:0) L , (cid:0) β αα γ (cid:1)(cid:1) : n X r =1 − αc rij D sr + βc srj D ri + γc sir D rj = 0 , ∀ i, j, s ∈ { , . . . , n } , and similarly for q > 1. Since L ε ∼ = L holds for all 0 < ε ≤ 1, we see from Corollary 3.2 that(19) dim Z q ( L , ad L , κ ) = dim Z q ( L ε , ad L ε , κ ) , < ε ≤ , q ∈ N . Since the relation(20) dim C q ( L , L ) = dim C q ( L ε , L ε ) = dim C q ( L , L ) , < ε ≤ , q ∈ N holds, the relations (17), (19) then imply that(21) rank S q ( L , κ ) = rank S q ( L ε , κ ) , < ε ≤ , q ∈ N . The rank of the matrix S q ( L , κ ) is equal to r if and only if there exists a non-zero minor of the order r and every minor of order higher than r is zero. It follows from (21) that all minors of the orders higherthan r of the matrices S q ( L ε , κ ) are zeros. Since the equality (16) holds, all minors of the matrices S q ( L ε , κ ) converge to the minors of the matrix S q ( L , κ ). Thus, as the limits of zero functions, allminors of order higher than r of the matrix S q ( L , κ ) are also zero. Therefore rank S q ( L , κ ) ≤ r andthe statement of the theorem follows from (17) and (20). (cid:3) There exist other necessary contraction criteria, similar to (15) – certain inequalities betweeninvariants. However, one very powerful criterion is quite unique. This highly non-trivial theorem,very useful in [11, 5, 13], was originally proved in [15]. Theorem 4.2. If L is a proper contraction of a complex Lie algebra L then it holds:(22) dim der L < dim der L . Corollary 4.3. If L is a proper contraction of a complex Lie algebra L then it holds:(1) ψ L ≤ ψ L (2) ψ L (1) < ψ L (1). Proof. Since ψ L ( α ) = dim Z ( L , ad L , ( αα )) the first inequality follows from (15) and the secondfrom ψ L (1) = dim der L and (22). (cid:3) Corollary 4.4. If L is a contraction of a complex Lie algebra L then it holds:(1) ϕ L ≤ ϕ L (2) ϕ L ≤ ϕ L . Proof. Since ϕ L ( α ) = dim Z (cid:16) L , ad L , (cid:16) α α 11 1 α (cid:17)(cid:17) the first inequality follows from (15); the proof ofthe second condition is analogous. (cid:3) Continuous Contractions of Low–dimensional Lie Algebras. We have used the invariantfunction ψ to classify all three–dimensional Lie algebras in Theorem 3.4. We now employ the necessarycontraction criterion of Corollary 4.3 to describe all possible contractions among these algebras. Thebehaviour of the function ψ determines the classification and contractions of three–dimensional Liealgebras. Contractions of three–dimensional algebras were the most recently classified in [13]: Theorem 4.5. Only the following proper contractions among three–dimensional Lie algebras exist:(1) g , ( − 1) is a contraction of sl(2 , C ),(2) g , is a contraction of g , ,(3) g , is a contraction of g , , g , ( a ), g , ⊕ g and sl(2 , C ).(4) all algebras contract to the Abelian algebraAnalysis of all possible pairs of three–dimensional Lie algebras leads us to the following theorem. Theorem 4.6 (Contractions of three–dimensional complex Lie algebras) . Let L , L be two three–dimensional complex Lie algebras. Then there exists a proper one–parametriccontinuous contraction L → L if and only if ψ L ≤ ψ L and ψ L (1) < ψ L (1) . Proof. ⇒ : This implication is, in fact, Corollary 4.3. ⇐ : This implication follows from a direct comparison of the tables of the invariant functions ψ ofthree–dimensional Lie algebras in Appendix A and Theorem 4.5. (cid:3) We discuss the application of the criteria of the Corollaries 4.3 and 4.4 to the four–dimensional Liealgebras in the following examples. Example . To demonstrate behaviour of the functions ψ, ϕ and ϕ in dimension four, we considerthe following sequence of contractions [5, 13] :sl(2 , C ) ⊕ g → g , ( − → g , ( − ⊕ g → g , → g , ⊕ g → . Note in Table 1, how the value of each invariant function is greater or equal than the value in theprevious row. As expected, the strict inequality for the values ψ (1) holds – in this case the sequenceof dimensions: 4 , , , , , 16. The strict increase of values is also identified in the following cases: ψ (2), ’generic values’ of ψ , ϕ (1 / 2) and ’generic values’ of ϕ . These conjectures of strict inequalitiesare, however, not valid for the general case of a contraction in dimension four. Example . Consider the pair of parametric four–dimensional Lie algebras g , ( a ) , a = 0 , ± , − , ( a ′ , a ′ = 0 , ± , − 2. There are two possibilities, how the corresponding tables of the invariantfunctions ϕ , in Appendix A cases (g-11) and (g-20), can satisfy ϕ g , ( a ) ≤ ϕ g , ( a ′ , a ′ + 1 = 2 /a and a + 1 = 2 /a ′ – these have solutions a = a ′ = 1 , − a = a ′ . The necessary condition 1. of Corollary 4.4therefore admits only the contraction g , ( a ) → g , ( a, ψ, ϕ and ϕ . Note that the function ψ growsonly at the points 1 , a, a and the function ϕ only at one(!) point 1 + a . Example . Consider the pair of four–dimensional Lie algebras g , and g , (1). The necessaryconditions ψ g , ≤ ψ g , (1) , [ ψ g , ](1) < [ ψ g , (1)](1) and ϕ g , ≤ ϕ g , (1) are satisfied. But sinceit holds 1 = [ ϕ g , ] (cid:18) (cid:19) > [ ϕ g , (1)] (cid:18) (cid:19) = 0 , a contraction is not possible. Table 1. Invariant functions ψ, ϕ and ϕ of the contraction sequence: sl(2 , C ) ⊕ g → g , ( − → g , ( − ⊕ g → g , → g , ⊕ g → . ψ ( α ) ϕ ( α ) ϕ ( α ) α -1 0 1 2 -1 0 1 , C ) ⊕ g , ( − 1) 6 4 5 4 4 14 12 13 12 12 0 0 1 0g , ( − ⊕ g , , ⊕ g 10 11 10 10 10 19 20 19 19 19 8 11 8 84 g 16 16 16 16 16 24 24 24 24 24 24 24 24 24 Table 2. Invariant functions ψ, ϕ and ϕ of the contraction: g , ( a ) → g , ( a, ψ ( α ) ϕ ( α ) ϕ ( α ) α a a a a a a g , ( a ) 6 5 5 4 13 13 12 3 1 1 0g , ( a, 1) 8 6 6 4 15 13 12 7 2 2 0 Graded Contractions of Lie algebras. Consider a graded contraction of the Pauli gradedsl(3 , C ) which was in [7] denoted by ε , ( a ), a = 0. We may determine this graded contraction bylisting its non–zero commutation relations in Z –labeled basis ( l , l , l , l , l , l , l , l ): L , ( a ) : [ l , l ] = − al , [ l , l ] = l , [ l , l ] = l , [ l , l ] = l , [ l , l ] = l , [ l , l ] = l , [ l , l ] = l , a = 0 . Lie algebras L , ( a ) are all indecomposable, nilpotent and their derived series, lower central series,upper central series and the number of formal Casimir invariants [16] coincide.The invariant function ψ has the following form: α ψ L , ( a )( α ) 20 19 18 In this case, the function ψ completely fails – does not depend on a = 0. We are able, however, toadvance by calculation of the function ϕ : a = 1 α ϕ L , (1)( α ) 112 83 81 80 a = − α ϕ L , ( − α ) 104 83 81 80 a = + √ iα − − √ iϕ L , (cid:16) + √ i (cid:17) ( α ) 104 82 82 80 a = − √ iα − + √ iϕ L , (cid:16) − √ i (cid:17) ( α ) 104 82 82 80 a = α − ϕ L , (cid:0) (cid:1) ( α ) 104 83 81 80 a = 0 , ± , , ± √ iα − a − + a ϕ L , ( a )( α ) 104 82 81 81 80 In order to verify that(23) ϕ L , ( a ) : 104 , , , , a = 0 , ± , , ± √ i we have to check the equality − a = − 12 + 12 a which has the solutions ± √ i . Thus, (23) is verified.We proceed to solve the relation ϕ L , ( a ) = ϕ L , ( a ′ ) , a, a ′ = 0 , ± , , ± √ i and we obtain(1) If − a = − a ′ , − + a = + a ′ then a = a ′ .(2) If − a = − + a ′ , − a ′ = − + a then a = a ′ = ± √ i .The second case is not possible. Observing that all other tables of the function ϕ L , ( a ) , a = ± , , ± √ i are mutually different, we have: if ϕ L , ( a ) = ϕ L , ( a ′ ) , a, a ′ = 0 then a = a ′ . We conclude thateven though the function ψ did not distinguish the algebras in nilpotent parametric continuum, thefunction ϕ provided their complete description.5. Concluding Remarks • The invariant functions ψ and ϕ are able to classify all four–dimensional Lie algebras andprovide also necessary contraction criteria. In order to obtain stronger contraction criteria,we also defined the function ϕ – a supplement to the functions ψ and ϕ (see Example 4.3).However, the combined forces of the Corollaries 4.3 and 4.4 do not provide us with a completeclassification of contractions of four–dimensional Lie algebras. • Existence of invariant functions, arising from the concept of two–dimensional twisted cocyclesand allowing classification of continuous contractions of four–dimensional complex Lie alge-bras, remains an open problem. The complete description of the spaces of two–dimensionaltwisted cocycles for four–dimensional Lie algebras would solve the existence of such functionsexplicitly. However, such a complete description seems, at the moment, out of reach. • The contraction criterion formulated in Theorem 4.1 is a natural generalization of contractioncriteria which involve standard cohomology cocycles – see e.g. [11, 14]. • The invariant function ψ can be easily generalized and used as an invariant of an arbitraryanti-commutative or commutative algebra – it can, for example, describe all two–dimensionalcomplex Jordan algebras and their contractions [17]. • In contrast to algebraic approach of the ’trace’ invariants C pq and χ i [5, 6], the conceptof invariant cocycles provides a new knowledge about the dimensions of highly non–trivialstructures, interlaced with given Lie algebra. Considering parametric continua of nilpotentalgebras, which firstly appear in dimension 7, the invariant functions ψ, ϕ, ϕ seem to be evenmore important. In these cases, the behaviour of these invariant functions is quite unique andirreplaceable. Acknowledgements The authors are grateful to J. Tolar for numerous stimulating discussions. Partial support bythe Ministry of Education of Czech Republic (projects MSM6840770039 and LC06002) is gratefullyacknowledged. Appendix A Appendix A contains the classification of complex Lie algebras up to dimension four and the invari-ant functions ψ, ϕ, ϕ . We basically follow the notation of [13]. Instead of the symbols ψ L , ϕ L , ϕ L ,abbreviated symbols ψ, ϕ, ϕ are used. Blank spaces in the tables of the functions ψ, ϕ, ϕ denotegeneral complex numbers, different from all previously listed values, e. g. it holds: ψ g , ( − α ) = 3 , α ∈ C , α = ± . Two–dimensional Complex Lie Algebras. : Abelian α ψ ( α ) 4 4 αϕ ( α ) 2 αϕ ( α ) 2g , : [ e , e ] = e α ψ ( α ) 2 3 2 αϕ ( α ) 2 α ϕ ( α ) 1 0 Three–dimensional Complex Lie Algebras. : Abelian α ψ ( α ) 9 9 αϕ ( α ) 9 αϕ ( α ) 9g , ⊕ g : [ e , e ] = e α ψ ( α ) 4 6 4 αϕ ( α ) 6 α ϕ ( α ) 2 2 1g , : [ e , e ] = e α ψ ( α ) 6 6 α ϕ ( α ) 9 8 αϕ ( α ) 3g , : [ e , e ] = e , [ e , e ] = e + e α ψ ( α ) 4 3 αϕ ( α ) 6 α ϕ ( α ) 2 0g , : [ e , e ] = e , [ e , e ] = e α ψ ( α ) 6 3 αϕ ( α ) 6 α ϕ ( α ) 6 0g , ( − 1) : [ e , e ] = e , [ e , e ] = − e α ψ ( α ) 4 5 3 α ϕ ( α ) 9 7 α ϕ ( α ) 2 2 0g , ( a ) : [ e , e ] = e , [ e , e ] = ae , a = 0 , ± α a a ψ ( α ) 4 4 4 3 αϕ ( α ) 6 α a a ϕ ( α ) 2 1 1 0sl(2 , C ) : [ e , e ] = e , [ e , e ] = e , [ e , e ] = 2 e α − ψ ( α ) 3 5 1 0 α ϕ ( α ) 9 6 α ϕ ( α ) 1 0 Four–dimensional Complex Lie Algebras. (g-1) 4 g : Abelian α ψ ( α ) 16 16 αϕ ( α ) 24 αϕ ( α ) 24(g-2) g , ⊕ : [ e , e ] = e α ψ ( α ) 8 11 8 α ϕ ( α ) 16 14 α ϕ ( α ) 8 7 6(g-3) g , ⊕ g , : [ e , e ] = e , [ e , e ] = e α ψ ( α ) 4 6 4 α ϕ ( α ) 12 12 10 α ϕ ( α ) 2 2 0(g-4) g , ⊕ g : [ e , e ] = e α ψ ( α ) 10 11 10 α ϕ ( α ) 20 19 α ϕ ( α ) 11 8(g-5) g , ⊕ g : [ e , e ] = e , [ e , e ] = e + e α ψ ( α ) 6 7 5 α ϕ ( α ) 13 12 α ϕ ( α ) 3 3 1 (g-6) g , ⊕ g : [ e , e ] = e , [ e , e ] = e α ψ ( α ) 8 7 5 α ϕ ( α ) 15 12 α ϕ ( α ) 3 7 1(g-7) g , ( − ⊕ g : [ e , e ] = e , [ e , e ] = − e α − ψ ( α ) 6 7 7 5 α − ϕ ( α ) 15 16 16 14 α ϕ ( α ) 3 3 3 1(g-8) g , ( a ) ⊕ g : [ e , e ] = e , [ e , e ] = ae , a = 0 , ± α a a ψ ( α ) 6 7 6 6 5 α a a ϕ ( α ) 13 13 13 12 α a a ϕ ( α ) 3 3 2 2 1(g-9) sl(2 , C ) ⊕ g : [ e , e ] = e , [ e , e ] = e , [ e , e ] = 2 e α − ψ ( α ) 4 4 6 2 1 α − ϕ ( α ) 12 12 14 10 9 α ϕ ( α ) 1 0(g-10) g , : [ e , e ] = e , [ e , e ] = e α ψ ( α ) 7 7 α -1 0 ϕ ( α ) 16 16 15 αϕ ( α ) 3(g-11) g , ( a ) : [ e , e ] = ae , [ e , e ] = e , [ e , e ] = e + e , a = 0 , ± , − α a a ψ ( α ) 6 5 5 4 α a a ϕ ( α ) 13 13 12 α a a ϕ ( α ) 3 1 1 0(g-12) g , (1) : [ e , e ] = e , [ e , e ] = e , [ e , e ] = e + e α ψ ( α ) 8 4 α ϕ ( α ) 15 12 α ϕ ( α ) 7 0(g-13) g , ( − 2) : [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e + e α − − ψ ( α ) 6 5 5 4 α -1 ϕ ( α ) 15 12 α − ϕ ( α ) 3 1 1 0(g-14) g , ( − 1) : [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e + e α − ψ ( α ) 6 6 4 α − ϕ ( α ) 13 16 12 α ϕ ( α ) 2 3 0(g-15) g , : [ e , e ] = e , [ e , e ] = e α ψ ( α ) 6 7 6 α ϕ ( α ) 16 13 α ϕ ( α ) 3 3 2 (g-16) g , : [ e , e ] = e , [ e , e ] = e + e , [ e , e ] = e + e α ψ ( α ) 6 4 α ϕ ( α ) 13 12 α ϕ ( α ) 0 3 0(g-17) g , ( a, b ) : [ e , e ] = ae , [ e , e ] = be , [ e , e ] = e ,a = 0 , ± , ± b, /b, b , − − b, b = 0 , ± , ± a, /a, a , − − aα a a b b ab ba ψ ( α ) 6 5 5 5 5 5 5 4 α a + b ab ba ϕ ( α ) 13 13 13 12 α a b a b ab ba ϕ ( α ) 3 1 1 1 1 1 1 0(g-18) g , ( a, − − a ) : [ e , e ] = ae , [ e , e ] = ( − − a ) e , [ e , e ] = e ,a = 0 , ± , − , − / , − / ± i √ / α a a − − a − a − aa +1 − a +1 a ψ ( α ) 6 5 5 5 5 5 5 4 α -1 ϕ ( α ) 15 12 α a − a a aa +1 1 a +1 − a ϕ ( α ) 3 1 1 1 1 1 1 0(g-19) g , ( a, a ) : [ e , e ] = ae , [ e , e ] = a e , [ e , e ] = e ,a = 0 , ± , ± i, − / ± i √ / α a a a a ψ ( α ) 6 6 6 5 5 4 α a + a a +1 a a +1 a ϕ ( α ) 13 13 13 12 α a a a a ϕ ( α ) 3 2 1 2 1 0(g-20) g , ( a, 1) : [ e , e ] = ae , [ e , e ] = e , [ e , e ] = e ,a = 0 , ± , − α a a ψ ( α ) 8 6 6 4 α a a ϕ ( α ) 15 13 12 α a a ϕ ( α ) 7 2 2 0(g-21) g , ( a, − 1) : [ e , e ] = ae , [ e , e ] = − e , [ e , e ] = e ,a = 0 , ± , ± iα a a − − a − a ψ ( α ) 6 5 5 6 5 5 4 α − a − − a ϕ ( α ) 13 13 16 12 α a a − a − a ϕ ( α ) 3 1 2 1 1 1 0 (g-22) g , (1 , 1) : [ e , e ] = e , [ e , e ] = e , [ e , e ] = e α ψ ( α ) 12 4 α ϕ ( α ) 18 12 α ϕ ( α ) 18 0(g-23) g , ( − , 1) : [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e α − ψ ( α ) 8 8 4 α − ϕ ( α ) 20 13 12 α ϕ ( α ) 7 4 0(g-24) g , ( − , 1) : [ e , e ] = − e , [ e , e ] = e , [ e , e ] = e α − − ψ ( α ) 8 6 6 4 α − ϕ ( α ) 16 12 α − ϕ ( α ) 7 2 2 0(g-25) g , ( − + √ i, − − √ i ) : [ e , e ] = ( − + √ i ) e , [ e , e ] = ( − − √ ) ie , [ e , e ] = e α − + √ i − − √ iψ ( α ) 6 7 7 4 α − ϕ ( α ) 15 12 α − √ i + √ iϕ ( α ) 3 3 3 0(g-26) g , ( i, − 1) : [ e , e ] = ie , [ e , e ] = − e , [ e , e ] = e α i − i − ψ ( α ) 6 6 6 6 4 α − i − − i ϕ ( α ) 13 13 16 12 α i − iϕ ( α ) 3 2 2 2 0(g-27) g , : [ e , e ] = e , [ e , e ] = 2 e , [ e , e ] = e , [ e , e ] = e + e α ψ ( α ) 5 4 3 α ϕ ( α ) 12 12 12 11 α ϕ ( α ) 1 1 0(g-28) g , ( a ) : [ e , e ] = e , [ e , e ] = (1 + a ) e , [ e , e ] = e , [ e , e ] = ae a = 0 , ± , ± , ± / , − / ± √ i/ α a a ψ ( α ) 5 4 4 4 3 α a a + ( a + a ) ϕ ( α ) 12 12 12 12 12 11 α a a a a ϕ ( α ) 1 1 1 0(g-29) g , (1) : [ e , e ] = e , [ e , e ] = 2 e , [ e , e ] = e , [ e , e ] = e α ψ ( α ) 7 4 3 α ϕ ( α ) 12 12 14 11 α ϕ ( α ) 1 2 0(g-30) g , (2) : [ e , e ] = e , [ e , e ] = 3 e , [ e , e ] = e , [ e , e ] = 2 e α ψ ( α ) 5 5 4 3 α ϕ ( α ) 12 12 12 12 12 11 α 53 43 ϕ ( α ) 1 1 1 0(g-31) g , (0) : [ e , e ] = e , [ e , e ] = e , [ e , e ] = e α ψ ( α ) 5 6 4 α ϕ ( α ) 12 13 11 α ϕ ( α ) 2 2 0(g-32) g , ( − 1) : [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e α − ψ ( α ) 5 6 4 α − ϕ ( α ) 13 14 12 α ϕ ( α ) 1 0(g-33) g , ( − 2) : [ e , e ] = e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = − e α − − ψ ( α ) 5 4 4 4 3 α − ϕ ( α ) 16 12 12 11 α ϕ ( α ) 1 1 1 0(g-34) g , ( − + √ i ) : [ e , e ] = e , [ e , e ] = ( + √ i ) e , [ e , e ] = e ,[ e , e ] = ( − + √ i ) e α − + √ i − − √ iψ ( α ) 5 4 4 4 3 α √ i −√ iϕ ( α ) 12 12 12 12 11 α + √ i − √ iϕ ( α ) 1 1 1 0 Appendix B: Proof of Theorem 3.6 Lemma 5.1. For the following complex four–dimensional Lie algebras defined in Appendix A itholds: (g-8) g , ( a ) ⊕ g , a = 0 , ± ψ g , ( a ) ⊕ g : 7 , , ϕ g , ( a ) ⊕ g : 13 , , ( a ) , a = 0 , ± , − ψ g , ( a ) : 6 , , ϕ g , ( a ) : 13 , , ( a, b ) , a = 0 , ± , ± b, b , b , − − b, b = 0 , ± , ± a, a , a , − − aψ g , ( a, b ) : 6 , , ϕ g , ( a, b ) : 13 , (g-18) g , ( a, − − a ) , a = 0 , ± , − , − , − ± √ iψ g , ( a, − − a ) : 6 , , ϕ g , ( a, − − a ) : 15 , , ( a, a ) , a = 0 , ± , ± i, − ± √ iψ g , ( a, a ) : 6 , , ϕ g , ( a, a ) : 13 , , ( a, , a = 0 , ± , − ψ g , ( a, 1) : 8 , , ϕ g , ( a, 1) : 15 , , , ( a, − , a = 0 , ± , ± i ψ g , ( a, − 1) : 6 , , ϕ g , ( a, − 1) : 16 , , , ( a ) , a = 0 , ± , ± , ± , − ± √ iψ g , ( a ) : 5 , , ϕ g , ( a ) : 12 , Proof. Let us give the detailed proof for the case (g-18). We have to check for solutions each of 15possible equalities a = 1 aa = − − aa = − aa + 1... − a a = − a + 1 a . These equations have all solutions in the set n , ± , − , − , − ± i √ o – these values we excludedfrom the beginning. The rest of the proof is analogous. (cid:3) Lemma 5.2. For the four–dimensional Lie algebras from Lemma 5.1 it holds:(g-8) If ψ g , ( a ) ⊕ g = ψ g , ( a ′ ) ⊕ g then a ′ = a, a .(g-11) If ϕ g , ( a ) = ϕ g , ( a ′ ) then a ′ = a .(g-17) If ϕ g , ( a, b ) = ϕ g , ( a ′ , b ′ ) then( a ′ , b ′ ) = ( a, b ) , ( b, a ) , (cid:18) a , ba (cid:19) , (cid:18) ba , a (cid:19) , (cid:18) b , ab (cid:19) , (cid:18) ab , b (cid:19) . (g-18) If ψ g , ( a, − − a ) = ψ g , ( a ′ , − − a ′ ) then a ′ = a, a , − a a , − − a , − − a, − 11 + a . (g-19) If ψ g , ( a, a ) = ψ g , ( a ′ , a ′ ) then a ′ = a, a .(g-20) If ϕ g , ( a, 1) = ϕ g , ( a ′ , 1) then a ′ = a .(g-21) If ϕ g , ( a, − 1) = ϕ g , ( a ′ , − 1) then a ′ = a, − a .(g-28) If ψ g , ( a ) = ψ g , ( a ′ ) then a ′ = a, a . Proof. Cases (g-8), (g-19), (g-20), (g-21) and (g-28) are obvious. Case (g-11). The function ϕ of g , ( a ′ ) has the form(24) α a ′ a ′ ϕ g , ( a ′ )( α ) 13 13 12and there are two possibilities:(12) If a + 1 = a ′ + 1 , a = a ′ then a ′ = a .(21) If a + 1 = a ′ , a ′ + 1 = a then a = a ′ = 1 , − 2, which is not possible. Case (g-17). The function ϕ of g , ( a ′ , b ′ ) has the form(25) α a ′ + b ′ a ′ b ′ b ′ a ′ ϕ g , ( a ′ , b ′ )( α ) 13 13 13 12and there are six possible correspondences between this table and (g-17). We obtain:(123) If z = ba , z = a + b then a ′ = a, b ′ = b .(213) If z = ab , z = a + b then a ′ = b, b ′ = a .(132) If z = a + b, z = ba then a ′ = a , b ′ = ba .(312) If z = ab , z = ba then a ′ = ba , b ′ = a .(231) If z = a + 1 , z = ab then a ′ = b , b ′ = ab .(312) If z = ba , z = ab then a ′ = ab , b ′ = b . Case (g-18). It is convenient to note that six values in the table (g-18) can be arranged in thetriple of pairs { a, a } , {− − a, − a } and {− aa , − a a } . Then one checks directly only 6 · = 48permutations and obtains the solutions like in the previous case. (cid:3) Corollary 5.3. For the four–dimensional Lie algebras from Lemma 5.1 it holds:(g-8) If ψ g , ( a ) ⊕ g = ψ g , ( a ′ ) ⊕ g then g , ( a ) ⊕ g ∼ = g , ( a ′ ) ⊕ g .(g-17) If ϕ g , ( a, b ) = ϕ g , ( a ′ , b ′ ) then g , ( a, b ) ∼ = g , ( a ′ , b ′ ).(g-18) If ψ g , ( a, − − a ) = ψ g , ( a ′ , − − a ′ ) then g , ( a, − − a ) ∼ = g , ( a ′ , − − a ′ ).(g-19) If ψ g , ( a, a ) = ψ g , ( a ′ , a ′ ) then g , ( a, a ) ∼ = g , ( a ′ , a ′ ).(g-21) If ϕ g , ( a, − 1) = ϕ g , ( a ′ , − 1) then g , ( a, − ∼ = g , ( a ′ , − ψ g , ( a ) = ψ g , ( a ′ ) then g , ( a ) ∼ = g , ( a ′ ). Proof. The statement follows from Lemma 5.2 and from the relationsg , ( a ) ⊕ g ∼ = g , (1 /a ) ⊕ g (26) g , ( a, b ) ∼ = g , ( b, a ) ∼ = g , (cid:18) a , ba (cid:19) ∼ = g , (cid:18) ba , a (cid:19) ∼ = g , (cid:18) b , ab (cid:19) ∼ = g , (cid:18) ab , b (cid:19) (27) g , ( a ) ∼ = g , (1 /a ) , (28)which hold for all a, b = 0 and can be directly verified. (cid:3) Proof of Theorem 3.6: ⇒ : See Corollary 3.3. ⇐ : According to Lemmas 5.1, 5.2, Corollary 5.3 and observing the tables in Appendix A, weconclude that all non–isomorphic four-dimensional complex Lie algebras differ at least in one of thefunctions ψ or ϕ . References [1] P. Novotn´y, J. Hrivn´ak, On ( α, β, γ ) –derivations of Lie algebras and corresponding invariant functions. J. Geom.& Phys. , Issue 2 (2008), 208–217.[2] N. Jacobson, Lie Algebras , Dover, New York, (1979).[3] D. Rand, P. Winternitz, H. Zassenhaus, On the identification of Lie algebra given by its structure constants I. 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